Numerical Methods for Partial Differential ... - SAM - ETH Zürich

Numerical Methods for
Partial Differential Equations
Numerical
Methods
for PDEs
1.5.1 Galerkin discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.5.1.1 Spectral Galerkin scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
1.5.1.2 Linear finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
1.5.2 Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
1.5.2.1 Spectral collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
1.5.2.2 Spline collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
1.5.3 Finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
1.6 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
1.6.1 Norms on function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
1.6.2 Algebraic and exponential convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Numerica
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Contents
Prof. Ralf Hiptmair, Prof. Christoph Schwab,
Prof. H. Harbrecht, Dr. V. Gradinaru, and Dr. A. Chernov
Draft version December 14, 2010, SVN rev. 30831
(C) Seminar für Angewandte Mathematik, ETH Zürich
URL: http://www.sam.math.ethz.ch/~hiptmair/tmp/NPDE10.pdf
1 Case Study: A Two-point Boundary Value Problem 21
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2 A model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.1 Linear elastic string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.2 Mass-spring model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.2.3 Continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.3 Variational approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3.1 Virtual work equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3.2 Regularity requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.3.3 Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.4 Simplified model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.5 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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2 Second-order Scalar Elliptic Boundary Value Problems 168 R. Hiptm
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2.1 Equilibrium models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 H.
Harbrech
2.1.1 Taut membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
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2.1.2 Electrostatic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
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2.1.3 Quadratic minimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
2.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
2.3 Variational formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
2.3.1 Linear variational problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
2.3.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
2.4 Equilibrium models: Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
2.5 Diffusion models (Stationary heat conduction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
2.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
2.7 Characteristics of elliptic boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
2.8 Second-order elliptic variational problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
2.9 Essential and natural boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
3 Finite Element Methods (FEM) 272
3.1 Galerkin discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
3.2 Case study: Triangular linear FEM in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
3.2.1 Triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
3.2.2 Linear finite element space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
3.2.3 Nodal basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
3.2.4 Sparse Galerkin matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
3.2.5 Computation of Galerkin matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
3.2.6 Computation of right hand side vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
3.3 Building blocks of general FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
3.3.1 Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
3.3.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
3.3.3 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
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3.4 Lagrangian FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Harbrech
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3.4.1 Simplicial Lagrangian FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Gradinar
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3.4.2 Tensor-product Lagrangian FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
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3.5 Implementation of FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
3.5.1 Mesh file format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
3.5.2 Mesh data structures [6, Sect. 1.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
3.5.3 Assembly [6, Sect. 5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
3.5.4 Local computations and quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
3.5.5 Incorporation of essential boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
3.6 Parametric finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
3.6.1 Affine equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
3.6.2 Example: Quadrilaterial Lagrangian finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
3.6.3 Transformation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
3.6.4 Boundary approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
3.7 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
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4 Finite Differences (FD) and Finite Volume Methods (FV) 432
4.1 Finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
4.2 Finite volume methods (FVM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
4.2.1 Gist of FVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
4.2.2 Dual meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
4.2.3 Relationship of finite elements and finite volume methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
5 Convergence and Accuracy 461
5.1 Galerkin error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
5.2 Empirical Convergence of FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
5.3 A priori finite element error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
5.3.1 Estimates for linear interpolation in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
5.3.2 Error estimates for linear interpolation in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
5.3.3 The Sobolev scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
5.3.4 Anisotropic interpolation error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 DRAFT
5.3.5 General approximation error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
5.4 Elliptic regularity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
5.5 Variational crimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
5.5.1 Impact of numerical quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
5.5.2 Approximation of boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
5.6 Duality techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
5.6.1 Linear output functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
5.6.2 Case study: Boundary flux computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
5.6.3 L 2 -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
5.7 Discrete maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
6 2nd-Order Linear Evolution Problems 604
6.1 Parabolic initial-boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
6.1.1 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
6.1.2 Spatial variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
6.1.3 Method of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
6.1.4 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
6.1.4.1 Single step methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
6.1.4.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
6.1.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
6.2 Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662
6.2.1 Vibrating membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
6.2.2 Wave propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671
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6.2.3 Method of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678
6.2.4 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681
6.2.5 CFL-condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
7 Convection-Diffusion Problems 705
7.1 Heat conduction in a fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
7.1.1 Modelling fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
7.1.2 Heat convection and diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
7.1.3 Incompressible fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
7.1.4 Transient heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
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7.2 Stationary convection-diffusion problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721
7.2.1 Singular perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
7.2.2 Upwinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
7.2.2.1 Upwind quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
7.2.2.2 Streamline diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751
7.3 Transient convection-diffusion BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767
7.3.1 Method of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
7.3.2 Transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774
7.3.3 Lagrangian split-step method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778
7.3.3.1 Split-step timestepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778
7.3.3.2 Particle method for advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
7.3.3.3 Particle mesh method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793
DRAFT
7.3.4 Semi-Lagrangian method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807
8 Numerical Methods for Conservation Laws 820
8.1 Conservation laws: Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822
8.1.1 Linear advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824
8.1.2 Inviscid gas flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829
8.2 Scalar conservation laws in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834
8.2.1 Integral and differential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834
8.2.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839
8.2.3 Weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846
8.2.4 Jump conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850
8.2.5 Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853
8.2.6 Entropy condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863
8.2.7 Properties of entropy solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868
8.3 Conservative finite volume discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872
8.3.1 Semi-discrete conservation form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875
8.3.2 Discrete conservation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880
8.3.3 Numerical flux functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884
8.3.3.1 Central flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884
8.3.3.2 Lax-Friedrichs flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889
8.3.3.3 Upwind flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892
8.3.3.4 Godunov flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899
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8.3.4 Montone schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908 DRAFT
8.4 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919
8.4.1 CFL-condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923
8.4.2 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929
8.4.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943
8.5 Higher-order conservative schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954
8.5.1 Piecewise linear reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956
8.5.2 Slope limiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
8.5.3 MUSCL scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 980
8.6 Outlook: systems of conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986
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9 Finite Elements for the Stokes Equations 987
9.1 Viscous fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988
9.2 The Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997
9.2.1 Constrained variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997
9.2.2 Saddle point problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003
9.2.3 Stokes system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014
9.3 Saddle point problems: Galerkin discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017
9.3.1 Pressure instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022
9.3.2 Stable Galerkin discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034
9.3.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048
9.4 The Taylor-Hood element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054
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10 Adaptive Finite Element Discretization 1060 H.
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10.1 Concept of adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10.2 A priori hp-adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
DRAFT
10.2.1 Graded meshes in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.2.2 Triangular graded meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.2.3 hp-approximation in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.2.4 hp-adapted Lagrangian finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.3 A posteriori mesh adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.3.1 Equidistribution principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.3.2 Local mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.3.3 Refinement control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.4 A posteriori error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.4.1 Hierarchical error estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
10.4.2 Goal oriented error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061
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11 Multilevel iterative solvers 1062
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11.1 Solving finite element linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063 Methods
for PDEs
11.2 Subspace correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.2.1 Successive subspace correction algorithm (SSC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.2.2 Gauss-Seidel iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.2.3 Hierarchical basis multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.2.3.1 Hierarchical basis in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.2.3.2 Hierarchical transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.2.3.3 Hierarchical basis in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.2.3.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.3 Multigrid method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063 R. Hiptmair
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11.3.1 Multilevel subspace corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063 H.
Harbrecht
11.3.2 Multigrid algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11.3.2.1 Transfer operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
DRAFT
11.3.2.2 Local relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.3.3 Nested iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.4 Algebraic multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
11.5 Multilevel preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12 Sparse Grids Galerkin Methods 1064
12.1 The curse of dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.2 Hierarchical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.3 Sparse grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.4 Approximation on sparse grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.5 Sparse grids algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
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Index 1065
Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083
MATLAB-CODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085
Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
Course history
• Summer semester 04, R. Hiptmair (for RW/CSE undergraduates)
• Winter semester 04/05, C. Schwab (for RW/CSE undergraduates)
• Winter semester 05/06, H. Harbrecht (for RW/CSE undergraduates)
• Winter semester 06/07, C. Schwab (for BSc RW/CSE)
• Autumn semester 07, A. Chernov (for BSc RW/CSE)
• Autumn semester 08, C. Schwab (for BSc RW/CSE)
• Autumn semester 09, V. Gradinaru (for BSc RW/CSE, Subversion Revision: 22844)
• Spring semester 10, R. Hiptmair (for BSc Computer Science)
• Autumn semester 10, R. Hiptmair (for BSc RW/CSE, Subversion Revision: 30025)
Preamble
This lecture is a core course for
• BSc in Computational Science and Engineering (RW/CSE),
• BSC in Computer Science with focus Computational Science.
Main skills to be acquired in this course:
Ability to implement advanced numerical methods for the solution of partial differential equations
in MATLAB efficiently
Ability to modify and adapt numerical algorithms guided by awareness of their mathematical foundations
Ability to select and assess numerical methods in light of the predictions of theory
Ability to identify features of a PDE (= partial differential equation) based model that are relevant
for the selection and performance of a numerical algorithm
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Ability to understand research publications on theoretical and practical aspects of numerical methods
for partial differential equations.
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Numerica
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Numerical analysis of PDE (→ mathematics curriculum)
TODOs
This course
≠
(401-3651-00V Numerical methods for elliptic and parabolic partial differential
equations, )
Cleanse index and code index.
Instruction on how to apply software packages
State “learning outcomes” at the end of each section/chapter.
Reading instructions
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What to expect
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This course materials are neither a textbook nor lecture notes.
They are meant to be supplemented by explanations given in class.
The course is difficult and demanding (ie. ETH level)
Do not expect to understand everything in class. The average student will
Some pieces of advice:
these lecture slides are not designed to be self-contained, but to supplement explanations in class.
this document is not meant for mere reading, but for working with,
turn pages all the time and follow the numerous cross-references,
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• understand about one third of the material when attending the lectures,
• understand another third when making a serious effort to solve the homework problems,
• hopefully understand the remaining third when studying for the examnination after the end of
the course.
Perseverance will be rewarded!
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study the relevant section of the course material when doing homework problems.
Practical information
Teaching instructions
Explanations in class should haevily rely on the blackboard: a 2 : 1 split between developing
material at the blackboard and discussing formulas and examples given on the slides has emerged
as proper balance.
Many technical details given in supplementary parts of the lecture material can be skipped or
simplified.
Use slides for repetition: after having sketched a method at the blackboard, again go through the
presentation on the slides.
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Course:
401-3663-00L Numerical Methods for Partial Differential Equations
Lectures: Mon 13-15 HG D 5.2
Wed 10-12 HG D 5.2
Tutorials: Mon 08-10 HG D 5.2
Mon 10-12 HG D 5.2
Lecturer: Prof. Ralf Hiptmair, office: HG G 58.2, e-mail: hiptmair@sam.math.ethz.ch
Assistants: Eivind Fonn, office: HG J 59, e-mail: eivind.fonn@sam.math.ethz.ch
Dr. Erhan Turan,
(2nd year PhD student at Seminar of Applied Mathematics, D-MATH)
office: CAB F 81, e-mail: erhan.turan@inf.ethz.ch
(Postdoctoral researcher at Chair of Computational Science, D-INFK)
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Assignments: 12 weekly assignment sheets, handed out on Monday, discussed in tutorial
classes in the following week.
“Testat” requirement:
• Take part in the tutorial classes. The attendance is mandatory: at most
two tutorial classes can be skipped during the semester.
• Take active part during the tutorial. Every student will be asked to present
the solution of a few exercises at the blackboard.
• Work extensively on the exercises and hand in the approaches/solutions. In
order to obtain the access to the examination, about 33% of the exercises
proposed during the semester should be solved and handed in.
Examination: “Sessionsprüfung”: computer based examination, programming and theoretical
tasks
✗
✖
3 hour examination on Friday, January 28, 2011
✔
✕
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to be found at the URL
Numerical Methods for Partial Differential Equations
http://forum.vis.ethz.ch/forumdisplay.php?f=79
This forum has been set up so that you can post questions on the programming exercises that accompany
the course. One of the assistants will look at the entries in this forum and
• write an answer in this forum or
• discuss the question in a consulting session and post an answer later.
A second purpose of this forum is that the assistants can collect FAQs and post answers here.
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Web page: http://www.math.ethz.ch/education/bachelor/lectures/hs2010/math/numpdecse
Reporting errors
Please report errors in the electronic lecture notes via a wiki page !
http://elbanet.ethz.ch/wikifarm/rhiptmair/index.php?n=Main.NPDECourse
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Literature
This course is designed to be rather self-contained and additional study of literature should not be
crucial to follow the lecture. These references point to futher sources of information, mainly devoted
mathematical theory.
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(Password: NPDE, please choose EDIT menu to enter information)
• D. Braess. Finite Elements. Cambridge University Press, 2nd edition, 2001.
Please supply the following information:
• (sub)section where the error has been found,
• precise location (e.g, after Equation (4), Thm. 2.3.3, etc. ). Refrain from giving page numbers,
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• S. Brenner and R. Scott. Mathematical theory of finite element methods. Texts in Applied
Mathematics. Springer–Verlag, New York, 1994.
• A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements, volume 159 of Applied
Mathematical Sciences. Springer, New York, 2004.
• Ch. Großmann and H.-G. Roos. Numerik partieller Differentialgleichungen. Teubner
Studienbücher Mathematik. Teubner, Stuttgart, 1992.
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• brief description of the error.
• W. Hackbusch. Elliptic Differential Equations. Theory and Numerical Treatment, volume 18 of
Springer Series in Computational Mathematics. Springer, Berlin, 1992.
• P. Knabner and L. Angermann. Numerical Methods for Elliptic and Parabolic Partial Differential
Equations, volume 44 of Texts in Applied Mathematics. Springer, Heidelberg, 2003.
Online discussion forum
• S. Larsson and V. Thomée. Partial Differential Equations with Numerical Methods, volume 45 of
Texts in Applied Mathematics. Springer, Heidelberg, 2003.
Contribute to the forum
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• R. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied
Mathematics. Cambridge University Press, Cambridge, UK, 2002.
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1
Case Study: A Two-point Boundary
Value Problem
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1.1 Introduction
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OPTIONAL (WITH QUALIFICATIONS)
This chapter provides a quick tour of modelling and discretization in the case of a concrete mechanical
system, of which everybody should have some intuitive grasp. The modelling focus is on extremal
principles and variational calculus. Different discretization approaches are introduced and discussed
in 1D, which enables students to develop implementations from scratch. Thus, they can conduct
numerical simulations right from the outset of the course.
Most topics covered in this chapter are revisited later, but students benefit a lot from this repetition:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
The term “partial differential equation” (PDE) usually conjures up formulas like
√
div( 1+‖gradu(x)‖ 2 gradu(x))+v·gradu = f(x) , x ∈ Ω ⊂ R d .
This chapter aims to wean you off this impulse and instil an appreciation that
★
✧
a meaningful PDE encodes structural principles
(like equilibrium, conservation, etc.)
✥
✦
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• The elastic string model discussed in Sects. 1.2 and Sect. 1.4 is instrumental in understanding
the membrane model of Sect. 2.1.1
1.0
p. 21
!
The design and selection of numerical methods has to take into account these governing
principles.
1.1
p. 23
• The brief introduction into variational calculus in Sect. 1.3 prepares for the in-depth treatment
given in Ch. 2
Numerical
Methods
for PDEs
Remark 1.1.1 (Mathematical modelling).
Numerica
Methods
for PDEs
• Galerkin discretization introduced in Sect. 1.5.1 is addressed again in 3.1.
Prerequisite for numerical simulation:
Mathematical modelling
• Finite differences are covered in Sect. 1.5.3 and, again, in 4.1.
Physics Biology ··· Economics
Appropriate & necessary simplifications
This chapter offers a brief tour of
mathematical modelling of a physical system based on variational principles,
the derivation of differential equations from these variational principles,
the discretization of the variational problems and/or of the differential equations using various
approaches.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
(PDE based) mathematical model
Necessary simplification:
{ }
system
described by a few variables/functions in configuration space
phenomenon
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
The art of modelling:
devise “faithful model”
1.1
p. 22
Essential/relevant traits of
{ }
system
phenomenon
−→ structural properties of model
1.1
p. 24

△
Numerical
Methods
for PDEs
Numerica
Methods
for PDEs
Remark 1.1.2 (“PDEs” for univariate functions).
The classical concept of a PDE inherently involves functions of several independent variables. However,
when one embraces the concept of a PDE as encoding fundamental structural properties of a
model, then simple representatives in a univariate setting can be discussed.
☞
ordinary differential equations (ODEs) offer simple specimens of important classes of PDEs!
Thus, in this chapter we examine ODEs that are related to the important class of elliptic PDEs.
△
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Fig. 2
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Note: The developments below cannot live up to standards of mathematical rigour, because what
has deliberately omitted is the discussion of the functional analytic framework (function space
theory) required for a complete statement of, for instance, minimization problems and variational
problems.
Sought:
(Approximation of) “shape” of elastic string
1.2 A model problem
1.2.1 Linear elastic string
1.2
p. 25
Numerical
Methods
for PDEs
u 2
a
(a,u a )
(b,u b )
Configuration space
= space of curves ‘w
☞ shape of string
↕
curveu : [0,1] ↦→ R 2 ,u = u(ξ)
(physical units[u] = 1m)
1.2
p. 27
Numerica
Methods
for PDEs
Static mechanical problem:
Deformation of elastic “1D” string (rubber
band) under its own weight
Constraint: string pinned at endpoints
✄
Rubber band
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
force fieldf
b
Fig. 3
u 1 ☞ Pinning conditions (boundary conditions):
( ) ( )
a b
u a u b
u(0) =
∈ R 2 , u(1) =
∈ R 2 .
(1.2.1)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Gravity
Fig. 1
Note:
description of a curve in the plane by a mapping[0,1] ↦→ R 2 requires coordinate system. Of
course, the choice of the coordinate system must not affect the shape obtained from the mathematical
model, a property called frame indifference.
1.2
p. 26
Current concept of force field f: force “pulls” at elastic string. Alternative: force due to a potential.
Gravitational force (i.e., a constant force field) allows both descriptions.
1.2
p. 28

Terminology: [0,1] ˆ= parameter domain, ✎ notationΩ
Remark 1.2.2 (Parametrization of a curve). → [29, Sect. 7.4]
Numerical
Methods
for PDEs
Remark 1.2.3 (Material coordinate).
Numerica
Methods
for PDEs
Interpretation of curve parameterξ:
We consider a curve in R 2
u ∈ (C 0 ([0,1])) 2
⇕
u : [0,1] ↦→ R 2 ✄
u 2
u(0)
u(ξ)
u ′ (ξ)
connected curve u
Fig. 4 1
u(1)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
ξ: unique identifier for each infinitesimal section of the string,
a label for each “material point” on the string
ξ ˆ= material coordinate, unrelated to “position in space” (= physical coordinate),
ξ has no physical dimension ➤ ′ does not affect dimension.
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✎ notation: C k ([a,b]) ˆ= k-times continuously differentiable functions on [a,b] ⊂ R , see [29,
Sect. 5.4]
(C k ([a,b])) 2 ˆ= k-times continuously differentiable curvesu : [a,b] ↦→ R 2 , that is, if
u = ( u 1
u 2
)
, thenu1 ,u 2 ∈ C k ([a,b]).
Remark 1.2.4 (Non-dimensional equations).
By fixing reference values for the basic physical units occurring in a model (“scaling”), one can switch
to a non-dimensional form of the model equations.
Geometric intuition:
u(ξ) moves along the curve asξ increases from0to1.
1.2
p. 29
In the case of the elastic string model the basic units are
1.2
p. 31
Interpretation of curve parameterξ: “virtual time”
➣
∥ ∥u ′∥ ∥ ˆ= “speed” with which curve is traversed
∫ 1 ∥
➣ ∥u ′ (ξ) ∥ dξ ˆ= length of curve
0
✎ notation: ‖·‖ ˆ= Euclidean norm of a vector∈ R n
➣ parametrization is supposed to be locally injective:
∀ξ ∈]0,1[: ∃ǫ > 0: ∀η, |η −ξ| < ǫ: u(η) ≠ u(ξ) .
Foru ∈ (C 1 ([0,1])) 2 we expect u ′ (ξ) ≠ 0 for all0 ≤ ξ ≤ 1
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
• unit of length1m,
• unit of force1N.
Thus, non-dimensional equations arise from fixing a reference lengthl 0 and a reference forcef 0 .
Below, following a (bad) habit of mathematicians, physical units will be routinely dropped, which tacitly
assumes a priori scaling.
Quantities that have to be specified to allow the unique determination of a comfiguration in a mathematical
model are called problem data/parameters. In the elastic string model the problem parameters
are
△
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✎
notation:
′ ˆ= derivative w.r.t. curve parameter, hereξ
the boundary conditions (1.2.1),
the force fieldf : [0,1] ↦→ R 2 ,[f] = 1N,f(ξ) ˆ= force “pulling at” a material pointξ.
△
1.2
p. 30
Special case: gravitational forcef(ξ) := −gρ(ξ) ( 0
1
) ,0 ≤ ξ ≤ 1,g = 9.81ms −2
with densityρ : [0,1] ↦→ R + ,[ρ] = kg,
1.2
p. 32

local elastic material properties, see Sect. 1.2.3.
Numerical
Methods
for PDEs
Models, for which configurations can be described by means of finitely many real numbers are called
discrete. Hence, the mass-spring model is a discrete model, see Sect. 1.5.
Numerica
Methods
for PDEs
1.2.2 Mass-spring model
Idea:
model string as a system of many simple components that interact in simple ways
Rubber band
elastic string
Fig. 5
Assumption: linear springs ↔ Hooke’s law
↔
m 8
R
m m 7 1
m 2 m 6
m 3 m4
m 5
Fig. 6
n point masses connected withn+1 springs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.2
p. 33
Total elastic energy of mass-spring model in configuration (u 1 ,...,u n ) ∈ (R 2 ) n :
(1.2.6) ⇒ J (n)
el = J (n) n∑
el (u 1 ,...,u n ) := 2
1 κ i ∥
( ∥u i+1 −u i∥ ∥ ∥−li ) 2 , (1.2.7)
l
i=0} i
{{ }
elastic energy ofi-th spring
where u 0 := ( a) ua ,u n+1 := ( b)
ub (pinning positions (1.2.1)),
κ i ˆ= spring constant ofi-th spring,i = 0,...,n,
l i > 0 ˆ= equilibrium length ofi-th spring.
where
Total “gravitational” energy of mass-spring model in configuration(u 1 ,...,u n ) due to external
force field:
J (n)
f = J (n)
f (u 1 ,...,u n ) := −
f i ˆ= force acting oni-th mass,i = 1,...,n.
n∑
f i·u i , (1.2.8)
i=1
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.2
p. 35
Force
( ) l
F(l) = κ −1
l 0
(relative elongation) . (1.2.5)
Numerical
Methods
for PDEs
✎ notation: u·v := u T ∑
v = n u i v i ˆ= inner product of vectors in R n .
j=1
Numerica
Methods
for PDEs
κ ˆ= spring constant (stiffness),[κ] = 1N,κ > 0,
l 0 ˆ= equilibrium length of (relaxed) spring.
Known from classical mechanics, static case:
equilibrium principle
elastic energy stored in linear spring at lengthl > 0
∫ l
E el = F(τ)dτ = 1 κ
(l−l
l 0
2l 0 ) 2 R. Hiptmair
, [E el ] = 1J . (1.2.6)
0
DRAFT
Configuration space for mass-spring model:
u i ∈ R 2 ˆ= position ofi-th mass point,i = 1,...,n
➣ finite-dimensional configuration space = (R 2 ) n
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
1.2
p. 34
systems attains configuration(s) of minimal (potential) energy
J (n) := J (n)
el +J (n)
f
equilibrium configurationu 1 ∗,...,u n ∗ of mass-spring system solves
(u 1 ∗,...,u n ∗) = argmin
(u 1 ,...,u n )∈R 2n J (n) (u 1 ,...,u n ) . (1.2.9)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.2
p. 36

J
Plot ofJ (1) (u 1 )
✄
150
Numerical
Methods
for PDEs
Assumption: equal equilibrium lengths of all springs l i = L ,L > 0,
n+1
➣ L ˆ= equilibrium length of elastic string: L = ∑ i l i,[L] = 1m.
Numerica
Methods
for PDEs
100
Mass-spring system with only one point mass
(non-dimensional l 1 = l 2 = 1, κ 1 = κ 2 = 1,
u 0 = ( 0) 0 ,u 2 = ( 1 )
0.2 ,f 1 = ( 0 )
−1 )
50
0
−50
10
5
0
y
−5
−10
−10
−5
x
0
5
10
Fig. 7
Equilibrium configuration of mass-spring system✄
0.2
0.1
n = 5
n = 10
n = 20
n = 40
n = 80
0
Note: solutions of (1.2.9) need not be unique !
∑
To see this, consider the case L := n ∥
l i > ∥u n+1 −u 0∥ ∥ and f ≡ 0 (slack ensemble of springs
i=0
without external forcing). In this situation many crooked arrangements of the masses will have zero
total potential energy.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
(non-dimensionall i = n+1 L , κ i = 1, m i = n 1 , f i =
1
( 0
n
)
−1 ,L = 1,nvarying)
−0.2
−0.3
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
u(1)
−0.1
u(0)
Fig. 9
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.2
p. 37
1.2
p. 39
u(1)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
u(0)
1.2.3 Continuum limit
L = 0.5
L = 1
L = 1.5
Fig. 8
✁ minimal energy configuration of a mass spring
system for variableL.
(n = 10, non-dimensional κ i = 1, l i = L/n, i =
1,...,10)
Heuristics: elastic string = spring-mass system with “infinitely many infinitesimal masses”
Policy:
and “infinitesimally short” springs.
consider sequence(SMM n ) n∈N of spring-mass systems withnmasses,
identify material coordinate (→ Rem. 1.2.3) of point masses,
choose system parameters with meaningful limits,
derive expressions for energies asn → ∞,
use them to define the “continuous elastic string model”.
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.2
p. 38
u 2
m 8
m 1
m 7
m 2 m 6
m 3 m 4
m 5
u 1
0
Fig.
ξ 1 ξ 2 ξ 3 ξ 4 ξ 5 ξ 6 ξ 7 ξ 8
m 1
10ξ
masses are uniformly spaced on string u :
[0,1] ↦→ R 2
➣ material coordinate ofi-th mass inSMM n :
ξ (n)
i := i
n+1 : ui := u(ξ (n)
i )
In the spring-mass model each spring has its own stiffness κ i and every mass point its own force f i
acting on it. When considering the “limit” of a sequence of spring-mass models, we have to detach
stiffness and force from springs and masses and attach them to material points, cf. Rem. 1.2.3. In
other words stiffnessκ i and forcef i have to be induced by a stiffness functionκ(ξ) and force function
f(ξ). This linkage has to be done in a way to allow for a meaningful limit n → ∞ for the potential
energy.
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.2
p. 40

“Limit-compatible” system parameters:
(ξ (n)
i+1/2 := 1 2 (ξ(n) i+1 +ξ(n) i
))
κ i = κ(ξ (n)
i+1/2 ) with integrable stiffness functionκ : [0,1] ↦→ R+ ,
Numerical
Methods
for PDEs
Tool: Taylor expansion: foru = ( u ) 1
u ∈ C 2 with derivativeu
′ ,1 ≫ η → 0
2
√
‖u(ξ +η)−u(ξ −η)‖ = (u 1 (ξ +η)−u 1 (ξ −η)) 2 +(u 2 (ξ +η)−u 2 (ξ −η)) 2
√
= (2u ′ 1 (ξ)η +O(η3 )) 2 +(2u ′ 2 (ξ)η +O(η3 )) 2
(1.2.13)
Numerica
Methods
for PDEs
f i =
ξ (n)
i+1/2
∫
ξ (n)
i−1/2
f(ξ)dξ “lumped force”, integrable force fieldf : [0,1] ↦→ R 2
energies, see (1.2.7), (1.2.8)
J (n) n∑
(
el (u) = 2
1 n+1 ∥∥∥u(ξ L κ(ξ(n) i+1/2 ) (n)
i+1 )−u(ξ(n) ∥
i ) ∥− L ) 2
, (1.2.10)
n+1
i=0
J (n) n∑
f (u) = −
i=1
ξ (n)
i+1/2
∫
ξ (n)
i−1/2
f(ξ)dξ ·u(ξ (n)
i ) . (1.2.11)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.2
p. 41
Apply this to (1.2.10) withη = 1 2 1
n+1 forn → ∞
J (n) n∑
el (u) = 1 2
i=0
= 1 1
n∑
2Ln+1
i=0
= 2η ∥ ∥u(ξ) ′∥ ∥ √ 1+O(η 2 ) = 2η ∥ ∥u ′ (ξ) ∥ ∥+O(η 2 ) .
∥
∥u ′ (ξ (n) ∥ ∥∥+O(
n+1 i+1/2 ) 1 L ) 2
(n+1) 2)− n+1
( ∥∥∥u
κ(ξ (n)
i+1/2 ) ′ (ξ (n) ∥ ) (1.2.14)
∥∥+O(
i+1/2 ) 1 2
n+1 )−L
n+1
L κ(ξ(n) i+1/2 ) ( 1
Consideration: integral as limit of Riemann sums, see [29, Sect. 6.2]:
q ∈ C 0 1
n∑
([0,1]): lim q( j +1/2 ∫ 1
n→∞n+1
n+1 ) = q(ξ)dξ . (1.2.15)
j=0
0
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.2
p. 43
Assumption:
u ∈ (C 2 ([0,1])) 2 (twice continuously differentiable)
Numerical
Methods
for PDEs
⇒ J el (u) = lim
n→∞ J(n) el (u) = 1 ∫1
κ(ξ) (∥ ∥u ′ (ξ) ∥ ∥−L ) 2 dξ . (1.2.16)
2L
0
Numerica
Methods
for PDEs
➊
Simple limit for potential energy due to external force:
Equilibrium condition for limit model
(minimal total potential energy):
➋
n J f (u) = lim
n→∞ J(n) f (u) = lim
n→∞ − ∑
i=1
Limit of elastic energy:
ξ (n)
i+1/2
∫
ξ (n)
i−1/2
∫1
f(ξ)dξ ·u(ξi n ) = − f(ξ)·u(ξ)dξ . (1.2.12)
0
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
∫1
κ(ξ) (∥
u ∗ = argmin ∥ u ′ (ξ) ∥ ∥−L ) 2 −f(ξ)·u(ξ)dξ
u∈(C 1 ([0,1])) 2 & (1.2.1)
2L
0
} {{ }
=:J(u)
. (1.2.17)
total potential energy functional,[J] = 1J
= a minimization problem in a function space !
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Example 1.2.18 (Tense string without external forcing).
1.2
p. 42
1.2
p. 44

Setting:
no external force: f ≡ 0
Numerical
Methods
for PDEs
1.3 Variational approach
Numerica
Methods
for PDEs
homogeneous string:
κ = κ 0 = const
tense string:
L < ‖u(0)−u(1)‖
(➣ positive elastic energy) Fig. 11
∫1
κ
(1.2.17) ⇔ u ∗ = argmin 0 (∥ ∥ u ′ (ξ) ∥ ∥−L ) 2 dξ . (1.2.19)
u∈(C 1 ([0,1])) 2 & (1.2.1)
2L
0
Note: in (1.2.19)uentersJ only throughu ′ !
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
We face the task of minimizing a functional over an ∞-dimensional function space. In this section
necessary conditions for the minimizer will formally be derived in the form of variational equations.
This idea is one of the cornerstone of a branch of analysis called calculus of variations.
We will not dip into this theory, but perform manipulations at a formal level. Yet, all considerations
below can be justified rigorously.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Constraint onu ′ : by triangle inequality for integrals, see [29, Sect. 6.3]
∫ 1
l := ‖u(1)−u(0)‖ =
u ′ ∫ 1
(ξ)dξ
∥ 0 ∥ ≤ ∥
∥u ′ (ξ) ∥ dξ . (1.2.20)
0
1.3.1 Virtual work equation
➥ Consider related minimization problem
⎧
⎪⎨ ∫1 w ∈ (C
κ 0 ([0,1])) 2 ⎫
, ⎪⎬
w ∗ = argmin 0
(w−L) 2 dξ:
∫ 1
w ⎪⎩ 2L w(ξ)dξ ≥ l ⎪⎭ . (1.2.21)
0 0
⇒ unique solution w ∗ (ξ) = l (constant solution)
∥
∥u ′ (ξ) ∥ = l and the boundary conditions (1.2.1) are satisfied for the straight line solution of (1.2.19)
u ∗ (ξ) = (1−ξ)u(0)+ξu(1) .
1.2
p. 45
Numerical
Methods
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✎ notation: C k 0 ([0,1]) := {v ∈ Ck ([0,1]):v(0) = v(1) = 0},k ∈ N 0
Main “idea of calculus of variations”:
u ∗ solves (1.2.17) ⇒ J(u ∗ ) ≤ J(u ∗ +tv) ∀t ∈ R, v ∈ (C 2 0 ([0,1]))2 . (1.3.1)
ϕ(t) := J(u ∗ +tv) has global minimum fort = 0
Ifϕdifferentiable, then
dϕ
dt (0) = 0
Note: v(0) = v(1) = 0, because we must not tamper with the pinning conditions (1.2.1).
1.3
p. 47
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
It is exactly the “straight string” solution that physical intuition suggests.
Rule:
Variationvmust vanish where argument functionuis fixed.
✸
1.3
p. 46
Computation of dϕ
dt (0) for J from (1.2.17) amounts to computing a “configurational derivative” in directionv.
1.3
p. 48

We pursue a separate treatment of energy contributions (This also demonstrates a simple formal
approach to computing configurational derivatives.):
➊
Potential energy (1.2.12) due to external force:
∫1 ∫1
J
lim f (u ∗ +tv)−J f (u ∗ ) 1
= − lim f(ξ)·tv(ξ)dξ = − f(ξ)·v(ξ)dξ . (1.3.2)
t→0 t t→0 t
0
0
➋ Elastic energy (1.2.16): more difficult, tool: Taylor expansion [29, Sect. 5.5]
Analoguous to (1.2.13),x ∈ R 2 \{0},h ∈ R 2 , for R ∋ t → 0
√
√
‖x+th‖ = (x 1 +th 1 ) 2 +(x 2 +th 2 ) 2 = ‖x‖ 2 +2tx·h+t 2 ‖h‖ 2
= ‖x‖ √ x·h 1+2t
‖x‖ 2 +t2‖h‖2
‖x‖ 2 = ‖x‖+tx·h
‖x‖ +O(t2 ) ,
(1.3.3)
√ where we used the truncated Taylor series for 1+x
√
1+δ = 1+ 1
2 δ +O(δ 2 ) for δ → 0 . (1.3.4)
Use (1.3.3) in the perturbation analysis for the elastic energy:
( ∥ ∥u ′ (ξ)+tv ′ (ξ) ∥ ( ∥∥
∥−L) 2 = u ′ (ξ) ∥ ∥+t u′ (ξ)·v ′ ) 2
(ξ)
‖u ′ +O(t 2 )−L
(ξ)‖
= (∥ ∥u ′ (ξ) ∥ ∥−L ) 2 (∥
+2t ∥ u ′ (ξ) ∥ ∥−L ) u ′ (ξ)·v ′ (ξ)
‖u ′ +O(t 2 ) .
(ξ)‖
J el (u+tv)−J el (u) = t ∫1
κ(ξ) (∥ ∥u ′ (ξ) ∥ ∥−L )u′ (ξ)·v ′ (ξ)
L
‖u ′ +O(t 2 )dξ . (1.3.5)
(ξ)‖
0
∫1
J
lim el (u ∗ +tv)−J el (u ∗ )
=
t→0 t
0
κ(ξ)
L
(∥ ∥ u ′ (ξ) ∥ ∥−L )u′ (ξ)·v ′ (ξ)
‖u ′ dξ . (1.3.6)
(ξ)‖
Here we take for granted ∥ ∥u ′ (ξ) ∥ ∥ ≠ 0, which is an essential property of a meaningful parameterization
of the elastic string, see Rem. 1.2.2.
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C. Schwab,
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DRAFT
1.3
p. 49
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Necessary condition for u ∗ solving (1.2.17)
∫1
0
κ(ξ)
L
(∥ ∥ u ′ ∗(ξ) ∥ ∥−L )u′ ∗(ξ)·v ′ (ξ)
∥
∥u ′ ∗(ξ) ∥ −f(ξ)·v(ξ)dξ = 0 ∀v ∈ (C0 2 ([0,1]))2 . (1.3.7)
This is a non-linear variational equation on domainΩ = [0,1]
Remark 1.3.8 (Differentiating a functional on a space of curves).
For aC 2 -function F : R d ×R d ↦→ R,d ∈ N, consider the functional
∫ 1
J : (Cpw([0,1])) 1 d ↦→ R , J(u) := F(u ′ (ξ),u(ξ))dξ .
0
Use the multidimensional Taylor’s formula [29, Satz 7.5.2]
F(u+δu,v+δv) = F(u,v)+D 1 F(u,v)δu+D 2 F(u,v)δv+O(‖δu‖ 2 +‖δv‖ 2 ) . (1.3.9)
Here, D 1 F and D 2 F are the partial derivatives of F w.r.t the first and second vector argument,
respctively. These are row vectors.
∫ 1
J(u+tv) = J(u)+t D 1 F(u ′ (ξ),u(ξ))v ′ (ξ)+D 2 F(u ′ (ξ),u(ξ))v(ξ)dξ+O(t ) . (1.3.10)
}
0
{{ }
“directional derivative” (D u J)(u)(v)
The derivativesu ′ ,v ′ are just regular 1D derivatives w.r.t. the parameterξ. They yield column vectors.
Hence, we deal with a scalar integrand.
Remark 1.3.11 (Virtual work principle).
In statics, the derivation of variational equations from energy minimization (equilibrium principle, see
(1.2.9)) is known as the method of virtual work: Small admissible changes of the equilibrium configuration
of the system invariably entail active work.
Remark 1.3.12 (Non-linear variational equation).
△
△
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DRAFT
1.3
p. 51
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DRAFT
Now we unravel the structure behind the non-linear variational problem (1.3.7).
1.3
p. 50
Recall essential terminology from linear algebra:
1.3
p. 52

Definition 1.3.13 ((Bi-)linear forms).
Given an R-vector space V , a linear form (linear functional) l is a mapping f : V ↦→ R that
satisfies
l(αu+βv) = αl(u)+βl(v) ∀u,v ∈ V , ∀α,β ∈ R .
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Explanation of terminology:
• trial space ˆ= the function space in which we seek the solution
• test space ˆ= the space of eligible test functionsv in a variational problem like (1.3.14)
Numerica
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A bilinear formaonV is a mappinga : V ×V ↦→ R, for which
a(α 1 v 1 +β 1 u 1 ,α 2 v 2 +β 2 u 2 ) =
= α 1 α 2 a(v 1 ,v 2 )+α 1 β 2 a(v 1 ,u 2 )+β 1 α 2 a(u 1 ,v 2 )+β 1 β 2 a(u 1 ,u 2 )
for allu i ,v i ∈ V ,α i ,β i ∈ R,i = 1,2.
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DRAFT
The two spaces need not be the same:
V ≠ V 0 is common and already indicated by the notation.
For many variational problem, which are not examined in this course, they may even comprise
functions with different smoothness properties.
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DRAFT
✎ notation: a,b,... ˆ= bilinear forms
In the case of (1.3.7) we make a very important observation, namely that, keeping u ∗ fixed, the left
hand side is a linear functional (linear form) in the test functionv:
Structure of
∫1
(∥ ∥ u ′ ∗(ξ) ∥∥ −L )u′ ∗(ξ)·v ′ (ξ)
∥
∥u ′ ∗(ξ) ∥ −f(ξ)·v(ξ)dξ = 0 ∀v ∈ (C0 2 ([0,1]))2 : (1.3.7)
κ(ξ)
L
0
✄ abstract non-linear variational equation
u ∈ V: a(u;v) = l(v) ∀v ∈ V 0 , (1.3.14)
1.3
p. 53
Numerical
Methods
for PDEs
In concrete terms (for elastic string continuum model):
V 0 := (C 2 0 ([0,1]))2 ,
V := {u ∈ (C 2 ([0,1])) 2 : u(0) =
(
a
u a
)
, u(1) =
(
b
u b
)
= [ξ ↦→ (1−ξ)u(0)+ξu(1)] +V
} {{ } 0 , (1.3.16)
=:u 0
∫ 1
l(v) := f(ξ)·v(ξ)dξ , (1.3.17)
0
∫1
a(u;v) :=
0
κ(ξ)
L
(∥ ∥ u ′ (ξ) ∥ ∥−L )u′ (ξ)·v ′ (ξ)
‖u ′ dξ . (1.3.18)
(ξ)‖
Thus, for variational problem (1.3.7) arising from the elastic string model we find the common pattern
V = V 0 +u 0 , that is, the trial space is an affine space, arising from the test space V 0 by adding an
offset functionu 0 .
}
1.3
p. 55
Numerica
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V 0 ˆ= (real) vector space of functions,
V ˆ= affine space of functions: V = u 0 +V 0 , with offset functionu 0 ∈ V ,
l ˆ= a linear mappingV 0 ↦→ R, a linear form,
a ˆ= a mappingV ×V 0 ↦→ R, linear in the second argument, that is
a(u;αv +βw) = αa(u;v)+βa(u;w) ∀u ∈ V , v,w ∈ V 0 , α,β ∈ R . (1.3.15)
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DRAFT
IfV = V 0 +u 0 , then there is a way to recast (1.3.14) as a variational problem with the same trial and
test spaceV 0 :
Rewriting (1.3.14) using the offset functionu 0 ∈ V :
(1.3.14) ⇒ w ∈ V 0 : a(u 0 +w;v) = l(v) ∀v ∈ V 0 and u = u 0 +w . (1.3.19)
△
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H.
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DRAFT
Terminlogy related to variational problem (1.3.14): V is called trial space
V 0 is called
test space
1.3
p. 54
1.3.2 Regularity requirements
1.3
p. 56

Issue:
The derivation of the continuum models (1.2.17) (→ Sect. 1.2.3) and (1.3.7) was based on
the assumptionu ∈ (C 2 ([0,1])) 2 .
Is u ∈ (C 2 ([0,1])) 2 required to render the minimization problem (1.2.17)/variational problem
1.2.3) meaningful?
We will find that curves with less smoothness can still yield relevant solutions of
(1.2.17)/(1.3.7).
Obvious (→ Rem. 1.2.2): u ∈ (C 0 ([0,1])) 2
Observation:
u ∗ =
argmin
∫1
(string must not be torn)
κ(ξ) (∥ ∥ u ′ (ξ) ∥ ∥−L ) 2 −f(ξ)·u(ξ)dξ . (1.2.17)
2L
u∈(C 1 ([0,1])) 2 & (1.2.1)
0
J(u) from (1.2.17),a,lfrom (1.3.18) well defined for
merely continuous, piecewise continuously differentiable functionsu,v : [0,1] ↦→ R 2 ,
➣ u ′ will be piecewise continuous and can be integrated.
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➩
u 2 /f(ξ)
(a,u a )
force
u 1 /ξ
a/0 b/1
✔
discontinuousf
u ∗ ∉ (C 2 ([0,1])) 2 physically meaningful:
(b,u b )
Fig. 14
u ∗ ∈ (C 1 ([0,1])) 2 for discontinuousf
merelyu ∗ ∈ (C 0 ([0,1])) 2 for point force concentrated
inξ 0) : kink atξ 0 !
u 2 /f(ξ)
(a,u a )
(b,u b )
u 1 /ξ
a/0 ξ 0 b/1
force
Fig. 15
✔ point forcef(x) = δ(ξ −ξ 0 ) ( 0)
1
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mere integrability ofκ,f sufficient.
✎ notation: Cpw([a,b]) k ˆ= globallyC k−1 and piecewisek-times continuously differentiable functions
on [a,b] ⊂ R: for each v ∈ Cpw([a,b]) k there is a finite partition {a =
τ 0 < τ 1 < ··· < τ m = b} such that v |]τi−1 ,τ i [ can be extended to a function
∈ C k ([τ i−1 ,τ i ]). Cpw([a,b]) 0 ˆ= piecewise continuous functions with only a finite
number of discontinuities.
u ∈ C 1 pw([0,1])
u ′ ∈ Cpw([0,1])
0
0 x 1 x 2 x 3 ··· 1
0 x 1 x 2 x 3 ···
Fig. 1 12 Fig. 13
1.3
p. 57
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1.3.3 Differential equation
Consider non-linear variational equation (1.3.7):
∫1
0
κ(ξ)
L
Fig. 16
✸
(∥ ∥ u ′ (ξ) ∥ ∥−L )u′ (ξ)·v ′ (ξ)
‖u ′ −f(ξ)·v(ξ)dξ = 0 ∀v ∈ (C 1 (ξ)‖
0,pw ([0,1]))2 . (1.3.7)
Assumption: u ∈ (C 2 ([0,1])) 2 &κ ∈ C 1 ([0,1]) &f ∈ (C 0 ([0,1])) 2 (1.3.21)
1.3
p. 59
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Example 1.3.20 (Non-smooth external forcing).
Setting:
κ = const (homogeneous string)
1.3
p. 58
Recall: integration by parts formula [29, Satz 6.1.2]:
∫1
0
∫1
u(ξ)v ′ (ξ)dξ = −
0
u ′ (ξ)v(ξ)+(u(1)v(1)−u(0)v(0))
} {{ }
boundary terms
∀u,v ∈ C 1 pw([0,1]) . (1.3.22)
1.3
p. 60

Apply to elastic energy contribution in (1.3.7):
∫1
0
( κ(ξ) (∥ ∥ u ′ (ξ) ∥ ∥−L ) u′ )
(ξ)
L ‖u ′ ·v ′ (ξ)−f(ξ)·v(ξ)dξ
(ξ)‖
∫1
=
0
{
− d dξ
( κ(ξ) (∥ ∥ u ′ (ξ) ∥ ∥−L ) u ′ ) }
(ξ)
L ‖u ′ −f(ξ) ·v(ξ)dξ .
(ξ)‖
Note: v(0) = v(1) = 0 ⇒ boundary terms vanish !
(1.3.7) ⇒
∫1 {
− d ( κ(ξ) (∥ ∥ u ′ (ξ) ∥ ∥−L ) u′ ) }
(ξ)
dξ L ‖u ′ −f(ξ) ·v(ξ)dξ = 0
(ξ)‖
0 } {{ }
∈Cpw([0,1])
0
∀v ∈ (C 1 0 ([0,1]))2
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Minimization problem
(1.2.17)
u ∗ = argminJ(v)
v∈V
❶
⇒
Variational problem
(1.3.7)
a(u;v) = f(v) ∀v
❷
⇒
❶: equivalence (“⇔”) holds if minimization problem has unique solution
Two-point BVP
F(u,u ′ ,u ′′ ) = f ,
u(0),u(1) fixed .
❷: meaningful two-point BVP stipulates extra regularity (smoothness) ofu, see Rem. 1.3.25.
Terminology:
{ }
minimization problem (1.2.17)
variational problem (1.3.7)
is called the weak form of the string model,
Two-point boundary value problem (1.3.24), (1.2.1) is called the strong form of the
string model.
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A solutionuof (1.3.24), for which all occurring derivatives are continuous is called
a classical solution of the two-point BVP.
✬
Lemma 1.3.23 (fundamental lemma of the calculus of variations).
Letf ∈ C 0 pw([a,b]),−∞ < a < b < ∞, satisfy
∫ b
f(ξ)v(ξ)dξ = 0 ∀v ∈ C k ([a,b]), v(a) = v(b) = 0 .
a
for somek ∈ N 0 . This impliesf ≡ 0.
✫
✩
✪
1.3
p. 61
Numerical
Methods
for PDEs
Remark 1.3.25 (Extra regularity requirements). → Ex. 1.3.20
Minimization problem
(1.2.17):
=
κ,f integrable,
u piecewiseC 1
Variational problem
(1.3.7):
κ,f integrable,
u piecewiseC 1
≠
Two-point BVP:
κ ∈ C 1 ([0,1]),
f ∈ (C 0 ([0,1]) 2 ,
u ∈ (C 2 ([0,1])) 2 .
1.3
p. 63
Numerica
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Ass. (1.3.21) & (1.3.7)
Lemma 1.3.23
=⇒ − d dξ
( κ(ξ) (∥ ∥ u ′ (ξ) ∥ ∥−L ) u′ )
(ξ)
L ‖u ′ = f(ξ) 0 ≤ ξ ≤ 1 .
(ξ)‖
R. Hiptmair
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☞
formulation as a classical two-point BVP imposes (unduly) restrictive smoothness on solution
and coefficient functions.
✬
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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✩DRAFT
Ifκ ∈ C 1 ,f ∈ C 0 , then aC 2 -minimizer ofJ/aC 2 -solution of (1.3.7) solve the 2nd-order ODE
− d ( κ(ξ) (∥ ∥ u ′∥ ∥−L ) u′ )
dξ L ‖u ′ = f on [0;1] . (1.3.24)
‖
Lemma 1.3.26 (Classical solutions are weak solutions).
For ˜κ ∈ C 1 ([0,1]), any classical solution of (1.3.24) also solves (1.3.7).
✫
Proof. (“Derivation of (1.3.24) reversed”)
✪
ODE (1.3.24) + boundary conditions (1.2.1) = two-point boundary value problem
(on domainΩ = [0,1])
1.3
p. 62
1.3
p. 64

Multiply (1.3.24) withv ∈ C0,pw 1 ([0,1]) and integrate over[0,1]. The push a derivative ontov by using
(1.3.22). ✷
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☛ variational problem corresponding to (1.4.2): use
‖x+th‖ 2 = ‖x‖+2tx·h+t 2 ‖h‖ 2 = ‖x‖+2tx·h+O(t 2 ) .
Numerica
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1.4 Simplified model
Setting:
➥
expected:
taut string
L ≪ ‖u(0)−u(1)‖ . (1.4.1)
∥
∥u ′ ∗(ξ) ∥ ≫ L for all0 ≤ ξ ≤ 1 for solutionu ∗ of (1.2.17)
“Intuitive asymptotics”: renormalize stiffness κ → ˜κ := L κ ,[˜κ] = Nm−1
suppress equilibrium length: L = 0 in (1.2.17).
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DRAFT
˜J(u+tv)−
lim
˜J(u) ∫1
= ˜κ(ξ)u ′ (ξ)·v ′ (ξ)−f(ξ)·v(ξ) dξ = 0 , v ∈ (C
t→0 t
pw,0 1 ([0,1]))2 .
0
Variational equation satisfied by solutionũ ∗ of (1.4.2):
∫ 1
˜κ(ξ)u ′ ∗(ξ)·v ′ (ξ)−f(ξ)·v(ξ)dξ = 0 ∀v ∈ (Cpw,0 1 ([0,1]))2 . (1.4.4)
0
Remark 1.4.5 (Linear variational problems). → Rem. 1.3.12
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Simplified equilibrium model:
ũ ∗ =
∫1
argmin
2˜κ(ξ)∥ 1
∥u ′ (ξ) ∥ 2 −f(ξ)·u(ξ)dξ
u∈(Cpw([0,1])) 1 2 & (1.2.1)
0
} {{ }
=:˜J(u)
. (1.4.2)
1.4
p. 65
Numerical
Methods
for PDEs
(1.4.4) has the structure (1.3.14)
where now
u ∈ V: a(u,v) = l(v) ∀v ∈ V 0 , (1.4.6)
a : V 0 ×V 0 ↦→ R is a bilinear form (→ Def. 1.3.13), that is, linear in both arguments.
1.4
p. 67
Numerica
Methods
for PDEs
= a quadratic minimization problem in a function space !
Remark 1.4.3 (Quadratic minimization problem).
The functional (= mapping from a function space to R) ˜J from (1.4.2) has the structure
˜J(u) = 1 2 a(u,u)−l(u) ,
with a symmetric bilinear forma : V ×V ↦→ R and a linear forml : V ↦→ R.
Minimization problems for functionals of this form have been dubbed quadratic minimization problems.
△
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DRAFT
1.4
p. 66
In general, quadratic minimization problems give rise to linear variational problems .
This can be confirmed by an elementary computation:
J(u) = 1 2 a(u,u)−l(u)
J(u+tv)−J(u) ta(u,v)+ 1 lim = lim 2 t2 a(v,v)−tl(v)
= a(u,v)−l(v) ,
t→0 t t→0 t
where the bilinearity ofaand the linearity oflwas crucial, see Rem. 1.4.3.
△
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DRAFT
1.4
p. 68

☛ Corresponding two-point boundary value problem: by integration by parts, see (1.3.22),
∫1
∫1
˜κ(ξ)u ′ ∗(ξ)·v ′ (ξ)−f(ξ)·v(ξ)dξ =
0
Then use Lemma 1.3.23.
0
{
− d (˜κ(ξ) d ) }
dξ dξ u(ξ) −f(ξ) ·v(ξ)dξ
∀v ∈ (C 1 pw,0 ([0,1]))2 .
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R. Hiptmair
Ifκ ∈ C 1 ,f ∈ C 0 , then aC 2 -solution of (1.4.4) solves the two-point BVP
Special setting: “gravitational force” f(ξ) = −g(ξ)e 2
− d (
˜κ(ξ) du )
dξ dξ (ξ) = f(ξ) , 0 ≤ ξ ≤ 1 ,
( ) ( )
(1.4.7)
a b
u(0) = , u(1) = .
u a u b
DRAFT
(1.4.2) decouples into two minimizatin problems for the components ofu!
∫1
ũ 1,∗ = argmin 1
u∈Cpw([0,1]),u(0)=a,u(1)=b
1 2 ˜κ(ξ)(u ′ (ξ)) 2 dξ ,
0
(1.4.2) ⇒
∫1
ũ 2,∗ = argmin 2˜κ(ξ)(u′ 1 (ξ)) 2 +g(ξ)u(ξ) dξ .
u∈Cpw 1 ([0,1]),u(0)=u a,u(1)=u b 0
The minimization problem forũ 1,∗ has a closed-form solution:
(1.4.8)
∫
b−a
ξ
ũ 1,∗ (ξ) = a+ ∫ 10 ˜κ −1 ˜κ −1 (τ)dτ , 0 ≤ ξ ≤ 1 . (1.4.9)
(τ)dτ
0
This solution can easily be found by converting the minimization problem to a 2-point boundary value
problem as was done above, cf. (1.4.4), (1.4.7).
The minimization problem forũ 2,∗ leads to the linear variational problem, cf. (1.4.4)
ũ 2,∗ ∈ C 1 pw([0,1])
ũ 2,∗ (0) = u a , ũ 2,∗ (1) = u b
:
∫1
0
∫1
˜κ(ξ)ũ ′ 2,∗ (ξ)v′ (ξ)dξ = −
0
g(ξ)v(ξ)dξ ∀v ∈ C 1 0,pw ([0,1]) .
(1.4.10)
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p. 69
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1.4
p. 70
Remark 1.4.11 (Graph description of string shape).
Focus: situation with vertical gravitational force, see (1.4.9), (1.4.10)
u
a
(a,u l )
Load functionĝ(x)
u(x)
(b,u r )
b
Fig. 17
x
Describe shape of string through graph of displacement
function û ∗ = û ∗ (x), û : [a,b] ↦→ R
(physical units[û] = 1m).
boundary conditions:
u ∗ (a) = u a , u ∗ (b) = u b . (1.4.12)
û(x) = ũ 2,∗ (Φ −1 (x)) with Φ(ξ) := ũ 1,∗ (ξ) . (1.4.13)
Here ũ 1,∗ (ξ), ũ 1,∗ (ξ) are the components of the curve description of the equilibrium shape of the
string, see Sect. 1.2.1:
u ∗ (ξ) =
) (ũ1,∗ (ξ)
, 0 ≤ ξ ≤ 1 .
ũ 2,∗ (ξ)
Of course, the graph description is possible only for special string shapes. It also hinges on the choice
of suitable coordinates.
Note: ξ ↦→ Φ(ξ) is monotone,Φ ′ (ξ) ≠ 0 for all0 ≤ ξ ≤ 1,Φ(0) = a,Φ(1) = b.
By chain rule [29, Thm. 5.1.3]:
v(ξ) = ̂v(Φ(ξ)) ⇒ v ′ (ξ) = d̂v
dx (x)Φ′ (ξ) , x := Φ(ξ) . (1.4.14)
Recall: transformation formula for integrals in one dimension (substitution rule, x := Φ(ξ), “dx =
Φ ′ (ξ)dξ”):
q ∈ C 0 pw([0,1]):
∫1 b=Φ(1) ∫
1
q(ξ)dξ = ̂q(x)|
Φ ′ (Φ −1 (x)) |dx , ̂q(x) := g(Φ−1 (x)) . (1.4.15)
0 a=Φ(0)
← (1.4.14) & (1.4.15)
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Harbrech
V.
Gradinar
A. Chern
DRAFT
1.4
p. 71
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.4
p. 72

Variational problem of taut string problem described as a function of spatial coordinate:
∫1
0
∫b
˜κ(ξ)ũ ′ 2,∗ (ξ)v′ (ξ)dξ =
∫1
−
0
a
∫b
=
a
∫b
g(ξ)v(ξ)dξ = −
˜κ(Φ −1 (x))Φ ′ (ξ) dû
dx (x)Φ′ (ξ) d̂v
dx (x) 1
|Φ ′ (ξ)| dx
˜κ(Φ −1 (x))|Φ ′ (Φ −1 dû
(x))| (x)d̂v
} {{ } dx dx (x)dx ,
=:̂σ(x)
a
g(Φ −1 (x))
|Φ ′ (Φ −1 ̂v(x)dx .
(x))|
} {{ }
=:ĝ(x), [ĝ]=Nm −1
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✛
✚
Only option:
Numerical algorithm
Computer
−−−−−−→
Finitely many floating point operations
Numerical algorithms can only operate on discrete models
Continuous (PDE) model
(“∞-dimensional”)
Discretization
−−−−−−−−→
approximate solution
Discrete model
(“finitely many unknowns”)
Numerica
Methods
for PDEs
✘
✙
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Linear variational problem in physical space coordinate on spatial domainΩ = [a,b]:
û ∗ ∈ C 1 pw([a,b]),
û ∗ (a) = u a , û ∗ (b) = u b
:
∫b
(assuming ̂σ ∈ C 1 ([a,b]))
(1.4.16) ⇒
a
̂σ(x) dû ∗
(x)d̂v ∫b
dx dx (x)dx = −
a
ĝ(x)̂v(x)dx ∀̂v ∈ C 1 0,pw ([a,b]) .
Two-point BVP
{ (
d
̂σ(x) dû )
∗
dx dx (x) = ĝ(x) , a ≤ x ≤ b ,
û ∗ (a) = u a , û ∗ (b) = u b .
(1.4.16)
(1.4.17)
△
1.4
p. 73
Numerical
Methods
for PDEs
as small as possible
(only a few unknowns)
as accurate as possible
(good approximation (∗))
as faithful as as possible
(structure preserving)
(∗): needs a measure for quality of a solution, usually a norm of the error, error = difference of
exact/analytic and approximate solution.
Remark 1.5.1 (“Physics based” discretization).
Mass-spring model (→ Sect. 1.2.2) = discretization of the minimization problem (1.2.17) describing
the elastic string.
1.5
p. 75
Numerica
Methods
for PDEs
1.5 Discretization
Goal:
“computation” of a/the solutionu : [0,1] ↦→ R 2 of
⎧
⎨ minization problem (1.2.17)
variational problem (1.3.7)
⎩
two-point BVP (1.3.24) & (1.2.1)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
This discretization may be called “physics based”, because it is inspired by the (physical) context of
the model.
Note: Other approaches to discretization discussed below will lead to equations resembling the massspring
model, see 1.5.1.2.
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
a function: infinite amount of information, see [18, Rem. 3.1.3].
! Well, just provide a formula foru (analytic solution): in general elusive
for the above problems
1.5
p. 74
This section will present a few strategies on how to derive discrete models for the problem of computing
the shape of an elastic string. The different approaches start from different formulations, some
target the minization problem (1.2.17), or, equivalently, the variational problem (1.3.7), while others
tackle the ODE (1.3.24) together with the boundary conditions (1.2.1).
1.5
p. 76

Remark 1.5.2 (Timestepping for ODEs).
For initial value problems for ODEs, whose solutions are functions, too, we also face the problem of
discretization: timestepping methods compute a finite number of approximate values of the solutions
at discrete instances in time, see [18, Ch. 12].
△
Numerical
Methods
for PDEs
Note that a subscript tag N distinguishes “discrete functions/quantities”, that is, functions/operators
etc. that are associated with a finite dimensional space. In some contexts, N will also be an integer
designating the dimension of a finite dimensional space.
Formal presentation: V ,V 0 : (affine) function spaces,dimV 0 = ∞,
V N ,V N,0 : subspacesV N ⊂V ,V N,0 ⊂V 0 , N := dimV N,0 ,dimV N < ∞.
Numerica
Methods
for PDEs
Remark 1.5.3 (Coefficients/data in procedural form).
For the elastic string model (→ Sect. 1.2.3) the stiffness κ(ξ), and force field f may not be available
in closed form (as formulas).
Instead they are usually given in procedural form:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Galerkin discretization of minimization problem for functionalJ : V ↦→ R:
Continuous minimization problem
u = argminJ(v) . (1.5.4)
v∈V
Galerkin disc.
−−−−−−−−→
Discrete minimization problem
u N = argmin
v N ∈V N
J(v N ) . (1.5.5)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
function k = kappa(xi);,
function f = force(xi);,
1.5
p. 77
Galerkin discretization of abstract (non-linear) variational problem (1.3.14), see Rem. 1.3.12
1.5
p. 79
because they may be obtained
as results of another computation,
Numerical
Methods
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Continuous variational problem
u ∈ V: a(u;v) = l(v) ∀v ∈ V 0 .
(1.5.6)
Galerkin disc.
−−−−−−−−→
Discrete variational problem
u N ∈ V N : a(u N ;v N ) = l(v N )
∀v N ∈ V N,0 . (1.5.7)
Numerica
Methods
for PDEs
by interpolation from a table.
Terminology: u N ∈ V N satisfying (1.5.5)/(1.5.7) is called a Galerkin solution of (9.2.21)/(9.3.62)
viable discretizations must be able to deal with data in procedural form!
△
R. Hiptmair V N is called the (Galerkin) trial space,V N,0 is the (Galerkin) test space.
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 1.5.8 (Relationship between discrete minimization problem and discrete variational problem).
Simple idea of first step of Galerkin discretization
In Sect. 1.3.1 we discovered the equivalence
⎧
⎨ minization problem, e.g., (1.2.17)
replace function spaceV/V
In ⇕
0 with
⎩
finite dimensional subspaceV
variational problem, e.g. (1.3.7)
N /V N,0
Continuous minimization problem
Continuous variational problem
⇐⇒
1.5
(9.2.21)
(9.3.62)
1.5.1 Galerkin discretization
p. 78
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.5
p. 80

Now it seems that we have two different strategies for Galerkin discretization:
1. Galerkin discretization via the discrete minimization problem (1.5.5),
Numerical
Methods
for PDEs
Below we will always make the assumption V = u 0 +V 0 .
Numerica
Methods
for PDEs
2. Galerkin discretization based on the discrete variational problem (1.5.7).
However,
the above equivalence extends to the discrete problems!
△
More precisely, we have the commuting relationship:
minization problem
Galerkin discretization
−−−−−−−−−−−−−→ discrete minimization problem
⏐ ⏐
Configurational ⏐↓ ⏐↓ Configurational
derivative inV 0 derivative inV N,0
(1.5.9)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
However, a computer is clueless about a concept like “finite dimensional subspace”. What it can
process are arrays of floating point numbers.
Idea: choose basis B N = {b 1 N ,...,bN N } ofV N,0 : V N,0 = Span{B N }
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
variational problem
Galerkin discretization
−−−−−−−−−−−−−→ discrete variational problem .
The commuting diagram means that the same discrete variational problem is obtained no matter
whether
insert basis representation into minimization problem (1.5.5)
v N ∈ V N,0 ⇒ v N = u 0 +ν 1 b 1 N +···+ν Nb N N , ν i ∈ R , (1.5.13)
and variational equation (1.5.7)
1. the minimization problem is first restricted to a finite dimensional subspace and the result is converted
into a variational problem according to the recipe of Sect. 1.3.1.
1.5
p. 81
v N ∈ V N,0 ⇒ v N = ν 1 b 1 N +···+ν Nb N N , ν i ∈ R , (1.5.14)
u N ∈ V N ⇒ u N = u 0 +µ 1 b 1 N +···+µ Nb N N , µ i ∈ R . (1.5.15)
1.5
p. 83
2. or whether the variational problem derived from the minimization problem is restricted to the subspace.
Numerical
Methods
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Remark 1.5.16 (Ordered basis of test space).
Numerica
Methods
for PDEs
To see this, understand that the manipulations of Sect. 1.3.1 can be carried out for infinite and finite
dimensional function spaces alike.
△
Once we have choosen a basis B and ordered it, as already indicated in the notation above, the
test space V N,0 can be identified with R N : a coefficient vector ⃗µ = (µ 1 ,...,µ N ) T ∈ R N provides
a unique characterization of a functionu ∈ V N,0 (basis property)
Remark 1.5.10 (Offset functions and Galerkin discretization).
Often: V = u 0 +V 0 , with offset functionu 0 → Rem. 1.3.19, (1.3.16)
Ifu 0 is sufficiently simple, we msay choose a trial space V N = u 0 +V N,0
➣ Discrete variational problem analoguous to (1.3.19)
w N ∈ V N,0 : a(u 0 +w N ;v N ) = l(v N ) ∀v N ∈ V N,0 ➥ u N := w N +u 0 . (1.5.11)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Discrete minimization problem
u N = argminJ(v N ) . (1.5.5)
v N ∈V
N∑
u = µ j b j N . R. Hiptm
j=1
△
DRAFT
Minimization problem on R N
⃗µ = argminF(⃗ν) , (1.5.17)
⃗ν∈R N
F(⃗ν) := J(u 0 +ν 1 b 1 N +···+ν Nb N N ) .
Basis
−−−−−−−−→
representation
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
In the case of a linear variational problem (→ Rem. 1.4.5), that is, a bilinear forma, we have
(1.5.11) ⇔ a(w N ,v N ) = l(v N )−a(u 0 ,v N ) ∀v N ∈ V N,0 . (1.5.12)
1.5
p. 82
1.5
p. 84

amenable to classical optimization
techniques
✎ notation: ⃗ν, ⃗µ ˆ= vectors of coefficients (ν i ) N i=1 , (µ i) N i=1 , in basis representation of functions
v N ,u N ∈ V N according to (1.5.13).
Discrete variational problem
u N ∈ V N : a(u N ;v N ) = l(v N )
∀v N ∈ V N,0 . (1.5.7)
Basis
−−−−−−−−→
representation
System of equations
N∑
a(u 0 + µ j b j N ;bk N ) = l(bk N )
j=1
∀k = 1,...,N . (1.5.18)
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.5.1.1 Spectral Galerkin scheme
A simple function space (widely used for interpolation, see [18, Ch. 9], and approximation, see [18,
Sec. 9.1]): for intervalΩ ⊂ R
V N,0 = P p (R)∩C 0 0 (Ω)
ˆ= space of univariate polynomials of degree ≤ p vanishing at endpoints ofΩ ,
N := dimV N = p−1
(1.5.19)
[18, Sect. 3.2] for more information.
Obvious: choice (1.5.19) guarantees V N ⊂ C 1 pw,0 (Ω) (evenV N,0 ⊂ C ∞ (Ω))
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
use techniques for linear/non-linear
systems of equations, see [18, Ch. 2],
[18, Ch. 4].
Please note thatV N,0 is a space of global polynomials onΩ.
Example 1.5.20 (Spectral Galerkin discretization of linear variational problem).
The choice of the basisBhas no impact on the (set of) Galerkin solutions of (1.5.7)!
Below, we apply Galerkin approaches to
• (1.4.16) as an example for the treatment of a linear variational problem:
u ∈ C 1 pw([a,b]),
u(a) = u a , u(b) = u b
:
Here:
∫b
a
σ(x) du ∫b
dx (x)dv dx (x)dx = −
spatial domainΩ = [a,b], linear offset functionu 0 (x) = b−x
b−a u a+ x−a
b−a u a,
function space V 0 = C0,pw 1 ([a,b]).
• (1.3.7) to demonstrate its use in the case of a non-linear variational equation:
a
g(x)v(x)dx ∀v ∈ C 1 0,pw ([a,b]) .
(1.4.16)
1.5
p. 85
Numerical
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Targetted:
linear variational problem (1.4.16) with
a = 0,b = 1 ➣ domainΩ =]0,1[,
constant coefficient functionσ ≡ 1,
loadg(x) = −4π(cos(2πx 2 )−4πx 2 sin(2πx 2 )),
boundary valuesu a = u b = 0.
Concrete variational problem
∫1
u ∈ C0,pw 1 ([0,1]):
0
du
dx (x) dv
∫1
dx (x)dx = −
0
u(x) = sin(2πx 2 ) , 0 < x < 1 .
because d2 u
dx2(x) = g(x).
g(x)v(x)dx ∀v ∈ C 1 0,pw ([0,1]) .
1.5
p. 87
Numerica
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R. Hiptm
(1.5.21) C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
u ∈ Cpw([0,1])
1 ∫1
u(0),u(1) : from (1.2.1)
Here:
0
κ(ξ)
L
(∥ ∥ u ′ (ξ) ∥ ∥−L )u′ (ξ)·v ′ (ξ)
‖u ′ −f(ξ)·v(ξ)dξ = 0
(ξ)‖
∀v ∈ (C 1 0,pw ([0,1]))2 . (1.3.7)
parameter domainΩ = [0,1], linear offset functionu 0 (ξ) = ξu(0)+(1−ξ)u(1),
function space V 0 = (C 1 0,pw ([a,b]))2 .
1.5
p. 86
1.5
p. 88

1.5
1
Numerical
Methods
for PDEs
0.6
0.4
Integrated Legendre polynomials
n=1
n=2
n=3
n=4
n=5
Numerica
Methods
for PDEs
Polynomial spectral Galerkin discretization, degreep
∈ {4,5,6}.
Plots of approximate/exact solutions
✄
u
0.5
0
M n
(t)
0.2
0
−0.2
✁ integrated Legendre polynomials
M 1 ,...,M 5
Remark 1.5.22 (Choice of basis for polynomial spectral Galerkin methods).
−0.5
N=4
N=5
N=6
u(x)
−1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Fig. 18
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
−0.4
−0.6
−1 −0.5 0 0.5 1
t
Fig. 20
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Sought: (ordered) basis of V N,0 := C 1 0 ([−1,1])∩P p(R)
➊ “Tempting”:
monomial-type basis
{
V N,0 = Span 1−x 2 ,x(1−x 2 ),x 2 (1−x 2 ),...,x p−2 (1−x 2 }
) . (1.5.23)
1.5
p. 89
Definition 1.5.25 (Legendre polynomials). → [18, Def. 10.4.12]
Then-th Legendre polynomialP n ,n ∈ N 0 , is defined by (Rodriguez formula)
P n (x) := 1
n!2 n d n
dx n[(x2 −1) n ] .
1.5
p. 91
P n
(t)
Monomial basis polynomials (1−x 2 )x n
1
0.8
0.6
0.4
0.2
0
−0.2
n=0
n=1
−0.4
n=2
n=3
n=4
−0.6
n=5
n=6
−0.8
−1 −0.5 0 0.5 1
t
Fig. 19
➋ “Popular”:
✁ Monomial basis polynomials
→ Ex. 1.5.46 below
Beware: ill-conditioned !
integrated Legendre polynomials
{ ∫ x
}
V N,0 = Span x ↦→ M n (x) := P n (τ)dτ, n = 1,...,p−1 , (1.5.24)
−1
whereP n ˆ= n-th Legende polynomial.
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.5
p. 90
Legendre polynomialsP 0 ,...,P 5
P 0 (x) = 1 ,
P 1 (x) = x ,
P 2 (x) = 3 2 x2 − 1 2 ,
0.8
0.6
0.4
0.2
−0.2
P 3 (x) = 5 2 x3 − 3 2 x ,
−0.4
P 4 (x) = 35
8 x4 − 15
4 x2 + 3 8 . −0.6
−0.8
−1
−1 −0.5 0 0.5 1
Some facts about Legendre polynomials:
Symmetry:
P n is
Orthogonality
{
even
odd
for
P n
(t)
1
0
Legendre polynomials
t
n=0
n=1
n=2
n=3
n=4
n=5
Fig. 21
{
evenn
oddn , P n(1) = 1 , P n (−1) = (−1) n . (1.5.26)
∫ 1
P n (x)P m (x)dx =
−1
{ 2
2n+1 , ifm = n ,
0 else.
(1.5.27)
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.5
p. 92

3-term recursion
P n+1 (x) := 2n+1
n+1 xP n(t)− n
n+1 P n−1(x) , P 0 := 1 , P 1 (x) := x . (1.5.28)
This formula paves the way for the efficient evaluation of all Legendre polynomials at many (quadrature)
points, see Code 1.5.28.
Numerical
Methods
for PDEs
Note the effect of this transformation on the derivative (chain rule!):
db i N
dx (x) = dM (
i
2 x−a ) (
dx b−a −1 2
·
b−a = P i 2 x−a )
b−a −1 ·
2
b−a . (1.5.35)
Numerica
Methods
for PDEs
Code 1.5.29: Computation of Legendre polynomials based on 3-term recursion (1.5.28)
1 f u n c t i o n V= legendre(n,x)
2 % Computes values of Legendre polynomials up to degree n
3 % in the points x j passed in the row vector x.
4 % Exploits the 3-term recursion (1.5.28) for Legendre polynomials
5 V = ones( s i z e(x)); V = [V; x];
6 f o r j=1:n-1
7 V = [V; ((2*j+1)/(j+1)).*x.*V(end,:) - j/(j+1)*V(end-1,:)];
8 end
Representation of derivatives and primitives, cf. Code 1.5.31:
P n (x) = ( dx d P n+1(x)− dx d P n−1(x))/(2n+1) , n ∈ N , (1.5.30)
1
M n (x) =
2n+1 (P n+1(x)−P n−1 (x)) and dM n
dx = P n . (1.5.31)
Code 1.5.32: Computation of (integrated) Legendre polynomials using (1.5.28) and (1.5.31)
1 f u n c t i o n [V,M] = intlegpol(n,x)
2 % Computes values of the first n+1 Legendre polynomials P n (returned in
3 % matrix V) and the first n−1 integrated Legendre polynomials
4 % M n (returned in matrix M) in the points x j passed in the
5 % row vector x. Uses the recursion formulas (1.5.28) and
6 % (1.5.31)
7 V = ones( s i z e(x)); V = [V; x];
8 f o r j=1:n-1, V = [V; ((2*j+1)/(j+1)).*x.*V(end,:) -
j/(j+1)*V(end-1,:)]; end
9 M = diag(1./(2*(1:n-1)+1))*(V(3:n+1,:) - V(1:n-1,:));
Remark 1.5.33 (Transformation of basis functions).
On a “general domainΩ = [a,b]”, we obtain the basis function by a so-called affine transformation of
the basis functions on[0,1], cf. [18, Rem. 10.1.3], e.g., in the case of integrated Legendre polynomials
as basis functions onΩ = [a,b] we use the basis functions
(
b i N (x) = M i 2 x−a )
b−a −1 , a ≤ x ≤ b . (1.5.34)
△
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.5
p. 93
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C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.5
p. 94
Remark 1.5.36 (Spectral Galerkin discretization with quadrature).
Consider the linear variational problem, cf. (1.4.16),
u ∈ C 1 0,pw ([a,b]):
∫b
a
σ(x) du
dx (x)dv
∫b
dx (x)dx =
Assume: σ,g only given in procedural form, see Rem. 1.5.3.
a
g(x)v(x)dx ∀v ∈ C0,pw 1 ([a,b]) . (1.5.37)
Analytic evaluation of integrals becomes impossible even ifu,v polynomials !
Only remaning option: Numerical quadrature, see [18, Ch. 10]
Replace integral withm-point quadrature formula on[a,b],m ∈ N → [18, Sect. 10.1]:
∫ b
m∑
f(t)dt ≈ Q n (f) := ωj m f(ζm j ) . (1.5.38)
a
j=1
ωj n : quadrature weights , ζn j : quadrature nodes ∈ [a,b] .
(1.5.37) ➣ discrete variational problem with quadrature:
u N ∈ V N :
m∑
ωj m σ(ζm j )du N
dx (ζm j )dv N
dx (ζm j ) = ∑ m ωj m g(ζm j )v(ζm j ) ∀v ∈ V N . (1.5.39)
j=1
j=1
Popular (global) quadrature formulas: Gauss quadrature → [18, Sect. 10.4]
Important: Accuracy of quadrature formula and computational cost (no. m of quadrature nodes) have
to be balanced.
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.5
p. 95
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.5
p. 96

Remark 1.5.40 (Implementation of spectral Galerkin discretization for linear 2nd-order two-point
BVP).
Setting:
linear variational problem (1.5.37) ➣ u 0 = 0,
coefficientsσ,g in procedural form, see Rem. 1.5.3,
approximation of integrals byp-point Gaussian quadrature formula,
polynomial spectral Galerkin discretization, degree≤ p,p ≥ 2,
basisB: integrated Legendre polynomials, see (1.5.24):
V N,0 = Span{M n , n = 1,...,p−1} , M n ˆ= integrated Legendre polynomials .
Trial expression, cf. (1.5.13)
Note:
by definition
From (1.5.39) with (1.5.40)
u N = µ 1 M 1 +µ 2 M 2 +···+µ N M N , µ i ∈ R , N := p−1 .
d
dx M n = P n .
m∑
ωj m σ(ζm j ) ∑ N µ l P l (ζj m )P k(ζj m ) = ∑ m ωj m g(ζm j )M k(ζj m )
j=1 l=1 j=1
} {{ }
=:ϕ k
, k = 1,...,N . (1.5.41)
⇕
⎛
⎞
N∑ m∑
⎝ ωj m σ(ζm j )P l(ζj m )P k(ζj m ) ⎠µ l = ϕ k , k = 1,...,N . (1.5.42)
l=1 j=1
⇕
△
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Code 1.5.44: Polynomial spectral Galerkin solution of (1.5.37)
1 f u n c t i o n u = lin2pbvpspecgalquad(sigma,g,N,x)
2 % Polynomial spectral Galerkin discretization of linear 2nd-order two-point BVP
3 % − d ) = g(x), u(0) = u(1) = 0
dx (σ(x)du dx
4 % on Ω = [0,1]. Trial space of dimension N.
5 % Values of approximate solution in points x j are returned in the row vector u
6 m = N+1; % Number of quadrature nodes
7 [zeta,w] = gaussquad(m); % Obtain Gauss quadrature nodes w.r.t [−1,1]
8 % Compute values of (integrated) Legendre polynomials at Gauss nodes
9 [V,M] = intlegpol(N+1,zeta’);
10 omega = w’.*sigma((zeta’+1)/2)*2; % Modified quadrature weights
11 A = V(2:N+1,:)*diag(omega)*V(2:N+1,:)’; % Assemble Galerkin matrix
12 phi = M*(0.5*w’.*g((zeta’+1)/2))’; % Assemble right hand side
vector
13 mu = A\phi; % Solve linear system
14 % Compute values of integrated Legendre polynomials at output points
15 [V,M] = intlegpol(N+1,2*x-1); u = mu’*M;
Code 1.5.45: MATLAB driver script creating plots of Ex. 1.5.20
1 % MATLAB script: Driver routine for polynomial spectral Galerkin
discretization of linear 2nd-order
2 % two-point boundary value problem
3 c l e a r a l l ;
4 % Coefficient functions (function handles, see MATLAB help)
5 sigma = @(x) ones( s i z e (x));
6 g = @(x) -4* p i *(cos(2* p i *x.^2)-4* p i *x.^2.* s in(2* p i *x.^2));
7 x = 0:0.01:1; % Evaluation points
8 % Computation with trial space of dimension 4,5,6
9 N = 4; U = [lin2pbvpspecgalquad(sigma,g,N,x); ...
10 lin2pbvpspecgalquad(sigma,g,N+1,x); ...
11 lin2pbvpspecgalquad(sigma,g,N+2,x)];
12 % Graphical output
13 f i g u r e (’name’,’Polynomial spectral Galerkin’);
14 p l o t (x,U); hold on;
15 p l o t (x,sin (2* p i *x.^2),’g--’,’linewidth’,2);
16 x l a b e l (’{\bf x}’,’fontsize’,14);
17 y l a b e l (’{\bf u}’,’fontsize’,14);
18 legend(’N=4’,’N=5’,’N=6’,’u(x)’,’location’,’southwest’);
19 p r i n t -depsc2 ’../../../Slides/NPDEPics/specgallinsol.eps’;
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A⃗µ = ⃗ϕ with
m∑
(A) kl := ωj m σ(ζm j )P l(ζj m )P k(ζj m ), k,l = 1,...,N ,
j=1
⃗µ = (µ l ) N l=1 ∈ RN , ⃗ϕ = (ϕ k ) N k=1 ∈ RN .
(1.5.43)
Example 1.5.46 (Conditioning of spectral Galerkin system matrices).
△
A linear system of equations !
The Galerkin discretization of a linear variational problem leads to a linear system of equations.
1.5
p. 98
1.5
p. 10

Finally we can provide a rationale for preferring integrated Legendre polynomials to plain monomials
for polynomial spectral Galerkin discretization: the argument is based on condition number of the
system matrix from (1.5.43).
Linear variational problem (1.5.21) with bilinear form
∫ 1
a(u,v) = du
dx (x) dx dv(x)dx
, u,v ∈ C1 0,pw ([0,1]) .
0
Choice of basis functions for Galerkin trial/test space V N,0 := P p (R) ∩ C0 0 ([0,1]): monomial
basis (1.5.23), integrated Legendre polynomials (1.5.24).
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Example 1.5.47 (Implementation of spectral Galerkin discretization for elastic string problem).
Targetted: non-linear variational equation on domainΩ = [0,1]
∫1
0
(
κ(ξ)
1− L )
L ‖u ′ u ′ (ξ)·v ′ (ξ)−f(ξ)·v(ξ)dξ = 0 ∀v ∈ (C 1 (ξ)‖
0,pw ([0,1]))2 . (1.3.7)
Dataκ,f given in procedural form, see Rem. 1.5.3.
Spectral Galerkin discretization, basisB= {M n } K n=1 ,K ∈ N, consists of integrated Legendre
polynomials, see (1.5.24) ➣ basis representation, cf. (1.5.40)
( )
µ1
u N (ξ) = u(0)(1−ξ)+u(1)ξ+
M
} {{ } µ 1 (ξ)+···+
K+1
=:u 0 (ξ) (offset function)
( )
µK
M
µ K (ξ) . (1.5.48)
2K
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Approximate evaluation of integrals by m-point Gaussian quadrature on [0,1], m := K + 1
below: nodesζ j , weightsω j ,j = 1,...,m.
1.5
p. 10
Monitored: condition number (w.r.t. Euclidean
matrix norm → [18, Def. 2.5.26]) of Galerkin matrices
condition number
10 12 monomial basis
Legendre basis
10 10
10 8
10 6
10 4
10 2
10 0
2 3 4 5 6 7 8 9 10
N
Fig. 22
Recall that a condition number of 10 m involves a loss of m digits w.r.t. the precision guaranteed for
the right hand side of the linear system. Thus, using the monomial basis for N > 10 may no longer
produce reliable results.
✸
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1.5
p. 102
In analogy to (1.5.41) we arrive at the non-linear system of equations: (M ′ k = P k !)
m∑ K∑
m∑
s j (b−a+ µ l P l (ζ j ))·P k (ζ j ) = ω j f 1 (ζ j )·M k (ζ j ) , k = 1,...,K ,
j=1 l=1
j=1
m∑ K∑
m∑
s j (u b −u a + µ K+l P l (ζ j ))·P k (ζ j ) = ω j f 2 (ζ j )·M k (ζ j ) , k = 1,...,K ,
j=1 l=1
j=1
( )
1
with s j := ω j κ(ζ j )
L − 1 (s
∥
∥u ′ N (ζ j) ∥ ) = s(⃗µ) .
Rewrite in compact form:
( ) ( )
R(⃗µ) 0 ⃗ϕ1
⃗µ =
0 R(⃗µ) ⃗ϕ 2
, (1.5.49)
m∑
with (R) k,l := s j P l (ζ j )P k (ζ j ) ,
j=1
m∑
m∑
(⃗ϕ 1 ) k = ω j f 1 (ζ j )·M k (ζ j )−(b−a) s j P k (ζ j ) ,
j=1
j=1
m∑
m∑
(⃗ϕ 1 ) k = ω j f 2 (ζ j )·M k (ζ j )−(u b −u a ) s j P k (ζ j ) .
j=1
j=1
(1.5.50)
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1.5
p. 10

Iterative solution of (1.5.49) by fixed point iteration:
Initial guess ⃗µ (0) ∈ R N ; k = 0;
repeat
k ← k +1;
Solve the linear system of equations
∥
until ∥⃗µ (k) − ⃗µ (k−1)∥ ∥ ∥∥⃗µ ≤ tol·∥
(k) ∥ ∥
(
R(⃗µ (k−1) ) ( )
) 0
0 R(⃗µ (k−1) ⃗µ (k) ⃗ϕ1
= ;
) ⃗ϕ 2
Code 1.5.51: Polynomial spectral Galerkin discretization of elastic string variational problem
1 f u n c t i o n [vu,figsol] = stringspecgal(kappa,f,L,u0,u1,K,xi,tol)
2 % Solving the non-linear variational problem (1.3.7) for the elastic string by
means of polynomial
3 % spectral Galerkin discretization based on K integratted Legendre polynomials.
Approximate
4 % evaluation of integrals by means of Gaussian quadrature.
5 % kappa, f are handles of type @(xi) providing the coefficient function
6 % κ and the force field f. The column vectors u0 and u1 pass the
7 % pinning points. M is the number of mesh cells, tol specifies the tolerance
for the
8 % fixed point iteration. return value: 2×length(xi)-matrix of node
9 % positions for curve parameter values passed in the row vector xi.
10 i f (nargin < 8), tol = 1E-2; end
11 m = K+1; % Number of quadrature nodes
12 [zeta,w] = gaussquad(m); % Obtain Gauss quadrature nodes w.r.t [−1,1]
13 % Compute values of (integrated) Legendre polynomials at Gauss nodes and
evaluation points
14 [V,M] = intlegpol(K+1,zeta’);
15 [Vx,Mx] = intlegpol(K+1,2*xi-1); Mx = [1-xi;Mx;xi]; %
16 % Compute right hand side based on m-point Gaussian quadrature on [0,1].
17 force = f((zeta’+1)/2); phi = M*(0.5*[w’;w’].*force)’;
18 sv = kappa((zeta’+1)/2); % Values of coefficient function κ at Gauss
points in [0,1].
19 % mu is an 2×(K +2)-matrix, containing the vectorial basis expansion
coefficients
20 % of u N . The first and last column are contributions of the two functions
21 % ξ ↦→ (1−ξ) and ξ ↦→ ξ, which represent the offset function.
22 % Initial guess for fixed point iteration: straight string
23 mu = [u0,zeros(2,K),u1];
24 figsol = f i g u r e ; hold on;
25 f o r k=1:100 % loop for fixed point iteration, maximum 100 iterations
26 % Plot shape of string
27 vu = mu*Mx; p l o t (vu(1,:),vu(2,:),’--g’); drawnow;
28 t i t l e ( s p r i n t f (’K = %d, iteration #%d’,K,k));
29 x l a b e l (’{\bf x_1}’); y l a b e l (’{\bf x_2}’);
30 % Compute values of derivatives of u N and ‖u ′ N ‖ at Gauss points
31 up = mu(:,2:K)*V(2:K,:) + repmat(u1-u0,1,m);
32 lup = s q r t(up(1,:).^2 + up(2,:).^2);
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33 s = 0.5*(w’).*sv.*(1/L - 1./lup); % Initialization of s j
34 % Modification of right hand side due to offset function
35 phi1 = phi(:,1) + (2*(u1(1)-u0(1))*V(2:K+1,:)*s’);
36 phi2 = phi(:,2) + (2*(u1(2)-u0(2))*V(2:K+1,:)*s’);
37 % Assemble K ×K-matrix blocks R of linear system
38 R = 4*V(2:K+1,:)*diag(s)*V(2:K+1,:)’;
39 mu_new = [u0,[(R\phi1)’;(R\phi2)’],u1];
40 % Check simple termination criterion for fixed point iteration.
41 i f (norm(mu_new - mu,’fro’) < tol*norm(mu_new,’fro’)/K)
42 vu = mu*Mx; fig = p l o t (vu(1,:),vu(2,:),’r--’);
43 legend(fig,’spectral Galerkin
solution’,’location’,’southeast’);break; end
44 mu = mu_new;
45 end
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R. Hiptm
DRAFT
✸
Example 1.5.52 (Spectral Galerkin discretization for elastic string simulation).
Test of polynomial spectral Galerkin method for elastic string problem, algorithm of Ex. 1.5.47, Code 1.5.50,
with
1.5
p. 105 pinning positionsu(0) = ( 1.5
0) (
0 ,u(1) =
1 )
0.2 , p. 10
equilibrium lengthL = 0.5,
x 2
0.15
0.05
−0.05
−0.1
−0.15
constant coefficient functionκ ≡ 1N,
gravitational force fieldf(ξ) = − ( 0
2
)
.
0.2
0.1
0
K = 8, iteration #7
spectral Galerkin solution
−0.2
0.5
x 1
0.9 1
Fig. 23
0 0.1 0.2 0.3 0.4 0.6 0.7 0.8
iteration history
1.5.1.2 Linear finite elements
Two ways to approximate functions by polynomials:
x 2
0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
K=4
K=6
K=8
−0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 1
solutionsu N for different resolutions
Fig. 24
✸
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1.5
p. 10

global polynomials
[18, Ch. 9]
←→
piecewise polynomials
[18, Ch. ??]
Numerical
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db j 1
⎧⎪ ⎨hj , ifx j−1 ≤ x ≤ x j ,
N
dx (x) = −h 1 , ifx
⎪ j+1 j < x ≤ x j+1 ,
⎩
0 elsewhere. (piecewise derivative!)
(1.5.55)
Numerica
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The spectral polynomial Galerkin approach presented in Sect. 1.5.1.1 relies on global polynomials.
Now let us examine the use of piecewise polynomials.
Preliminaries: piecewise polynomials have to be defined w.r.t. partitioning of the domainΩ ⊂ R
➣ Ω = [a,b] equipped with nodes (M ∈ N)
X := {a = x 0 < x 1 < ··· < x M−1 < x M = b} .
➤
mesh/grid
M := {]x j−1 ,x j [: 1 ≤ j ≤ M} .
a
x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10x 11x 12x 13x 14
Special case:
equidistant mesh:
x j := a+jh , h := b−a
M .
b
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Remark 1.5.56 (Benefit of variational formulation of BVPs).
The possibility of using simple piecewise linear trial and test functions is a clear benefit of the variational
formulation that can accommodate merely piecewise continuously differentiable functions, see
Sect. 1.3.2.
Below, in Sect. 1.5.2 we will learn about a method that targets the strong form of the 2-point BVP and,
thus, has to impose more regularity on the trial functions.
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☞ [x j−1 ,x j ],j = 1,...,M, ˆ= cells ofM, cell size h j := |x j −x j−1 |,j = 1,...,M
meshwidth h M := max|x j −x j−1 |
j
Recall from Sect. 1.3.2:
merely continuous, piecewise C 1 trial and test functions provide valid trial/test
functions!
1
a
x 1 x 2 x 3 ···
⇑ function∈ S 0 1,0 (M)
Choice of (ordered) basisB N ofV N ?
1D “tent functions”
Fig. 25
B = {b 1 N ,...,bM−1 N } , (1.5.53) 1
{
b j 1 , ifi = j ,
N (x i) = δ ij :=
(1.5.54)
0 , ifi ≠ j ,
b
✄
Simplest choice for test space
➣
V N = S1,0 0 {
(M)
v ∈ C
:=
0 }
([0,1]): v |[xi−1 ,x i ] linear,
i = 1,...,M,v(a) = v(b) = 0
a
N := dimV N = M −1
x 1 x 2 x 3 ···
Fig. 26
b
1.5
p. 109
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1.5
p. 110
➊ simplest case: linear variational problem with constant stiffness coefficient
u ∈ C 1 0,pw ([a,b]):
∫b
a
du
dx (x)dv
∫b
dx (x)dx =
Discrete variational problem withu N = µ 1 b 1 N +···+µ Nb N N :
∫b
a
N∑
l=1
⎛
N∑
∫b
⎜
⎝
l=1 a
µ l
db l N
dx (x)dbk N
db l N
dx (x)dbk N
A⃗µ = ⃗ϕ with
∫b
dx (x)dx =
A linear system of equations, cf. Rem. 1.5.40!
a
g(x)v(x)dx ∀v ∈ C 1 0,pw ([a,b]) .
g(x)b k N (x)dx k = 1,...,N .
a
⇕
⎞
∫b
dx (x)dx ⎟
⎠µ l = g(x)b k N (x)dx ,k = 1,...,N .
a
} {{ }
=:ϕ k
⇕
∫b
db l (A) kl := N
dx (x)dbk N(x)dx, k,l = 1,...,N ,
dx
a
⃗µ = (µ l ) N l=1 ∈ RN , ⃗ϕ = (ϕ k ) N k=1 ∈ RN .
1.5
p. 11
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p. 11

∫b
✄ system matrix A = (a ij ) ∈ R M−1,M−1 , a ij :=
∫b
✄ r.h.s. vector ⃗ϕ ∈ R M−1 , ϕ k :=
The detailed computations start with the evident fact that
db i N
dx (x) dbj N
dx (x)dx,
a
piecewise derivatives
a
1 ≤ i,j ≤ N
g(x)b k N (x)dx, k = 1,...,N .
|i−j| ≥ 2 ⇒ bj N
dx (x)· bi N(x) = 0 ∀x ∈ [a,b] ,
dx
because there is no overlap of the supports of the two basis functions.
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Natural choice: piecewise polynomial trial/test spaces ←→ composite quadrature rule
∫1
e.g, composite trapezoidal rule: ϕ k =
0
g(x)b k N (x)dx ≈ 1 2 (h k +h k+1 )g(x k ), 1 ≤ k ≤ N .
(1.5.59)
For equidistant mesh with uniform cell sizeh > 0 we arrive at the linear system of equations:
⎛
⎞
2 −1 0 0 ⎛ ⎞ ⎛ ⎞
−1 2 −1
µ 1 g(x 1 )
1
0 . .. . .. ...
h
⎜ .
⎟
.
⎜
.. ... ... 0
= h ⎜ .
⎟ . ⎝ ⎠ ⎝ ⎠ (1.5.60)
⎟
⎝ −1 2 −1⎠
µ N g(x N )
0 0 −1 2
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Definition 1.5.57 (Support of a function).
The support of a functionf : Ω ↦→ R is defined as
supp(f) := {x ∈ Ω: f(x) ≠ 0} .
Exercise 1.5.1. Show that the 1D linear finite element Galerkin solution of −u ′′ = f agrees with the
exact solution at the grid points, provided that all integrals are evaluated exactly [?, Sect. 2].
In addition, we use that the gradients of the tent functions are piecewise constant, see (1.5.55).
➡
∫1
0
0 , if|i−j| ≥ 2 →
⎧⎪
db j ⎨ −
N
dx (x)dbi N
dx (x)dx = h 1 , ifj = i+1 →
i+1
−h 1 , ifj = i−1 →
i
⎪ ⎩
1
h + 1
i h
, if1 ≤ i = j ≤ M −1 →
i+1
A symmetric, positive definite and tridiagonal:
⎛
1
h + 1 h 1 − 1 ⎞
2 h 0 0
2 − 1 1
h 2 h2
+ h 1 − 1 3 h 3
0 . .. ... ...
A =
... 0
⎜
... ... − 1
⎝
h ⎟
M−1
0 0 −h 1 1
M−1 h + 1 ⎠
M−1 h M
✍ notation: h j := |x j −x j−1 | local meshwidth, cell size
Mandatory:
computation of right hand side vector by numerical quadrature
1
0 1
1 b i N b j N
0 1
1 b j N b i N
0 1
1
b j N
0 1
(1.5.58)
1.5
p. 113
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p. 114
➋ case: linear variational problem with variable stiffness, cf. (1.4.16)
u ∈ C 1 0,pw ([a,b]):
∫b
a
σ(x) du
dx (x)dv
∫b
dx (x)dx =
Discrete variational problem withu N = µ 1 b 1 N +···+µ Nb N N :
Here:
∫b N∑
σ(x)
a l=1
µ l
db l N
dx (x)dbk N
∫b
dx (x)dx =
a
a
g(x)v(x)dx ∀v ∈ C 1 0,pw ([a,b]) .
g(x)b k N (x)dx k = 1,...,N . (1.5.61)
numerical quadrature required for both integrals
Choice: composite midpoint rule for left hand side integral → [18, Sect. 10.3]
∫ b M∑
f(x)dx ≈ h j f(m j ) , m j := 2 1(x j +x j−1 ) . (1.5.62)
a
j=1
composite trapezoidal rule [18, Eq. 10.3.3] for right hand side integral, see (1.5.59).
1.5
p. 11
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p. 11

Assumption:
σ ∈ C 0 pw([a,b]) with jumps only at grid nodesx j
Numerical
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Benefits of this choice of offset function:
Numerica
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•u 0 is a simple function (since p.w. linear),
(1.5.61)
⇕
⎛
⎞
N∑ M∑
⎝ h j σ(m j ) dbl N
dx (m j) dbk N
dx (m j) ⎠µ l = 2 1(h k+1 +h k )g(x k ) , k = 1,...,N ,
} {{ }
l=1 j=1
} {{ } =:ϕ k
=(A) k,l
⇕
A⃗µ = ⃗ϕ .
Resulting linear system of equations equidistant mesh with uniform cell sizeh > 0
⎛
⎞
σ 1 +σ 2 −σ 2 0 0 ⎛ ⎞
−σ 2 σ 2 +σ 3 −σ 3
µ 1
1
0 ... ... ...
h
⎜ .
⎟
.
⎜
.. . .. ... 0
⎝ ⎠
⎟
⎝
−σ M−2 σ M−2 +σ M−1 −σ M−1
⎠ µ N
0 0 −σ M−1 σ M−1 +σ M
⎛ ⎞
g(x 1 )
= h
⎜ .
⎟ , ⎝ ⎠ (1.5.63)
g(x N )
with
σ j = σ(m j ),j = 1,...,m.
Remark 1.5.64 (Offset function for finite element Galerkin discretization).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.5
p. 117
Numerical
Methods
for PDEs
•u 0 is locally supported: contributions from u 0 will alter only first and last component of right hand
side vector.
Example 1.5.66 (Linear finite element Galerkin discretization for elastic string model).
Targetted: non-linear variational equation on domainΩ = [0,1]
∫1 (
κ(ξ)
1− L )
L ‖u ′ u ′ (ξ)·v ′ (ξ)−f(ξ)·v(ξ)dξ = 0 ∀v ∈ (C
(ξ)‖
0,pw 1 ([0,1]))2 . (1.3.7)
0
Dataκ,f given in procedural form, see Rem. 1.5.3.
trial spaceV N,0 = (S 0 1,0 (M))2 on equidistant meshM, meshwidthh := 1 M .
Basis:
1D tent functions, lexikographic ordering
{( ) ( ) ( ),( ) ( ) ( )}
b 1
B = N0 b 2
, N0 b
M−1
,..., N 0 0 0
0 b 1 ,
N b 2 ,...,
N bN
M−1 .
Evaluation of right hand side by composite trapezoidal rule (1.5.59).
Evaluation left hand side by composite midpoint rule (1.5.62).
△
R. Hiptm
C. Schwa
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DRAFT
1.5
p. 11
Numerica
Methods
for PDEs
In the case of general boundary conditions
1
u(a) = u a , u(b) = u b u a
use piecewise linear offset function
a x 1 x 2 x 3 ···
⎧
⎪⎨ u a (1− x−a
h ) , ifa ≤ x ≤ x 1 1 ,
u 0 (x) = u b (1− b−x
h ) , ifx M M−1 ≤ x ≤ b ,
⎪⎩ 0 elsewhere.
b
u b
Fig. 27
(1.5.65)
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
A. Chernov
DRAFT
1.5
p. 118
Preliminary consideration:
( )
b 1
u N := u 0 +µ N0 1
the derivative of
+···+µ M−1
(
b
M−1
N
0
)+µ M
(
0
b 1 N
with offset function according to Rem. 1.5.64 is piecewise constant onM:
)
( )
0
+···+µ 2M−2
bN
M−1
in]x j−1 ,x j [: s j (⃗µ) := u ′ N (ξ) = u(x j)−u(x j−1 )
⎧(
h )
µ j −µ j−1
= 1 ⎪⎨
, if2 ≤ j ≤ M −1 ,
h ·
µ j+M−1 −µ
( j+M−2
µ1
)
µ −u(0) , ifj = 1 ,
M
⎪⎩ u(1)− ( µ ) M−1
µ , ifj = M .
2M−2
(1.5.67)
(1.5.68)
R. Hiptm
C. Schwa
H.
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Gradinar
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DRAFT
1.5
p. 12

Set: r j = r j (⃗µ) := h κ(m (
j)
1−
L
)
L
∥
∥s j (⃗µ) ∥ Single row non-linear system of equations arising from Galerkin finite element discretization:
row1: (r 1 +r 2 )µ 1 −r 2 µ 2 = hf 1 (h)+r 1 a , (1.5.69)
rowj: −r j µ j +(r j +r j+1 )µ j+1 −r j+1 µ j+2 = f 1 (jh) , 2 ≤ j < M −1 , (1.5.70)
rowM −1: −r M−1 µ M−2 +(r M−1 +r M )µ M−1 = hf 1 ((M −1)h)+r M b , (1.5.71)
rowM: (r 1 +r 2 (⃗µ))µ M −r 2 µ M+1 = hf 2 (h)+r 1 u a , (1.5.72)
rowj: −r j µ j+M−1 +(r j +r j+1 )µ j+M −r j+1 µ j+M+1 = f 2 (jh) , 2 ≤ j < M −1 ,
(1.5.73)
rowM −1: −r M−1 µ 2M−3 +(r M−1 +r M )µ 2M−2 = hf 2 ((M −1)h)+r M u b . (1.5.74)
Here the dependencer j = r j (⃗µ) has been suppressed to simplify the notation.
Please study the derivation of (1.5.63) in order to understand how (1.5.69)-(1.5.74) arise.
These equations can be written in a more compact form:
with
(1.5.69)-(1.5.74) ⇔
( ) ( )
R(⃗µ) 0 ⃗ϕ1
⃗µ =
0 R(⃗µ) ⃗ϕ 2
. (1.5.75)
⎛
⎞
r 1 +r 2 −r 2 0 0
−r 2 r 2 +r 3 −r 3
0 ... ... ...
R(⃗µ) :=
∈ R M−1,M−1 ,
.
⎜
.. . .. . .. 0
⎟
⎝
−r M−2 r M−2 +r M−1 −r M−1
⎠
0 0 −r M−1 r M−1 +r M
(⃗ϕ 1 ) j := hf 1 (hj) , (⃗ϕ 2 ) j := hf 2 (hj) , j = 1,...,M −1 .
Iterative solution of (1.5.75) by fixed point iteration, see Ex. 1.5.47
Initial guess ⃗µ (0) ∈ R N ; k = 0;
repeat
k ← k +1;
Solve the linear system of equations
(
R(⃗µ (k−1) ) ( )
) 0
0 R(⃗µ (k−1) ⃗µ (k) ⃗ϕ1
= ;
) ⃗ϕ 2
Numerical
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R. Hiptmair
C. Schwab,
H.
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DRAFT
1.5
p. 121
Numerical
Methods
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R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
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DRAFT
1.5
p. 122
until
∥
∥⃗µ (k) − ⃗µ (k−1)∥ ∥ ∥ ≤ tol·∥ ∥∥⃗µ (k) ∥ ∥ ∥
Code 1.5.76: Linear finite element discretization of elastic string variational problem
1 f u n c t i o n [vu,Jrec,figsol,figerg] =
stringlinfem(kappa,f,L,u0,u1,M,tol)
2 % Solving the non-linear variational problem (1.3.7) for the elastic string by
means of piecewise
3 % linear finite elements on an equidistant mesh with M −1 interior nodes.
4 % kappa, f are handles of type @(xi) providing the coefficient function
5 % κ and the force field f. u0 and u1 pass the pinning points.
6 % M is the number of mesh cells, tol specifies the tolerance for the fixed
point
7 % iteration. return value: 2×(M +1)-matrix of node positions
8 i f (nargin < 7), tol = 1E-2; end
9 h = 1/M; % meshwidth
10 phi = h*f(h*(1:M-1)); % Right hand side vector
11
12 % Initial guess: straight string, condition L > ‖u(0)−u(1)‖.
13 i f (L >= norm(u1-u0)), e r r o r (’String must be tense’); end
14 vu_new = u0*(1-(0:1/M:1))+u1*(0:1/M:1);
15 % Meaning of components of vu:
µ M ,...,µ 2M−2 .
vu(1,2:M) ↔ µ 1 ,...,mu M−1 , vu(2,2:M) ↔
16 figsol = f i g u r e ; Jrec = []; hold on;
17 f o r k=1:100 % loop for fixed point iteration, maximum 100 iterations
18 vu = vu_new;
19 % Plot shape of string
20 p l o t (vu(1,:),vu(2,:),’--g’); drawnow;
21 t i t l e ( s p r i n t f (’M = %d, iteration #%d’,M,k));
22 x l a b e l (’{\bf x_1}’); y l a b e l (’{\bf x_2}’);
23 %Compute the cell values s j , r j , j = 1,...,M, see (1.5.68).
24 d = (vu(:,2:end) - vu(:,1:end-1))/h;
25 s = s q r t (d(1,:).^2 + d(2,:).^2);
26 r = kappa(h*((1:M)-0.5)).*(1/L - 1./s)/h;
27 % Compute total potential energy
28 Jel = h/(2*L)*kappa(h*((1:M)-0.5))*((s-L).^2)’;
29 Jf = - (phi(1,:)*vu(1,2:M)’+phi(2,:)*vu(2,2:M)’);
30 Jrec = [Jrec; k , Jel, Jf, Jel+Jf];
31 % Assemble triadiagonal matrix R = R(⃗µ)
32 R = g a l l e r y (’tridiag’,-r(2:M-1),r(1:M-1)+r(2:M),-r(2:M-1));
33 % modify right hand side in order to take into account pinning conditions
34 phi1 = phi(1,:); phi1(1) = phi1(1) + r(1)*u0(1); phi1(M-1) =
phi1(M-1) + r(M)*u1(1);
35 phi2 = phi(2,:); phi2(1) = phi2(1) + r(1)*u0(2); phi2(M-1) =
phi2(M-1) + r(M)*u1(2);
36 % Solve linear system and compute new iterate
37 vu_new = [u0,[(R\phi1’)’;(R\phi2’)’],u1];
38 % Check simple termination criterion for fixed point iteration.
39 i f (norm(vu_new - vu,’fro’) < tol*norm(vu_new,’fro’)/M)
40 p l o t (vu(1,:),vu(2,:),’r-*’); break; end
41 end
Numerica
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H.
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V.
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DRAFT
1.5
p. 12
Numerica
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for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
1.5
p. 12

42 % Plot of total potential energy in the course of the iteration
43 figerg = f i g u r e (’name’,’total potential energy’);
44 t i t l e ( s p r i n t f (’elastic string, M = %d’,M));
45 p l o t (Jrec(:,1),Jrec(:,4),’m-*’,...
46 Jrec(:,1),Jrec(:,2),’b-+’,...
47 Jrec(:,1),Jrec(:,3),’g-^’);
48 x l a b e l (’{\bf no. of iteration step}’); y l a b e l (’{\bf energy}’);
49 legend(’total potential energy’,’elastic energy’,’energy in force
field’,’location’,’east’);
Numerical
Methods
for PDEs
x 2
0.2
0.15
0.1
0.05
0
−0.05
−0.1
M = 20, iteration #23
energy
0.5
0.4
0.3
0.2
0.1
0
total potential energy
elastic energy
energy in force field
Numerica
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for PDEs
✸
R. Hiptmair
C. Schwab,
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−0.15
−0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 1
Fig. 29
−0.1
−0.2
0 5 10 15 20 25
no. of iteration step
Fig. 30 ✸
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C. Schwa
H.
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V.
Gradinar
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Example 1.5.77 (Elastic string shape by finite element discretization).
DRAFT
DRAFT
Linear finite element discretization of (1.3.7), see Ex. 1.5.66, Code 1.5.75.
κ ≡ 1,L = 0.5,u(0) = ( 0) (
0 ,u(1) =
1 )
0.2
gravitational force fieldf(ξ) = − ( 0
2
)
.
1.5.2 Collocation
1.5
p. 125
OPTIONAL Section: will only be references in Sect. 1.6 below, Ex. 1.6.19, but examples involving
collocation methods can safely be skipped.
1.5
p. 12
0.2
0.15
M=5
M=10
M=20
Numerical
Methods
for PDEs
Targetted: Two-point BVP = ODEL(u) = f + boundary conditions
Numerica
Methods
for PDEs
Piecewise linear finite element solution of (1.3.7),
equidistant meshes withM cells,M = 5,10,20✄
Convergence of fixed point iteration (M = 20):
x 2
0.1
0.05
0
−0.05
−0.1
−0.15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 1
Fig. 28
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C. Schwab,
H.
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V.
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DRAFT
Note: In contrast to the Galerkin approach, collocation techniques do not tackle the weak form of a
boundary value problem, but rather the “classical”/strong form.
Idea: ➊
➋
seek solution in finite-dimensional trial spaceV N,0 ,N := dimV N,0 < ∞
pick collocation nodesN := {x 1 ,...,x N } ⊂ Ω such that x
{
VN,0 ↦→ R N
“point evaluation”
is a bijective linear mapping.
v ↦→ ( v(x j ) ) N
j=1
(1.5.78)
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DRAFT
Collocation conditions: u N ∈ V N : L(u N )(x j ) = f(x j ) , j = 1,...,N .
(1.5.79)
1.5
p. 126
1.5
p. 12

➌ choose ordered basis B = {b 1 N ,...,bN N } ofV N,0 & plug basis representation
u N = u 0 +µ 1 b 1 N +···+µ Nb N N (u 0ˆ= offset function, cf. Rem. 1.5.10)
Numerical
Methods
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Remark 1.5.83 (Collocation: smoothness requirements for coefficients).
Numerica
Methods
for PDEs
into collocation conditions (1.5.79)
⃗µ = (µ l ) N l=1 : L(u 0+µ 1 b 1 N +···+µ Nb N N )(x j) = f(x j ) , j = 1,...,N . (1.5.80)
For 2-point BVP (1.5.81): σ must be differentiable in collocation collocation nodes, with known values
dσ
dx (x j ),j = 1,...,N, in the sense of Rem. 1.5.3: extra difficulty whenσ given in procedural form.
△
In general: (1.5.80) is a non-linear system of equation (N equations for N unknownsµ 1 ,...,µ N ).
Note: bijectivity of point evaluation (1.5.78) ⇒ ♯{nodes} = dimV N,0
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DRAFT
1.5.2.1 Spectral collocation
Focus: linear two point boundary value problem (1.5.81)
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DRAFT
trial space for polynomial spectral collocation:
V N,0 = P p (R)∩C0 2 ([a,b]), p ≥ 2 . (1.5.84)
1.5
p. 129
= polynomials of degree≤ p, vanishing at endpoints of domain, N := dimV N,0 = p−1 .
➣ same trial space as for polynomial spectral Galerkin approach, see Sect. 1.5.1.1.
1.5
p. 13
Below: detailed discussion for linear two point boundary value problem
L(u) := − d (
σ(x) du )
dx dx (x) = g(x) , a ≤ x ≤ b ,
u(a) = u a , u(b) = u b ,
on domainΩ = [a,b], related to variational problem (1.4.16).
(1.5.81)
Numerical
Methods
for PDEs
Discussion: polynomial spectral collocation for two-point BVP (1.5.81)
offset function
u 0 (x) := b−x
b−a u a+ x−a
b−a u b .
Numerica
Methods
for PDEs
Basis B := {b j N := M j} consisting of integrated Legendre polynomials, see (1.5.24).
Remark 1.5.82 (Smoothness requirements for collocation trial space).
For two-point BVP (1.5.81) consider space V N,0 := S1,0 0 (M) of M-piecewise linear finite element
functions. → Sect. 1.5.1.2
Note: v N ∈ S 0 1,0 (M) is not differentiable in nodesx j of the mesh.
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DRAFT
Note:
Terminology:
L from (1.5.81) is a linear differential operator!
A differential operator is a mapping on a function space involving only values of the
function argument and some of its derivatives in the same point.
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Natural choice collocation points = nodes of the mesh is not possible!
(because for v N ∈ S1,0 0 (M) the functionL(v N) is discontinuous in the nodes of the mesh)
A differential operatorLis linear, if
Assuming σ ∈ C 1 ([a,b]) global continuity of L(v N ) entails V N,0 ⊂ C 2 ([a,b]), cf.
Sect. 1.5.2.2.
△
1.5
p. 130
L(αu+βv) = αL(u)+βL(v) ∀α,β ∈ R, ∀functionsu,v (1.5.85)
1.5
p. 13

(1.5.80)
(1.5.85)
=⇒
A⃗µ = ⃗ϕ ,
N∑
L(b l N )(x k)µ l = f(x k )−L(u 0 )(x k ) , k = 1,...,N . (1.5.86)
l=1
⇕
(A) k,l := L(b l N )(x k) , k,l ∈ {1,...,N} ,
ϕ k := f(x k )−L(u 0 )(x k ) , k ∈ {1,...,N} .
AnN ×N linear system of equations
For BVPs featuring linear differential operators, collocation invariably leads to a linear system
of equations for the unknown coefficients of the basis representation of the collocation solution.
Remark 1.5.88 (Bases for polynomial polynomial spectral collocation).
Same choices as for spectral Galerkin methods, see Rem. 1.5.22.
(1.5.87)
△
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R. Hiptmair
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DRAFT
13 i f (nargout < 3)
14 D = [zeros( s i z e (x)); ones( s i z e (x))];
15 f o r j=1:n-1, D = [D;(2*j+1)*V(j+1,:)+D(j,:)]; end
16 end
Code 1.5.92: Spectral collocation for linear 2nd-order two-point BVP
1 f u n c t i o n u = linspeccol(g,N,x)
2 % Polynomial spectral collocation discretization of linear 2nd-order two-point
BVP
3 % − d2 u
dx 2 = g(x), u(0) = u(1) = 0
4 % on Ω = [0,1]. Trial space of dimension N, collocation in Chebychev nodes.
5 % Values of approximate solution in points x j are returned in the row vector u
6 cn = cos((2*(1:N)-1)* p i /(2*N)); % Chebychev nodes, see (1.5.90)
7 [V,M,D] = dilegpol(N+1,cn); % Obtain values of (2nd
derivatives) of M m
8 mu = (-4*D(2:N+1,:))’\(g(0.5*(cn+1))’); % Solve collocation system
9 % Compute values of integrated Legendre polynomials at output points
10 [V,M] = dilegpol(N+1,2*x-1); u = mu’*M;
Example 1.5.93 (Polynomial spectral collocation for 2-point BVP).
Numerica
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C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
Remark 1.5.89 (Collocation points for polynomial spectral collocation).
Rule of thumb (without further explanation, see [16]):
choose collocation points x j , j = 1,...,N such that the induced Lagrangian interpolation
operator (→ [18, Thm. 3.3.8]) has a small∞-norm, see [18, Lemma 3.3.17].
1.5
p. 133
Numerical
Methods
for PDEs
Setting of Ex. 1.5.20, spectral polynomial collocation, on ,N = 5,7,10, basis from integrated Legendre
polynomials, plot of solutionu N .
1.5
p. 13
Numerica
Methods
for PDEs
Popular choice (due to [18, Eq. 9.2.15]): Chebychev nodes
( )
x k := a+ 2 1(b−a) (2k −1
cos
2N π) +1 , k = 1,...,N . (1.5.90)
1.5
1
2.5
2
Spectral collocation: equidistant nodes
△
0.5
1.5
1
Code 1.5.91: Computation of derivatives of Legendre polynomials using (1.5.30)
1 f u n c t i o n [V,M,D] = dilegpol(n,x)
2 % Computes values of the first n+1 Legendre polynomials (returned in matrix V)
3 % the first n−1 integrated Legendre polynomials (returned in matrix M), and
the
4 % first n+1 first derivatives of Legendre polyomials in the points x j passed
5 % in the row vector x.
6 % Uses the recursion formulas (1.5.28) and (1.5.24)
7 V = ones( s i z e(x)); V = [V; x];
8 % recursion (1.5.28) for Legendre polynomials
9 f o r j=1:n-1, V = [V; ((2*j+1)/(j+1)).*x.*V(end,:) -
j/(j+1)*V(end-1,:)]; end
10 % Formula (1.5.24) for integrated Legendre polynomials
11 M = diag(1./(2*(1:n-1)+1))*(V(3:n+1,:) - V(1:n-1,:));
12 % Recursion formula (1.5.30) for derivatives of Legendre polynomials
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DRAFT
1.5
p. 134
u
0
−0.5
−1
N=4
N=5
N=6
u(x)
−1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Fig. 31
Collocation in Chebychev nodes
u
0.5
0
−0.5
−1
N=4
N=5
N=6
u(x)
−1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Fig. 32
Collocation in equidistant nodes
✸
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DRAFT
1.5
p. 13

1.5.2.2 Spline collocation
Numerical
Methods
for PDEs
Example 1.5.95 (Cubic spline collocation discretization of 2-point BVP).
1.5
Numerica
Methods
for PDEs
1
Analoguous to Sect. 1.5.1.2:
now collocation based on piecewise polynomials
Setting of Ex. 1.5.20
0.5
Rem. 1.5.82 ➣ for BVP (1.5.81) smoothnessV N,0 ⊂ C 2 ([a,b]) is required.
Which piecewise polynomial spaces offer this kind of smoothness ?
Recall [18, Def. 3.7.1], cf. [18, Sect. 3.7.1]:
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C. Schwab,
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DRAFT
Cubic spline collocation with equidistant nodes,
M = 5,7,12
Solutionu N ✄
u
0
−0.5
−1
exact
M = 5
M = 7
M = 12
−1.5
0 0.2 0.4 0.6 0.8 1
x
Fig. 33
✸
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C. Schwa
H.
Harbrech
V.
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DRAFT
1.5.3 Finite differences
Definition 1.5.94 (Cubic spline).
s :]a,b[↦→ R is a cubic spline function w.r.t. the node set T := {a = x 0 < x 1 < x 2 < ... <
x M−1 < x M = b},if
(i) s ∈ C 2 ([a,b]) (twice continuously differentiable),
(ii) s |]xj−1 ,x j [ ∈ P 3(R) (piecewise cubic polynomial)
✎ notation: S 3,T ˆ= vector space of cubic splines on node setX
Known: dimS 3,T = ♯T +2 = M +3
Trial space for collocation for 2-point BVP (1.5.81)
{
natural cubic splines: V N,0 := s ∈ S 3,T : s′′ (a) = s ′′ }
(b) = 0 ,
s(a) = s(b) = 0
Choice of collocation nodes:
⇒ dimN := V N = M −1 ,
1.5
p. 137
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Focus:
2nd-order linear two-point BVP
L(u) := − d (
σ(x) du )
dx dx (x) = g(x) , a ≤ x ≤ b , (1.5.81)
u(a) = u a , u(b) = u b ,
Idea:
E.g.
Replace
Setting as in Sect. 1.5.1.2:
➣ Ω = [a,b] equipped with nodes (M ∈ N)
X := {a = x 0 < x 1 < ··· < x M−1 < x M = b} .
➤
mesh/grid
M := {]x j−1 ,x j [: 1 ≤ j ≤ M} .
derivatives−→ difference quotients
(in finitely many special points = nodes of a mesh)
d 2 u u(x+h)−2u(x)+u(x−h)
dx2(x) ≈
h 2 , h > 0 “small” . (1.5.96)
a
x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10x 11x 12x 13x 14
Special case:
equidistant mesh:
x j := a+jh , h := b−a
M .
b
1.5
p. 13
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
collocation nodes for cubic spline collocation = spline nodesx j :
N = T
1.5
p. 138
☞ [x j−1 ,x j ],j = 1,...,M, ˆ= cells ofM, cell size h j := |x j −x j−1 ,j = 1,...,M
meshwidth h M := max|x j −x j−1 |
j
1.5
p. 14

➊ replacement of outer derivative (x j−1/2 = 1 2 (x j +x j−1 )):
(
d
σ(x) du )
dx dx (x)
|x=x j
≈
(
2
σ(x
h j−1 +h j+1/2 ) du
)
j dx (x j+1/2 )−σ(x j−1/2 )du dx (x j−1/2 ) .
Numerical
Methods
for PDEs
(Up to scaling with h) the finite difference approach and the linear finite element Galerkin
scheme (→ Sect. 1.5.1.2) yield the same system matrix for the BVP (1.5.81) and its associated
variational problem (1.4.16), cf. (1.5.100) and (1.5.63).
Numerica
Methods
for PDEs
➋
replacement of inner derivative, e.g.,
du
dx (x j+1/2 ) ≈ u(x j+1)−u(x j )
.
h j
1.6 Convergence
− d (
σ(x) du )
dx dx (x) =
|x=x j
σ(x j−1/2 ) u(x j)−u(x j−1 )
−σ(x
h j+1/2 ) u(x j+1)−u(x j )
j−1 h j
1
2 (h . (1.5.97)
j−1+h j )
On equidistant mesh, uniform meshwidthh j = h > 0,j = 1,...,M:
− d (
σ(x) du )
dx dx (x) |x=x j
= 1 )
(−σ(x
h 2 j+1/2 )u(x j+1 )+(σ(x j+1/2 )+σ(x j−1/2 ))u(x j )−σ(x j−1/2 )u(x j−1 ) . (1.5.98)
Unknowns in finite difference method: µ l = u(x l ), l = 1,...,M −1
− d (
σ(x) du )
dx dx (x) = g(x) ,a ≤ x ≤ b .
← restriction toX , use (1.5.98)
−σ(x j+1/2 )µ j+1 +(σ(x j+1/2 )+σ(x j−1/2 ))µ j −σ(x j−1/2 )µ j−1
h 2 = g(x j ) , j = 1,...,M −1 .
A⃗µ = ⃗ϕ , with
⇕
⎧
0 , if|j −l| > 1 ,
(A) jl = h −2·
⎪⎨ −σ(x j+1/2 ) , ifj = l−1 ,
σ(x j−1/2 )+σ(x j−1/2 ) , ifj = l ,
⎪⎩
−σ(x l+1/2 ) , ifl = j −1 .
⎧
⎪⎨ g(x 1 )+σ(x 1/2 )u a , ifj = 1 ,
ϕ j = g(x j ) , if1 < j < M −1 ,
⎪⎩ g(x M−1 )+σ(x M−1/2 )u b , ifj = M −1 .
(1.5.99)
(1.5.100)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.5
p. 141
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
For elastic string model (1.2.17)/(1.3.7), taut string model in graph description (1.4.16) with exact
solutionu : [0,1] ↦→ R 2 oru : [a,b] ↦→ R, respectively:
Discretization schemes
(Galerkin approach, Sect. 1.5.1
collocation methods, Sect. 1.5.2)
−→
Approximate solution
u N : [0,1] ↦→ R 2 /u N : [a,b] ↦→ R
(functions∈ V N )
Desirable: approximationu N “close to” exact solutionu: rigorous meaning ?
↕
Remark 1.6.1 (Grid functions).
Note:
How to measure discretization erroru−u N ?
for finite differences (→ Sect. 1.5.3) we get no solution function, only grid function X ↦→ R
(“point values”)
☞
Remark 1.6.2.
reconstruction of a function through postprocessing, e.g., linear interpolation
We encountered the issues of convergence of approximate solutions before:
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.6
p. 14
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
An (M −1)×(M −1) linear system of equations
Numerical quadrature [18, Ch. 10]: study of asymptotic behavior of quadrature error
Numerical integration [18, Ch. 12]: discretization error of single step methods
1.5
p. 142
△
1.6
p. 14

1.6.1 Norms on function spaces
Numerical
Methods
for PDEs
Definition 1.6.6 (Mean square norm/L 2 -norm). → Def. 2.2.5
For a functionu ∈ (C 0 pw(Ω)) n the mean square norm/L 2 -norm is given by
Numerica
Methods
for PDEs
Tools for measuring discretization errors:
norms on function spaces/grid function spaces
‖u‖ 0
(‖u‖ L 2 (Ω)
) (∫ 1/2
:= ‖u(x)‖ dx) 2 , u ∈ (L 2 (Ω)) n .
Ω
Reminder → [18, Sect. 2.5.1]
Definition 1.6.3 (Norm).
A norm‖·‖ V on an R-vector spaceV is a mapping‖·‖ V : V ↦→ R + 0 , such that
(definiteness) ‖v‖ V = 0 ⇐⇒ v = 0 ∀v ∈ V (N1)
(homogeneity) ‖λv‖ V = |λ|‖v‖ V ∀λ ∈ R, ∀v ∈ V , (N2)
(triangle inequality) ‖w +v‖ V ≤ ‖w‖ V +‖v‖ V ∀w,v ∈ V . (N3)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
L 2 (Ω) designates the vector space of square integrable functions, another L p -space (for p = 2)
and a Hilbert space.
The “0” in the notation‖·‖ 0 refers to the absense of derivatives in the definition of the norm.
Obviously, theL 2 -norm is weaker than the supremum norm:
‖v‖ L 2 ([a,b]) ≤ √|b−a|‖v‖ L ∞ ([a,b]) ∀v ∈ C 0 pw([a,b]) .
In particular, the L 2 -norm of the discretization error may be small despite large deviations of u N
fromu, provided that these deviations are very much localized.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Next: important norms on function spaces, cf. [18, Eq. 3.3.14], [18, Eq. 3.3.15], [18, Eq. 3.3.16]:
1.6
p. 145
Numerical
Methods
for PDEs
✗
✖
Remark 1.6.7 (Energy norm).
Relevant error norms suggested by application context/physics!
1.6
p. 14
✔
Numerica
Methods
✕for PDEs
Definition 1.6.4 (Supremum norm).
The supremum norm of an (essentially) bounded functionu : Ω ↦→ R n is defined as
(
‖u‖ ∞ = ‖u‖L (Ω)) ∞ := sup ‖u(x)‖ , u ∈ (L ∞ (Ω)) n . (1.6.5)
x∈Ω
1.6
p. 146
We consider the model for a homogeneous taut string in physical space, see (1.4.16), with associated
total potential energy functional
∫b
R. Hiptmair J(u) :=
C. Schwab,
H.
Harbrecht
a
V.
Gradinaru where, for the sake of simplicity, we assumeu
A. Chernov
a = u b = 0.
L ∞ DRAFT
(Ω) denotes the vector space of essentially bounded functions. It is the instance forp = ∞ of
anL p A manifestly relevant error quantity of interest is the deviation of energies
-space.
E J := |J(u)−J(u N )| .
The notation ‖·‖ ∞ hints at the relationship between the supremum norm of functions and the
maximum norm for vectors in R n .
Forn = 1 the Euclidean vector norm in the definition reduces to the modulus|u(x)|.
The norm‖u−u N ‖ L ∞ (Ω) measures the maximum distance of the function values ofuandu N .
∣ ∣
1 ∣∣∣ du ∣∣∣ 2
2 dx (x) +ĝ(x)u(x)dx , u ∈ C0,pw 1 ([a,b]) , (1.6.8)
We adopt the concise notations introduced for abstract (linear) variational problems in
Rems. 1.3.12, 1.4.5:
∫b
du
a(u,v) :=
dx (x)dv dx (x)dx ,
J(u) = 1 2 a(u,u)−l(u) , a
∫b
l(v) := − ĝ(x)v(x)dx ,
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.6
p. 14

whereais a bilinear form, see Def. 1.3.13.
Assumption: u N ∈ V N,0 ˆ= Galerkin solution based on discrete trial spaceV N,0 ⊂V 0 .
a(u,v) = l(v) ∀v ∈ V 0 := C0,pw 1 ([a,b]) ,
a(u N ,v N ) = l(v N ) ∀v N ∈ V N,0 ⊂ V 0 .
We can use the defining variational equations foruandu N to express
(1.6.9)
Numerical
Methods
for PDEs
From (1.6.11)
☛
‖u−u N ‖ H 1 (Ω) ≤ ǫ
|J(u)−J(u N)| ≤ |u+u N | H 1 (Ω) |u−u N| H 1 (Ω) (1.6.14)
(N3)
≤ (2|u| H 1 (Ω) +ǫ)ǫ .
estimate of the energy norm of the discretization error paves the way for bounding the energy
deviation.
Numerica
Methods
for PDEs
J(u)−J(u N ) = − 1 2 (a(u,u)−a(u N,u N )) (∗) = − 1 2 a(u+u N,u−u N ) . (1.6.10)
(∗): a straightforward consequence of the bilinearity ofa, see Def. 1.3.13, c.f. a 2 −b 2 = (a+b)(a−b)
fora,b ∈ R.
Concretely,
∫ |J(u)−J(u N )| = 2
1 b d
∣ a dx (u+u d
N)·
dx (u−u N)dx
∣
(
(∗) ∫ b
≤ 1 2 | d
) 1/2 ( ∫ b
a dx (u+u N)| 2 dx | d
) 1/2
a dx (u−u N)| 2 dx .
(∗): due to Cauchy-Schwarz inquality (2.2.15)
(1.6.11)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.6
p. 149
Remark 1.6.15 (Norms on grid function spaces).
To measure the discretization error for finite difference schemes (→ Sect. 1.5.3) one may resort to
mesh dependent norms
(discrete)l 2 -norm : ‖⃗µ‖ 2 M l 2 (X) := ∑
1
2 (h j +h j+1 )|µ j | 2 , (1.6.16)
j=0
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.6
p. 15
Definition 1.6.12 (H 1 -seminorm). → Def. 2.2.12
For a functionu ∈ C 1 pw([a,b]) theH 1 -seminorm reads
|u| 2 H 1 ([a,b]) := ∫ b
a
| du
dx (x)|2 dx . (1.6.13)
Numerical
Methods
for PDEs
(discrete) maximum norm : ‖⃗µ‖ l ∞ (X)
(under convention h 0 := 0, h M+1 := 0) ,
:= max
j=0,...,M |µ j| . (1.6.17)
△
Numerica
Methods
for PDEs
|·| H 1 ([a,b]) is merely a semi-norm, because it only satisfies norm axioms (N2) and (N3), but fails
to be definite: |·| H 1 ([a,b]) = 0 for constant functions.
In the setting of the homogeneous taut string model, we have
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 1.6.18 (Approximate computation of norms).
Standard testing of implementations of numerical methods for 2-point BVP: Examine norm of
discretization error for test cases with (analytically) known exact solutionu.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
|u| 2 H 1 (iab) = a(u,u) |·| H 1 ([a,b]) is called the energy norm for the model.
More explanations in Sect. 2.1.3.
On C0,pw 1 ([a,b]) the semi-norm |·| H 1 (iab) is a genuine norm → Def. 1.6.3. See proof of
Thm. 2.2.16.
Even for numerical methods computingu N ∈ V N ⊂ V (Galerkin methods→Sect. 1.5.1, collocation
methods→Sect. 1.5.2):
usually exact computation of ‖u−u N ‖ is impossible/very difficult.
1.6
p. 150
Option: approximate evaluation of norm‖u−u N ‖
1.6
p. 15

supremum norm‖·‖ ∞ : approximation by sampling on discrete point set.
L 2 -norm, energy norm: numerical quadrature
(Gauss quadrature for spectral schemes, composite quadrature for mesh based schemes)
Numerical
Methods
for PDEs
0.35
0.3
0.25
maximum norm
L 2 norm
energy norm
10 −1
maximum norm
L 2 norm
energy norm
Numerica
Methods
for PDEs
! Eror intrroduced by approximation of norm must be smaller than discretization error
(➣ use “overkill” quadrature/sampling, cost does not matter much in testing).
△
error norms
0.2
0.15
0.1
error norms
10 −2
10 −3
10 −4
1.6.2 Algebraic and exponential convergence
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
0.05
0
0 50 100 150 200 250 300 350 400 450 500
M
Fig. 34
10 0 M
10 −5
10 0 10 1 10 2 10 3
Fig. 35
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✬
Crucial: convergence is an asymptotic notion !
✩
What is plotted are the discrete versions of theL 2 -norm and supremum norm, see Rem. 1.6.15.
The energy norm of the error was computed according to the formula
✫
sequence of discrete models ⇒ sequence of approximate solutions(u (i)
N ) i∈N
⇒ study sequence(‖u (i)
N −u‖) i∈N
✪
∣
M∑ ∣∣∣∣
energy norm(error) 2 µ j −µ j−1
:= h j
h
j=1 j
− du
dx (x j−1/2 ) ∣ ∣∣∣∣ 2
.
created by variation of a discretization parameter:
1.6
p. 153
❷ Spectral collocation, polynomial degreep ∈ N → Sect. 1.5.2.1
1.6
p. 15
Discretization parameters:
meshwidth h > 0 for finite differences (→ Sect. 1.5.3), p.w. linear finite elements
(→ Sect. 1.5.1.2), spline collocation (→ Sect. 1.5.2.2)
polynomial degree for spectral collocation (→ Sect. 1.5.2.1),
spectral Galerkin discretization (→ Sect. 1.5.1.1)
Numerical
Methods
for PDEs
Monitored:
supremum norm (1.6.5),L 2 -norm (1.6.6) of discretization erroru−u N (approximated
by “overkill” Gaussian quadrature with10 4 nodes)
Numerica
Methods
for PDEs
Example 1.6.19 (Numerical studies of convergence).
Spectral collocation, Chebychev points
L ∞ −norm
L 2 −norm
energy norm
10 0
Spectral collocation, Chebychev points
L ∞ −norm
L 2 −norm
energy norm
Focus: Linear 2-point boundary value problem − d2 u
dx2 = g(x), u(0) = u(1) = 0 on Ω =]0,1[,
variational form (1.5.21),
exact solutionu(x) = sin(2πx 2 ) (→ setting of Ex. 1.5.20)
❶ finite difference discretization on equidistant mesh, meshwidthh > 0 (→ Sect. 1.5.3)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
error norm
10 0 N
10 −2
10 −4
10 −6
error norm
10 1 N
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
10 −8
10 −7
10 −8
10 −10
10 1
Fig. 36
10 −9
3 4 5 6 7 8 9 10 11 12
Fig. 37
❸ Spline collocation on equidistant mesh, meshwidthh > 0 (→ Sect. 1.5.2.2)
1.6
p. 154
Monitored:
supremum norm (1.6.5),L 2 -norm (1.6.6) ofu−u N (approximated by sampling on fine
grid with10 4 points)
1.6
p. 15

0 50 100 150 200 250 300 350 400 450 500
0.8
0.7
0.6
maximum norm
L 2 norm
10 0
maximum norm
L 2 norm
O(M −2 )
Numerical
Methods
for PDEs
Unified view:
✗
✖
Study‖u−u N ‖ as function of number N of unknowns (degrees of freedom)
measure for costs incurred by method
✔
Numerica
Methods
for PDEs
✕
10 −1
0.5
error norms
0.4
0.3
0.2
error norms
10 −2
10 −3
Definition 1.6.20 (Convergence rate). → [18, Sect. 9.1], [18, Eq. 9.1.3]
‖u−u N ‖ = O(N −α ),α > 0 :⇐⇒ algebraic convergence with rateα
‖u−u N ‖ = O(exp(−γN δ )), withγ,δ > 0 :⇐⇒ exponential convergence
10 −4
0.1
0
M
Fig. 38
10 1 M
10 −5
10 0 10 1 10 2 10 3
❹ Spectral Galerkin based on degreep ∈ N polynomials → Sect. 1.5.1.1
Fig. 39
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✍
recall notation (Landau-O):
f(N) = O(g(N)) :⇔
∃N 0 > 0,∃C > 0 independent ofN
such that|f(N)| ≤ Cg(N) forN > N 0 .
(1.6.21)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Monitored:
supremum norm (1.6.5),L 2 -norm (1.6.6) of discretization erroru−u N (approximated
by trapezoidal rule on fine grid with10 4 points)
1.6
p. 157
1.6
p. 15
error norm
1
0.9
0.8
0.7
0.6
0.5
0.4
Spectral Galerkin error convergence
L ∞ −norm
L 2 −norm
error norm
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
Spectral Galerkin error convergence
L ∞ −norm
L 2 −norm
Numerical
Methods
for PDEs
Φ(N)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
−2
N −1
exp(−N −1/2
1/3 exp(−N 1/5 )
exp(−N 1/3 /10)
Φ(N)
0.05
0.04
0.03
0.02
−1
−2
N
exp(−N −1/2
1/3 exp(−N 1/5 )
exp(−N 1/3 /10)
Numerica
Methods
for PDEs
0.3
0.2
0.1
2 4 6 8 10 12 14 16 18 20
N
Fig. 40
10 −7
10 −8
10 −9
10 −10
10 0 N
2 4 6 8 10 12 14 16 18 20
Fig. 41
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
0.2
0.1
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
N
0.01
0
100 200 300 400 500 600 700
N
Linear plot of qualitative convergence behavior: algebraic/exponential convergence rates
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
DRAFT
Observation:
✄ ‘Empiric convergence” in all cases
✄ different qualitative behavior (of norm of discretization error)
Exponential convergence will always win (asymptotically)
✸
How to compare different discretizations ?
1.6
p. 158
1.6
p. 16

Φ(N)
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
N −2
N −1
N −1/2
10 −9
exp(−N 1/3 )
exp(−N 1/5 )
exp(−N 1/3 /10)
10 −10
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
N
Log-linear plot of decrease of discretization error for algebraic/exponential convergence rates
10 0
10 −1
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1.6
p. 161
Numerical
Methods
for PDEs
Remark 1.6.22 (Exploring convergence experimentally). → [18, Rem. 9.1.4]
How to determine qualitative asymptotic convergence from raw norms of discretization error?
Given: data tuples(N i ,ǫ i ),i = 1,2,3,..., N i ˆ= problem sizes,ǫ i ˆ= error norms
1. Conjecture: algebraic convergence: ǫ i ≈ CN −α
i
➤
log(ǫ i ) ≈ log(C)−αlogN i
(affine linear in log-log scale).
linear regression on data(logN i ,logǫ i ),i = 1,2,3,... to determine rateα.
2. Conjecture: exponential convergence: ǫ i ≈ Cexp(−γN δ i )
➤
logǫ i ≈ log(C)−γN δ i .
non-linear least squares fit (→ [18, Sect. 7.5]) to determineδ:
⎧ ⎫
⎨∑
⎬
(c,γ,δ) = argmin |logǫ
⎩ i −c+γNi δ |2 ⎭ ,
i
residual↔validity of conjecture. This can be done by a short MATLAB code (→ exercise)
Example 1.6.23 (Asymptotic nature of convergence).
1
△
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1.6
p. 16
Numerica
Methods
for PDEs
10 −2
10 −3
0.8
0.6
0.4
0.2
2-point BVP − d2 u
dx 2 = g(x), u(0) = u(1) = 0,
Ω =]0,1[,
✁ u(x) = sin(50πx 2 )
Φ(N)
10 −4
10 −5
10 −6
10 −7
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
u
0
−0.2
−0.4
−0.6
−0.8
−1
0 0.2 0.4 0.6 0.8 1
x
Fig. 42
❶
❷
finite difference discretization on equidistant
mesh, meshwidthh > 0 (→ Sect. 1.5.3)
Spectral Galerkin based on degree p ∈ N
polynomials → Sect. 1.5.1.1
Evaluations as in Ex. 1.6.19
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
10 −8
10 −9
10 −10
N −2
N −1
N −1/2
exp(−N 1/3 )
exp(−N 1/5 )
exp(−N 1/3 /10)
10 1 10 2 10 3 10 4
N
Log-log plot of decrease of discretization error for algebraic/exponential convergence rates
1.6
p. 162
1.6
p. 16

10 3
maximum norm
L 2 norm
10 2
Spectral Galerkin error convergence
L ∞ −norm
L 2 −norm
Numerical
Methods
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Observations:
• no more exponential convergence of spectral Galerkin
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10 2
10 0
• FD: different rate of algebraic convergence for even/oddM !
10 1
10 −2
error norms
10 0
10 −1
error norm
10 −4
10 −6
✸
10 −2
10 −8
10 4 M
10 −3
10 −4
10 0 10 1 10 2 10 3
❶ Finite Difference Method
Fig. 43
10 4 N
10 −10
10 −12
0 20 40 60 80 100 120 140 160
❷ Spectral Galerkin Method
Fig. 44
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Qualitative asymptotic convergence also depends on data !
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Observation: for h → 0, p → ∞, algebraic convergence of the finite difference solution, and
exponential convergence of the spectral Galerkin solution emerge. This is the “typical” asymptotic
behavior of the discretization error norms for these discretization methods.
However, the onset of asymptotic convergence occurs only for rather small meshwidth or large p,
respectively, beyond thresholds that may never be reached in a computation. During a long preasymptotic
phase the error is hardly reduced when increasing the resolution of the discretization.
1.6
p. 165
1.6
p. 16
Example 1.6.24 (Convergence and smoothness of solution).
Ω =]0,1[ (for finite differences),Ω =]−1,1[ (for spectral Galerkin),
exact solution of 2-point BVP for ODE− d2 u
dx 2 = g(x),
✸
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Methods
for PDEs
2
Second-order Scalar Elliptic
Boundary Value Problems
Numerica
Methods
for PDEs
10 −2
u(x) =
{ 34 −x 2 , if|x| < 1 2 ,
1−|x| , if|x| ≥ 1 2 . ↔ g(x) =
{
2 , if|x| < 1
2 ,
0 elsewhere .
supremum norm
L 2 norm
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
CORE CHAPTER
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
DRAFT
10 −3
error norms
10 −4
10 −5
error norms
10 −2
Preface
10 −6
10 −1 M
maximum norm
L 2 norm
10 −7
10 0 10 1 10 2 10 3
❶ Finite Difference Method
10 −1 p
10 −3
10 20 30 40 50 60 70 80 90 100
❷ Spectral Galerkin Method
Fig. 45
1.6
p. 166
The previous chapter discussed the transformation of a minimization problem on a function space
via a variational problem to a differential equation. To begin with, in Sect. 2.1–Sect. 2.4, this chapter
2.0
p. 16

evisits this theme for models that naturlly rely on function spaces over domains in two and three
spatial dimensions. Thus the transformation leads to genuine partial differential equations.
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Terminology: boundary value problem is scalar :⇔ n = 1
(in this case the unknown is a real valued function)
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Methods
for PDEs
Sect. 2.2 ventures into the realm of Sobolev spaces, which provide the framework for rigorous mathematical
investigation of variational equations. However, we will approach Sobolev spaces as “spaces
of physically meaningful solutions” or “spaces of solutions with finite energy”. From this perspective
dealing with Sobolev spaces will be reduced to dealing with their norms.
What does elliptic mean ?
Mathematical theory of PDEs distinguishes three main classes of boundary value problems (BVPs)
for partial differential equations (PDE):
In Sect. 2.5, we change tack and consider a physical phenomenon (heat conduction) where modelling
naturally leads to partial differential equations. On this occasion, we embark on a general discussion
of boundary conditions in Sect. 2.6.
Then the fundamental class of second-order elliptic boundary value problems is introduced. Appealing
to “intuitive knowledge” about the physical systems underlying the models, key properties of their
solutions are presented in Sect. 2.7.
In 2.6 Sect.in the context of stationary heat conduction we introduce the whole range of standard
boundary conditions for 2nd-order elliptic boundary value problems. Their discussion in variational
context will be resumed in Sect. 2.9.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.0
p. 169
• Elliptic BVPs (➣ “equilibrium problems”, as discussed in Sects. 1.2.3, 2.1.1, 2.1.2)
• Parabolic initial boundary value problems (IBVPs) (➣ evolution towards equilibrium)
• Hyperbolic IBVPs, among them wave propagation problems and conservation laws (➣
transport/propagation)
The rigorous mathematical definition is complicated and often fails to reveal fundamental properties
of, e.g., solutions that are intuitively clear against the backdrop of the physics modelled by a certain
PDE. Further discussion of classification in [3, § 1] and [15, Ch. 1].
➣ In the spirit of Sect. 1.1
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.0
p. 17
Supplementary and further reading:
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for PDEs
Structural properties of a BPV inherited from the modelled system are more important than
formal mathematical classification.
Numerica
Methods
for PDEs
An excellent mathematical introduction to partial differential equations is Evans’ book [12]. Chapter
2 gives a very good idea about fundamenal properties of various simple PDEs. Chapters 6 and 7 fit
the scope of this course chapter, but go way beyond it in terms of mathematical depth.
△
Remark 2.0.1 (Boundary value problems (BVPs)).
The traditional concept of a boundary value problem for a partial differential equation:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.1 Equilibrium models
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Boundary value problem (BVP)
We only consider stationary systems. Then, frequently, see Sect. 1.2.2
equilibrium = minimal energy configuration of a system
Given a partial differential operatorL, a domainΩ ⊂ R d , a boundary differential operatorB,
boundary datag, and a source termf, seek a functionu : Ω ↦→ R n such that
L(u) = f in Ω ,
2.0
p. 170
Example: elastic string model of Sect. 1.2 (minimization of energy functionalJ(u), see (1.2.17))
2.1
p. 17

Now we study minimization problems for energy functional on spaces of functions Ω ↦→ R, where
Ω ⊂ R d is a bounded (spatial) domain andd = 2,3.
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2.1.1 Taut membrane
Recall: energy functional for pinned taut string under gravitational load ĝ, see (1.4.8), in terms of
displacement, see Fig. 17:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Shape of membrane
⇕
Graph ofu : Ω ↦→ R
“membrane” on spatial domainΩ =]0,1[ 2 ✄
Remark 2.1.1 (Spatial domains).
(– – – ˆ= boundary data)
u(x 1
,x 2
)
3
2.5
2
1.5
1
0.5
0
1
0.8
0.6
0.4
x 1
0.2
u(x)
0.4
0.6
0.8
1
0
x 2
0
0.2
Fig. 47
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
∫b
∣
J(u) := 2
1 ̂σ(x)
du ∣∣∣ 2
∣dx (x) u ∈ C
−ĝ(x)u(x)dx ,
pw([a,b]) 1 ,
u(a) = u a , u(b) = u
a
b .
“2D generalization” of an elastic string ➣ elastic membrane.
2.1
p. 173
2.1
p. 17
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General assumptions on spatial domainsΩ ⊂ R d :
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Taut drum membranes
✄
✎
d = 1,2,3 ˆ= “dimension” of domain
Ω
Ω is bounded
diam(Ω) := sup{‖x−y‖: x,y ∈ Ω} < ∞ ,
Fig. 46
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Ω has piecewise smooth boundary∂Ω
Fig. 48
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Pinning conditions (boundary conditions), cf. (1.2.1), (1.4.12):
fix
u(x) = g(x) x ∈ ∂Ω
⇕
u |∂Ω = g on∂Ω .
for some g ∈ C 0 (∂Ω) . (2.1.2)
2.1
p. 174
✎ notation: ∂Ω ˆ= boundary ofΩ
2.1
p. 17

(2.1.2) means that the displacement of the membrane over∂Ω is provided by a prescribed continuous
functiong : ∂Ω ↦→ R: the membrane is clamped into a rigid frame.
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with u : Ω ↦→ R ˆ= displacement function, see Fig. 47,[u] = m,
f : Ω ↦→ R ˆ= force density (pressure),[f] = Nm −2 ,
σ : Ω ↦→ R + ˆ= stiffness,[σ] = J.
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Intuition:
g has to be continuous, unless the membrane is to be torn!
(Further discussion in Rem. 2.9.4)
Displacement of taut membrane in equilibrium achieves minimal potential energy, cf. (1.2.17)
u ∗ = argminJ M (u) . (2.1.4)
u∈V
configuration space V =
{
continuous functionsu ∈ C 0 (Ω),
with u |∂Ω = g.
Think of the membrane as a grid of taut strings. Together with Rem. 1.4.11 this justifies the following
expression for its total potential energy.
}
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 2.1.5 (Minimal regularity of membrane displacement).
Smoothness required foru,f to renderJ M (u) from (2.1.3) meaningful, cf. Sect. 1.3.2:
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Potential energy of a taut membrane (described byu ∈ C 0 (Ω)) under vertical loading:
u ∈ C 1 pw(Ω) is sufficient for displacementu,
σ,f ∈ C 0 pw(Ω) already allows integration.
J M (u) := ∫ Ω
elastic energy
1
2 σ(x)‖gradu‖2 −f(x)u(x)dx , (2.1.3)
potential energy in force field
Recall the definition of the gradient of a function F : Ω ⊂ R d ↦→ R,F(x) = F(x 1 ,...,x d ), see [29,
Kap. 7], [18, Eq. 5.1.8]:
Note:
⎛ ⎞
∂F
∂x
⎜ 1
⎟
gradF(x) := ⎝ . ⎠ .
∂F
∂x d
the gradient atxis a column vector of first partial derivatives,
readgradF(x) as(gradF)(x);gradF is a vector valued functionΩ ↦→ R d .
Also in use (but not in this course) is the “∇-notation”:
∇F(x) := gradF(x).
2.1
p. 177
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
2.1.2 Electrostatic fields
metal body in metal box
prescribed voltage drop body—box
Sought: electric fieldE : Ω ↦→ R 3 inΩ ⊂ R 3
(Ω ˆ= blue region ✄)
✄
Γ 0
Γ 1
metal
Ω
△
2.1
p. 17
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
U 0
Fig. 49
DRAFT
Note that
σ(x)‖gradu‖ 2 = σ(x 1 ,x 2 )
∂u
2
∣ (x
∂x 1 ,x 2 )
∣ +σ(x 1 ,x 2 )
∂u
2
∣ (x
1 ∂x 1 ,x 2 )
∣ 2
which justifies calling the taut membrane a “two-dimensional string under tension”.
,
2.1
p. 178
2.1
p. 18

From Maxwell’s equations, static case:
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Methods
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Definition 2.1.11 (Uniformly positive (definite) tensor field).
An matrix-valued functionA : Ω ↦→ R n,n ,n ∈ N, is called uniformly positive definite, if
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Methods
for PDEs
E = −gradu , (2.1.6)
∃α − > 0: (A(x)z)·z ≥ α − ‖z‖ 2 ∀z ∈ R n (2.1.12)
where u : Ω ↦→ R ˆ= electric (scalar) potential,
[u] = 1V
for almost allx ∈ Ω, that is, only with the exception of a set of volume zero.
Fig. 50
✁ Electric potential in technical device
Electromagnetic field energy: (electrostatic setting)
∫
∫
J E (E) = 2
1 (ǫ(x)E(x))·E(x)dx = 2
1 (ǫ(x)gradu(x))·gradu(x)dx ,
Ω
Ω
(2.1.7)
where ǫ : Ω ↦→ R 3,3 ˆ= dielectric tensor,ǫ(x) symmetric, [ǫ] = Vm As R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
IfA(x) is symmetric, then we have the equivalence, cf. [18, Rem. 5.1.20],
(2.1.12) ⇔ A(x) s.p.d. (→ [18, Def. 2.7.9]) and λ min (A(x)) ≥ α − .
What is the set/spaceV of admissible electric scalar potentials ?
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.1
p. 181
Recall: in electrostatics surfaces of conducting bodies are equipotential surfaces
2.1
p. 18
• Symmetry of the dielectic tensor can always be assumed: ifǫ(x) was not symmetric, then
replacing it with 1 2 (ǫ(x)T +ǫ(x)) will yield exactly the same field energy.
• In terms of partial derivatives and tensor componentsǫ(x) = ( )
ǫ 3
ij i,j=1
we have
3∑ 3∑
(ǫ(x)gradu(x))·gradu(x) = ǫ ij (x) ∂u (x) ∂u (x) .
∂x
i=1 j=1 i ∂x j
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Methods
for PDEs
Γ 1
u = 0
Γ 0
Ω
In the situation of Fig. 49:
Boundary conditions
u = 0 onΓ 0 ,
u = U 0 onΓ 1 .
(2.1.13)
{
V = u ∈ Cpw(Ω), 1 }
u satisfies (2.1.13) .
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u = U 0
Fig. 51
Fundamental property of dielectric tensor
(for “normal” materials):
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
U 0
to renderJ E (u) well defined, cf. Sect. 1.3.2.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
∃0 < ǫ − ≤ ǫ + < ∞: ǫ − ‖z‖ 2 ≤ (ǫ(x)z)·z ≤ ǫ + ‖z‖ 2 ∀z ∈ R 3 , ∀x ∈ Ω . (2.1.8)
Terminology: (2.1.8) :⇔ ǫ is bounded and uniformly positive definite
Below, the notationu = U will designate the boundary conditions (2.1.13).
Equilibrium condition in electrostatic setting:
minimal electromagnetic field energy
2.1
p. 182
u ∗ = argminJ E (u) . (2.1.14)
u∈V
2.1
p. 18

2.1.3 Quadratic minimization problems
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Structure of minimization problems (equilibrium problems) encountered above:
∫
Sect. 2.1.1 ➣ u ∗ = argmin 1
2 σ(x)‖gradu(x)‖ 2 −f(x)u(x)dx , (2.1.15)
u∈Cpw(Ω)
1 Ω
u=g on∂Ω } {{ }
∫
=:J M (u), see (2.1.3)
1
2 (ǫ(x)gradu(x))·gradu(x)dx . (2.1.16)
Sect. 2.1.2 ➣ u ∗ = argmin
u∈C 1 pw(Ω)
u=U on∂Ω
Ω
} {{ }
=:J E (u), see (2.1.7)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Definition 2.1.21 (Quadratic minimization problem).
A minimization problem
w ∗ = argmin
w∈V 0
J(w)
is called a quadratic minimization problem, if J is a quadratic functional on a real vector space
V 0 .
Hey, both (2.1.15) and (2.1.16) are no genuine quadratic minimization problems, because they are
posed over affine spaces (= “vector space + offset function”, cf. (1.3.14))!
“Offset function trick”, c.f. (1.3.19), resolves the mismatch: for quadratic formJ from (2.1.18)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Evidently, (2.1.15) and (2.1.16) share a common structure. It is the same structure we have already
come across in the minimization problem (1.4.2) for the taut string model in Sect. 1.4.
J(u+u 0 ) = 1 2 a(u+u 0,u+u 0 )+l(u+u 0 )+c
= 2 1a(u,u)+a(u,u }
0)+l(u) ++ 1 {{ } 2 a(u 0,u 0 )+l(u 0 )+c =: ˜J(u) ,
} {{ }
=:˜l(u) =:˜c
due to the bilinearity ofaand the linearity ofl.
2.1
p. 185
argmin J(u) = u 0 +argminJ(w+u 0 ) = u 0 +argmin ˜J(w) . (2.1.22)
u∈u 0 +V 0 w∈V 0 w∈V 0
2.1
p. 18
Definition 2.1.17 (Quadratic functional).
A quadratic functional on a real vector spaceV 0 is a mappingJ : V 0 ↦→ R of the form
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Related: Rem. 1.5.10
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J(u) := 1 2 a(u,u)+l(u)+c , u ∈ V 0 , (2.1.18)
wherea : V 0 ×V 0 ↦→ R is a symmetric bilinear form (→ Def. 1.3.13),l : V 0 ↦→ R a linear form,
andc ∈ R.
For a discussion of quadratic functionals on R n → [18, Sect. 5.1.1]
Recall:
A bilinear forma : V 0 ×V 0 ↦→ R is symmetric, if
a(u,v) = a(v,u) ∀u,v ∈ V 0 . (2.1.19)
IfV 0 = R N (finite-dimensional case), then a quadratic functional has the general representation
J(u) = u T Au+b T u+c , A = A T ∈ R N,N , b ∈ R N , c ∈ R . (2.1.20)
Reminder: quadratic functionals of this forms occur in derivation of steepest descent and conjugate
gradient methods for linear systems of equations, see [18, Sect. 5.1.1].
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.1
p. 186
Both (2.1.15) and (2.1.16) involve quadratic functionals. To see this apply the “offset function trick”
from (2.1.22) in this concrete case: write u = u 0 + w with an offset function u 0 that satisfies the
boundary conditions andw ∈ C0,pw 1 R. Hiptm
(Ω), cf. (1.3.19).
In both cases: V 0 = C0,pw 1 (Ω) 2.1
(2.1.15) ➥ quadratic minimization problem (→ Def. 2.1.21) with, cf. (2.1.18),
DRAFT
∫
∫
a(w,v) = σ(x)gradw(x)·gradv(x)dx , l(v) := a(u 0 ,v)− f(x)v(x)dx . (2.1.23)
Ω
Ω
(2.1.16) ➥ quadratic minimization problem (→ Def. 2.1.21) with, cf. (2.1.18),
∫
a(w,v) = gradw(x) T ǫ(x)gradv(x)dx , l(v) := a(u 0 ,v) . (2.1.24)
Ω
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
p. 18

Can we conclude existence and uniqueness of solutions of the minimization problems (2.1.15) and
(2.1.16) ?
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Example 2.1.29 (Quadratic functionals with positive definite bilinear form in 2D).
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Let us first tackle the issue of uniqueness:
Analogy between quadratic functionals with positive
definite bilinear form and parabolas:
Definition 2.1.25 (Positive definite bilinear form).
A (symmetric) bilinear forma : V 0 ×V 0 ↦→ R on a real vector spaceV 0 is positive definite, if
u ∈ V 0 \{0} ⇐⇒ a(u,u) > 0 .
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
witha > 0!
J(v) = 2 1 a(v,v) − l(v)
↕ ↕ ↕
f(x) = 2 1ax2 + bx
graph of quadratic functionalR 2 ↦→ R
✄
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
For the special caseV 0 = R n any matrixA ∈ R n,n induces a bilinear form via
Fig. 52
✸
a(u,v) := u T Av = (Av)·u , u,v ∈ R n . (2.1.26)
This connects the concept of a symmetric positive definite bilinear form to the more familiar concept
of s.p.d. matrices (→ [18, Def. 2.7.9])
A s.p.d.
⇔ a from (2.1.26) is symmetric, positive definite.
2.1
2.1
p. 189
p. 19
✬
✩
Definition 2.1.27 (Energy norm). cf. [18, Def. 5.1.1]
A symmetric positive definite bilinear forma : V 0 ×V 0 ↦→ R (→ Def. 2.1.25) induces the energy
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Methods
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Theorem 2.1.30 (Uniqueness of solutions of quadratic minimization problems).
If the bilinear forma : V 0 ×V 0 ↦→ R is positive definite (→ Def. 2.1.25), then any solution of
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Methods
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norm
‖u‖ a := (a(u,u)) 1/2 .
is unique for any linear forml : V 0 ↦→ R.
u ∗ = argmin
u∈V 0
J(u) , J(u) = 1 2 a(u,u)+l(u)+c ,
✫
✪
Origin of the term “energy norm” is clear from the connection with potential energy (e.g., in membrane
model and in the case of electrostatic fields, see (2.1.23), (2.1.24)), see above.
Next, we have to verify the norm axioms (N1), (N2), and (N3) from Def. 1.6.3:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Proof. As in the proof of [18, Lemma 5.1.3], straightforward computations show
J(u)−J(u ∗ ) = 1 2 ‖u−u ∗‖ 2 a .
The assertion of the theorem follows from norm axiom (N1), wich holds for the energy norm.
✷
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• (N1) is immediate from Def. 2.1.25,
• (N2) follows from bilinearity ofa,
• (N3) is a consequence of the Cauchy-Schwarz inequality: for any symmetric positive definite
bilinear form
Under the assumptions of the theorem, the quadratic functional J is convex, which is easily seen by
considering the second derivative of the function
ϕ(t) := J(u+tv) ⇒ ¨ϕ(t) = a(v,v) > 0 , ifv ≠ 0 .
|a(u,v)| ≤ (a(u,u)) 1/2 (a(v,v)) 1/2 . (2.1.28)
2.1
p. 190
2.1
p. 19

∫
? Is a(u,v) := (ǫ(x)gradu(x))·gradv(x)dx positive definite onV 0 := C0,pw 1 (Ω)?
Ω
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Methods
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The function ϕ(ξ) = 1 2 ξ2 − ξ = 2 1ξ(1 − 2ξ) = 1 2 (ξ − 1)2 − 1 2 has a global minimum at ξ = 1 and
ϕ(ξ)−ϕ(1) = 1 2 (ξ −1)2 .
➤
|η −1| > |ξ −1| ⇒ ϕ(η) > ϕ(ξ).
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❶: Sinceǫbounded and uniformly positive definite (→ Def. 2.1.11, (2.1.8))
∫
∫
ǫ − ‖gradu(x)‖ 2 dx ≤ a(u,u) ≤ ǫ + ‖gradu(x)‖ 2 dx ∀u . (2.1.31)
Ω
Ω
Hence, it is sufficient to examine the simpler bilinear form
∫
d(u,v) := gradu(x)·gradv(x)dx , u,v ∈ C0,pw 1 (Ω) . (2.1.32)
Ω
❷: Obviously d(u,u) = 0 ⇒ gradu = 0 ⇒ u ≡ const inΩ
Observe: u = 0 on∂Ω ⇒ u = 0
Zero boundary conditions are essential; otherwise one could add constants to the arguments of a
without changing its value.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Assume that u ∈ V 0 is a global minimizer of J.
Then
w(x) := min{1,2max{u(x),0}} ,
0 ≤ x ≤ 1 ,
is another function ∈ C0,pw 0 ([0,1]), which satisfies
u(x) ≠ 1 ⇒ |w(x)−1| < |u(x)−1|
⇒ J(w) < J(u) !
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Fig. 53
Hence, whenever we think we have found a minimizer ∈ C0,pw 0 ([0,1]), the formula provides another
eligible function for which the value of the functional is even smaller!
u(x)
1
0.8
0.6
0.4
0.2
0
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.1
p. 193
The problem in this example seems to be that we have chosen “too small” a function space, c.f.
Sect. 2.2 below.
2.1
p. 19
What about existence?
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✸
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In a finite dimensional setting this is not a moot point, see Fig. 52 for a “visual proof”.
However, infinite dimensional spaces hold a lot of surprises and existence of solutions of quadratic
minimization problems becomes a subtle issue, even if the bilinear form is positive definite.
2.2 Sobolev spaces
Example 2.1.33 (Non-existence of solutions of positive definite quadratic minimization problem).
We consider the quadratic functional
∫ 1 ∫ 1
J(u) := 1
2 u 2 (x)−u(x)dx = 2
1 (u(ξ)−1) 2 −1dx ,
0
0
on the space V 0 := C0,pw 0 ([0,1])
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Mathematical theory is much concerned about proving existence of suitably defined solutions for
minimization problems. As demonstrated in Ex. 2.1.33 this can encounter profound problems.
In this section we will learn about a class of abstract function spaces that has been devised to
deal with the question of existence of solutions of quadratic minimization problems like (2.1.15) and
(2.1.16). We can only catch of glimps of the considerations; thorough investigation is done in the
mathemtical field of functional analysis.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
It fits the abstract form from Def. 2.1.17 with
∫ 1
∫ 1
a(u,v) = u(x)v(x)dx , l(v) = v(x)dx .
0
0
2.1
p. 194
2.2
p. 19

Consider a quadratic minimization problem (→ Def. 2.1.21) for a quadratic functional (→ Def. 2.1.17)
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We follow the above recipe, which suggests to choose
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J : V 0 ↦→ R , J(u) = 1 2 a(u,u)+l(u)+c ,
based on a symmetric positive definite (s.p.d.) bilinear forma → Def. 2.1.25.
∫
V 0 := {v : Ω ↦→ R integrable: |v(x)| 2 dx < ∞} (2.2.4)
Ω
It is clear thatJ(V 0 ) is bounded from below, if
∃C > 0: |l(u)| ≤ C‖u‖ a ∀u ∈ V 0 , (2.2.1)
where‖·‖ a is the energy norm induced bya, see Def. 2.1.27:
Remark:
J(u) = 1 2 a(u,u)−l(u) ≥ 1 2 ‖u‖2 a −C‖u‖ a ≥ −1 2 C2 .
In mathematical terms (2.2.1) means thatlis continuous w.r.t. ‖·‖ a
Under these conditions, the quadratic minimization problem for J should have a (unique, due to
Thm. 2.1.30) solution, if it is considered on a space that is “large enough”.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Definition 2.2.5 (SpaceL 2 (Ω)).
The function space defined in (2.2.4) is the space of square-integrable functions on Ω and
denoted byL 2 (Ω).
It is a normed space with norm
( ) (∫ 1/2
‖v‖ 0 := ‖v‖ L 2 (Ω) := |v(x)| dx) 2 .
Ω
Notation: L 2 (Ω) ← superscript “2”, because square in the definition of norm‖·‖ 0
Note: obviously C 0 pw(Ω) ⊂ L 2 (Ω).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Idea: for a quadratic minimization problem (→ Def. 2.1.21) with
symmetric positive definite (s.p.d.) bilinear forma,
a linear formlthat is continuous w.r.t. ‖·‖ a , see (2.2.1),
posed over a function space follow the advice:
consider it on the largest space of functions for whichastill makes sense !
2.2
p. 197
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Remark 2.2.6 (Boundary conditions andL 2 (Ω)).
Ex. 2.1.33 vs. Ex. 2.2.2: Crying foul! (boundary conditions u(0) = u(1) = 0 in Ex. 2.1.33, but
none in Ex. 2.2.2!)
✸
2.2
p. 19
Numerica
Methods
for PDEs
✗
✖
(and which complies with boundary conditions)
Choose “V 0 := {functionsv onΩ: a(v,v) < ∞}”
Example 2.2.2 (Space of square integrable functions). → Ex. 2.1.33
R. Hiptmair
✔
C. Schwab,
H.
Harbrecht
V.
Gradinaru
✕
A. Chernov
DRAFT
Consideru ∈ C 0 ([0,1]) and try to impose boundary valuesu 0 ,u 1 ∈ R by “altering”u:
⎧
⎪⎨ u(x)+(1−nx)(u 0 −u(0)) , for 0 ≤ x ≤ n 1 ,
ũ(x) = u(x) , for n 1 ⎪⎩
< x < 1− n 1 ,
u(x)−n(1− n 1 −x)(u 1−u(1)) , for 1− n 1 < x ≤ 1 .
ũ(0) = u 0 , ũ(1) = u 1 , ‖ũ−u‖ 2 L 2 (]0,1[) = 3n 1 (u 0+u 1 −u(0)−u(1))→ 0 forn → ∞ .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Quadratic functional (related toJ from Ex. 2.1.33):
∫
J(u) := 1
2 |u(x)| 2 −u(x)dx .
(
u ∈ Cpw(Ω) 0 )
?
(2.2.3)
Tiny perturbations of a function u ∈ L 2 (]0,1[) (in terms of changing its L 2 -norm) can make it
attain any value atx = 0 andx = 1.
Ω
2.2
p. 198
Boundary conditions cannot be imposed inL 2 (Ω) !
△
2.2
p. 20

Remark 2.2.7 (Quadratic minimization problems on Hilbert spaces).
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Remark 2.2.11 (Boundary conditions inH 1 0 (Ω)).
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On the function space V 0 = L 2 (Ω) the quadratic minimization problem for the quadratic functional
from (2.2.3) can be shown to possess a solution. Instrumental in the proof is the fact that L 2 (Ω) is a
Hilbert space, that is, a complete normed space.
Rem. 2.2.6 explained why imposing boundary conditions on functions inL 2 (Ω) does not make sense.
This theory is beyond the scope of this course. For more explanations see [12, Ch. 5 and Sect. 6.2].
△
Now consider a quadratic minimization problem for the functional, c.f. (2.1.15),
∫
J(u) := 1
2 ‖gradu‖ 2 −f(x)u(x)dx
(u ∈ C0,pw 1 )
(Ω) ?
Ω
(2.2.8)
What is the natural function space for this minimization problem? Again, we follow the above recipe,
which suggests that we choose
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.2
p. 201
Yet, in (2.2.9) zero boundary conditions are required forv !
Discussion parallel to Rem. 2.2.6, but now with the norm |·| H 1 (Ω) in mind: Consider u ∈ C1 ([0,1])
and try to impose boundary valuesu 0 ,u 1 ∈ R by “altering”u:
⎧
⎪⎨ u(x)+(1−nx)(u 0 −u(0)) , for 0 ≤ x ≤ n 1 ,
ũ(x) = u(x) , for n 1 ⎪⎩
< x < 1− n 1 ,
u(x)−n(1− n 1 −x)(u 1−u(1)) , for 1− n 1 < x ≤ 1 .
ũ(0) = u 0 , ũ(1) = u 1 , BUT |ũ−u| 2 H 1 (]0,1[) = n(u 0+u 1 −u(0)−u(1))→ ∞ forn → ∞ .
Enforcing boundary values atx = 0 andx = 1 cannot be done without significantly changing
the “energy” of the function.
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.2
p. 20
∫
V 0 := {v : Ω ↦→ R integrable: v = 0 on∂Ω, |gradv(x)| 2 dx < ∞} (2.2.9)
Ω
Numerical
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for PDEs
However, the solutions of the quadratic minimization problems (2.1.15), (2.1.16) are to satisfy nonzero
boundary conditions. They are sought in a larger Sobolev space, which arises from H0 1 (Ω) by
dispensing with the requirement “v = 0 on∂Ω”.
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Definition 2.2.10 (Sobolev spaceH 1 0 (Ω)).
The space defined in (2.2.9) is the Sobolev spaceH0 1 (Ω) with norm
(∫
1/2
|v| H 1 (Ω) := ‖gradv‖ dx) 2 .
Ω
Notation:
H 1 0
(Ω)
← superscript “1”, because first derivatives occur in norm
← subscript “0”, because zero on∂Ω
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Definition 2.2.12 (Sobolev spaceH 1 (Ω)).
The Sobolev space
∫
H 1 (Ω) := {v : Ω ↦→ R integrable: |gradv(x)| 2 dx < ∞}
Ω
is a normed function space with norm
‖v‖ 2 H 1 (Ω) := ‖v‖2 0 +|v|2 H 1 (Ω) .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Note:
☛
|·| H 1 (Ω) is the energy norm (→ Def. 2.1.27) associated with the bilinear form in the quadratic
functionalJ from (2.2.8), cf. (2.1.18).
See Rem. 1.6.7 for a discussion of the relevance of the energy norm.
2.2
p. 202
H 1 (Ω) is the “maximal function space” on which both J M and J E from (2.1.15), (2.1.16)
are defined.
2.2
p. 20

Remark 2.2.13 (|·| H 1 (Ω) -seminorm).
Note that|·| H 1 (Ω) alone is no longer a norm onH1 (Ω), because for v ≡ const obviously|v| H 1 (Ω) =
0, which violates (N1).
△
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Proof. The proof employs a powerful technique in the theoretical treatment of function spaces: exploit
density of smooth functions (which, by itself, is a deep result).
It boils down to the insight:
In order to establish inequalities between continuous functionals on Sobolev spaces of functions
on Ω it often suffices to show the target inequality for smooth functions in C ∞ 0 (Ω) or C∞ (Ω),
respectively.
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In the introduction to this section we saw that a quadratic functional with s.p.d. bilinear form a is
bounded from below, if its linear form l satisfies the continuity (2.2.1). Now, we discuss this for the
quadratic functionalJ from (2.2.8) in lieu ofJ M andJ E .
The quadratic functionalJ from (2.2.8) involves the linear form
∫
l(u) := f(x)u(x)dx . (2.2.14)
Ω
f ˆ= load function ➣ f ∈ C 0 pw(Ω) should be admitted.
Crucial question: Islfrom (2.2.14) continous onH 1 0 (Ω) ?
⇕ (c.f. (2.2.1))
∃C > 0: |l(u)| ≤ C|u| H 1 (Ω) ∀u ∈ H 1 0 (Ω) ? .
To begin with, we use the Cauchy-Schwarz inequality (2.1.28) for itegrals, which implies
⎛ ⎞
∫
1/2 ∫
⎛ ⎞1/2
∫
⎜
|l(u)| =
f(x)u(x)dx
≤ ⎝ |f(x)| 2 ⎟ ⎜
dx⎠
⎝ |u(x)| 2 ⎟
dx⎠
= ‖f‖ 0 ‖u‖
}{{} 0 . (2.2.15)
∣Ω
∣ Ω Ω 0: |l(u)| ≤ C|u| H 1 (Ω) ∀u ∈ H 1 0 (Ω) ,
✷
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.2
p. 20
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
(2.2.1)
✫
✪
2.2
p. 206
2.2
p. 20

Most concrete results about Sobolev spaces boil down to relationships between their norms. The
spaces themselves remain intangible, but the norms are very concrete and can be computed and
manipulated as demonstrated above.
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✬
Theorem 2.2.17. Compatibility conditions
for piecewise smooth functions in H 1 (Ω)]
✩
O
Ω 1
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✬
✫
Do not be afraid of Sobolev spaces!
It is only the norms that matter for us, the ‘spaces” are irrelevant!
Sobolev spaces = “concept of convenience”: the minimization problem seeks its own function
space.
✩
✪
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
LetΩbe partitioned into sub-domainsΩ 1 and
Ω 2 . A function that is continuously differentiable
in both sub-domains and continuous up
to their boundary, belongs to H 1 (Ω), if and
only ifuis continuous onΩ.
✫
✪
Ω 2
Fig. 54
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Minimization problem
u = argminJ(v)
v:Ω↦→R
“Maxmimal” function space
on whichJ is defined
(Sobolev space)
The proof of this theorem requires the notion of weak derivatives that will not be introduced in this
course.
Example 2.2.18 (Piecewise linear functions (not) inH 1 0 (]0,1[)).
2.2
2.2
p. 209
p. 21
Then, why do you bother me with these uncanny “Sobolev spaces” after all ?
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We conclude from Thm. 2.2.17:
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Anyone involved in CSE must be able to understand mathematical publications on numerical methods
for PDEs, Those regularly resort to the concept of Sobolev spaces to express their findings.
u
u
The statement that a function belongs to a certain Sobolev space can be regarded as a concise
way of describing quite a few of its essential properties.
Let us elucidate the second point:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1/2
u ∈ H 1 0 (]0,1[)
x
1
1/2
u∉H 1 0 (]0,1[)
x
1
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
From Thm. 2.2.17 we conclude
2.2
p. 210
C 1 pw([a,b]) ⊂ H 1 (]a,b[) andC 1 0,pw ([a,b]) ⊂ H1 0 (]a,b[)
2.2
p. 21

Thm. 2.2.17 provides a simple recipe for computing the norm |u| H 1 (Ω) of a piecewise C1 -function
that is continuous in all ofΩ.
✬
✩
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Methods
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If you are still feeling uneasy when dealing with Sobolev spaces, do not hesitate to think of the
following replacements
L 2 (Ω) → C 0 pw(Ω) , H 1 0 (Ω) → C1 0,pw (Ω) .
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Corollary 2.2.19 (H 1 -norm of piecewise smooth functions).
Under the assumptions of Thm. 2.2.17 we have for a continuous, piecewise smooth function
u ∈ C 0 (Ω)
∫ ∫
|u| 2 H 1 (Ω) = |u|2 H 1 (Ω 1 ) +|u|2 H 1 (Ω 2 ) = |gradu(x)| 2 dx+ |gradu(x)| 2 dx .
Ω 1 Ω 2
✫
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✪
2.3 Variational formulations
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Actually, this is not new, see Sect. 1.3.2: earlier we already evaluated the elastic energy functionals
(1.2.16), (1.4.2) for functions inC 1 pw([0,1]) by “piecewise differentiation” followed by integration of the
resulting discontinuous function.
2.3.1 Linear variational problems
Example 2.2.20 (Non-differentiable function inH 1 0 (]0,1[)).
d = 1,Ω =]0,1[:
“Tent function” u(x) =
{
2x for0 < x < 1/2 ,
2(1−x) for 1/2 < x < 1 .
∫ 1
Compute |u| 2 H 1 (Ω) = |u ′ (x)| 2 dx = 4 < ∞ .
0
Example for au ∈ H0 1 (]0,1[), which is not globally differentiable.
Recall: we cheerfully computed the derivative of a piecewise smooth function already in Sect. 1.5.1.2
when differentiating the basis functions, cf. (1.5.55). Now this “reckless” computations have found
their rigorous justification.
u
1/2
x
1
✸
2.2
p. 213
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Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Recall:
No surprise:
derivation of variational formulation (1.4.4) from taut string minimization problem (1.4.2) in
Sect. 1.4.
(2.1.15) & (2.1.16) are amenable to the same approach:
Calculus of variations→Sect. 1.3.1: “Directionale derivative” ofJ E :
∫
J E (u+tv)−J E (u) = 2
1 (ǫ(x)grad(u+tv))·grad(u+tv)dx
Ω ∫
− 1 2 (ǫ(x)gradu)·gradudx
R. Hiptm
∫
Ω
(∗)
= 2
1 (ǫ(x)gradu)·gradu+2t(ǫ(x)gradu)·gradv+
Ω t 2 DRAFT
(ǫ(x)gradv)·gradv −(ǫ(x)gradu)·gradudx
∫
= t (ǫ(x)gradu)·gradvdx+O(t 2 ) for t → 0 .
Ω
(∗): due to the symmetry ofǫ(x): (ǫgradu)·gradv = (ǫgradv)·gradu !
∫
J
lim E (u+tv)−J E (u)
= (ǫ(x)gradu(x))·gradv(x)dx ,
t→0 t
Ω
2.2
for perturbation functions v ∈ H0 1(Ω) , see Def. 2.2.10 2.3
p. 214
p. 21
2.3
p. 21
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C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern

The requirement v = 0 on ∂Ω reflects the fact that we may not perturb u on the boundary, lest the
prescribed boundary values be violated.
As explained in Sect. 1.3.1 (“idea of calculus of variations”), this leads to the following variational
problem equivalent to (2.1.16)
u ∈ H 1 ∫
(Ω) ,
u = U on ∂Ω :
For the membrane problem (2.1.15) we arrive at
u ∈ H 1 ∫
(Ω) ,
u = g on∂Ω :
Ω
Ω
(ǫ(x)gradu(x))·gradv(x)dx = 0 ∀v ∈ H0 1 (Ω) . (2.3.1)
∫
σ(x)gradu(x)·gradv(x)dx =
Ω
f(x)v(x) ∀v ∈ H0 1 (Ω) . (2.3.2)
Both, (2.3.1) and (2.3.2) have a common structure, expressed in the following variational problem:
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
for almost allx ∈ Ω.
∃0 < α − ≤ α + : α − ‖z‖ 2 ≤ (α(x)z)·z ≤ α + ‖z‖ 2 ∀z ∈ R d , (2.3.4)
Rewriting (2.3.3), using offset functionu 0 withu 0 = g on∂Ω, cf. (2.1.22),
∫
w ∈ H0 1 (Ω): (α(x)gradw(x))·gradv(x)dx
Ω
∫
=
f(x)v(x)−(α(x)gradu 0 (x))·gradv(x)dx ∀v ∈ H0 1 (Ω) . (2.3.5)
Ω
➥ (2.3.5) is a linear variational problem, see Rem. 1.4.5
We can lift the above discussion to an abstract level, cf. discussiion after (1.4.6):
Variational formulation of a quadratic minimization problem (→ Def. 2.1.21)
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.3
p. 217
J(u) := 1 2 a(u,u)+l(u)+c ⇒ J(u+tv) = J(u)+t(a(u,v)+l(v))+ 1 2 t2 a(v,v) ,
for allu,v ∈ V 0 .
2.3
p. 21
Variational formulation of 2nd-order elliptic (Dirichlet) minimization problems:
u ∈ H 1 ∫
∫
(Ω) ,
u = g on ∂Ω : (α(x)gradu(x))·gradv(x)dx = f(x)v(x)dx ∀v ∈ H0 1 (Ω) .
Ω
Ω
(2.3.3)
Symmetric uniformly positive definite material tensorα : Ω ↦→ R d,d
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For a quadratic functional (→ Def. 2.1.21) on real vector spaceV 0
J(u+tv)−J(u)
lim = a(u,v)+l(v) . (2.3.6)
t→0 t
Linear variational problem (→ Rem. 1.4.5) arising from quadratic minimization problem for functionalJ(u)
:= 1 2 a(u,u)+l(u)+c:
w ∈ V 0 : a(w,v)+l(v) = 0 ∀v ∈ V 0 . (2.3.7)
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The attribute “Dirichlet” refers to a setting, in which the functionuis prescribed on the entire boundary.
Some more explanations and terminology:
Ω ⊂ R d ,d = 2,3 ˆ= (spatial) domain, bounded, piecewise smooth boundary
g ∈ C 0 (∂Ω) ˆ= boundary values (Dirichlet data)
f ∈ C 0 pw(Ω) ˆ= loading function, source function
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Concretely, for (2.3.5):
V 0 = H0 1 (Ω) and
∫
a(u,v) = (α(x)gradw(x))·gradv(x)dx , (2.3.8)
∫
l(v) = −
Ω
Ω
f(x)v(x)+(α(x)gradu 0 (x))·gradv(x)dx . (2.3.9)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
α : Ω ↦→ R d,d ˆ= material tensor, stiffness function, diffusion coefficient
(uniformly positive definite, bounded → Def. 2.1.11)
2.3
p. 218
2.3
p. 22

2.3.2 Stability
Numerical
Methods
for PDEs
Consider the particular choice (2.3.8).
Numerica
Methods
for PDEs
Notion of stability for a (linear) variational problem (2.3.7):
How to choose the norms‖·‖ X (on solution space) and‖·‖ Y (on data space) ?
Lipschitz continuity of (linear) mapping
←→ Is there/what is a constantC stab > 0 such that
datal ↦→ solutionw
‖w‖ X ≤ C stab ‖l‖ Y , wherew solves (2.3.7), (2.3.10)
with suitable/relevant norms ‖·‖ X , ‖·‖ Y ? These norms will be suggested by the modelling background.
Their choice will determine existence and value ofC stab .
Remark 2.3.11 (Sensitivity of linear variational problems).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Norm on solution space: energy norm:
Norm on r.h.s:
‖·‖ a
Mean square norm (L 2 -norm,→Def. 2.2.5) forf,
H 1 -semi-norm (→ Def. 2.2.12) foru 0
What will be the impact of a perturbation ofl, if we use these norms?
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Recall a notion introduced in [18, Sect. 2.5.5]:
Sensitivity of a problem (for given data) gauges
impact of small perturbations of the data on the result.
2.3
p. 221
First use the Cauchy-Schwarz inequality (2.2.15) and the uniform positivity (→ Def. 2.1.11) ofα, see
(2.3.4):
|l(v)| ≤ ‖f‖ 0 ‖v‖ 0 +α + ‖gradu 0 ‖ 0 ‖gradv‖ 0
(
1/2‖v‖H
≤ ‖f‖ 2 0 +(α+ ) 2 ‖gradu 0 ‖ 2 0) 1 (Ω) ∀v ∈ H 1 (Ω) . (2.3.13)
2.3
p. 22
Remember: “Problem” = mapping from data space to solution space, see [18, Sect. 2.5.2].
Here, we define the “problem” as the mapping
{
{linear forms onV0 } ↦→ V 0
l ↦→ w ∈ V 0 : a(w,v) = −l(v) ∀v ∈ V 0 .
(2.3.12)
Undesirable: “sensitive dependence of solution on data”, that is small (in the norm of the data space)
perturbations ofltranslate into huge (in the norm of the solution space) or even “infinite” perturbations
of the solution. In this case of an “ill-posed problem” inevitable data errors (e.g., due to non-exact
measurements) will thwart any attempt to compute an “accurate” (in the norm of the solution space)
solution.
Desirable:
Note:
Lipschitz continuity of problem map with small Lipschitz constant (well-posed problem).
the problem map (2.3.12) is linear and its Lipschitz constant is given by the smallest value
forC stab in (2.3.10).
△
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.3
p. 222
Next, we appeal to the lower estimate in (2.3.4) and the first Poincaré-Friedrichs inequality of
Thm. 2.2.16:
Combine (2.3.13) and (2.3.14),
‖v‖ H 1 (Ω) ≤ √
1+diam 2 (Ω)|v| H 1 (Ω) ≤ √
1+diam 2 (Ω)
α − ‖v‖ a . (2.3.14)
(
|l(v)| ≤
) 1/2
√
1+diam 2 (Ω)
α −
‖f‖ 2 0 +(α+ ) 2 |u 0 | 2 H 1 (Ω)
} {{ }
=:K(f,u 0 )
This enters the estimate for the perturbation of the solution:
‖v‖ a .
a(w,v) = −l(v) ∀v ∈ V 0 ,
a(w+δw,v) = −(l+δl)(v) ∀v ∈ V 0 .
a bilinear
=⇒ a(δw,v) = −δl(v) ∀v ∈ V 0 ,
(2.3.10)
=⇒ ‖δw‖ a = √ a(δw,δw) = √ |δl(δw)| ≤ (K(δf,δu 0 )‖δw‖ a ) 1/2 ,
=⇒ ‖δw‖ a ≤ K(δf,δu 0 ) .
As in Rem. 1.6.7 for associated quadratic energy functionalJ:
|J(w +δw)−J(w)| = 1 2 |a(2w+δw,δw)| ≤ 1 2 ‖2w +δw‖ a ‖δw‖ a .
Numerica
Methods
for PDEs
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
R. Hiptm
DRAFT
(2.3.15)
2.3
p. 22

☛
Perturbation estimates in energy norm directly translate into perturbation
Remark 2.3.16 (Needle loading).
estimates for the equilibrium energy!
Now we inspect a stricking manifestation of instability for a 2nd-order elliptic variational problem
caused by a right hand side functional that fails to satisfy (2.2.1).
Numerical
Methods
for PDEs
Also recall integration in polar coordinates, see [29, Bsp. 8.5.3]:
Using these formulas we try to compute|v| H 1 (Ω) ,
Ω
1/2 2π
∫ ∫ ∫
v(x)dx = v(r,ϕ)rdϕdr . (2.3.20)
0
0
Numerica
Methods
for PDEs
Consider the taut membrane model, see Sect. 2.1.1 for details, (2.1.15) for the related minimization
problem, and (2.3.2) for the associated variational equation.
Let us assume that a needle is poked at the membrane: loading by a forcef “concentrated in a point
y”, often denoted byf = δ y ,y ∈ Ω, whereδ is the so-caled Dirac delta function (delta distribution).
In the variational formulation this can be taken into account as follows (u |∂Ω = 0,σ ≡ 1 is assumed):
∫
u ∈ H0 1 (Ω): gradu(x)·gradv(x)dx = v(y)
}
Ω
}{{}
{{ }
=:l(v)
=:a(u,v)
∀v ∈ H 1 0 (Ω) .
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
(2.3.17)
2.3
p. 225
1/2 2π
∫ ∫ ∫
‖gradv(x)‖ 2 dx =
∥ − 1
logrr e 2 ∫
r
∥ rdϕdr = 2π
Ω
0 0
0
= [−1/logr] 1/[2]
0 = 1
log2 < ∞ ,
1/2
1
log 2 r · 1
r dr
because the improper integral exists. This means thatv has “finite elastic energy” , that isv ∈ H 1 (Ω),
see Def. 2.2.12.
On the other hand, v(0) = ∞ !
H 1 (Ω) contains unbounded functions !
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.3
p. 22
Recall the discussion of Sect. 2.2: is the linear functional l on the right hand side continuous w.r.t.
theH0 1 (Ω)·-norm (= energy norm, see Def. 2.1.27) in the sense of (2.2.1)?
Consider the functionv(x) = log|log‖x‖|,x ≠ 0, onΩ = {x ∈ R 2 : ‖x‖ < 1 2 }.
Numerical
Methods
for PDEs
✬
Corollary 2.3.21 (Point evaluation onH 1 (Ω)).
The point evaluationv ↦→ v(y),y ∈ Ω is not a continuous linear form onH 1 (Ω).
✫
✩
Numerica
Methods
for PDEs
✪
First, we express this function in polar coordinates
(r,ϕ)
x 1 = rcosϕ , x 2 = rsinϕ v(r,ϕ) = log|logr| .
(2.3.18)
Then we recall the expression for the gradient in
polar coordinates
gradv(r,ϕ) = ∂v
∂r (r,ϕ)e r + 1 ∂v
r∂ϕ (r,ϕ)e ϕ ,
(2.3.19)
x 2
ϕ
r
e ϕ
x 1
e r
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
This is the mathematics behind the observation
that a needle can easily prick a taut membrane:
a point load leads to configurations with “infinite
elastic energy”.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
wheree r ande ϕ are orthogonal unit vectors in the
polar coordinate directions.
Fig. 55
2.3
p. 226
Another implication of Cor. 2.3.21:
2.3
p. 22

The quadratic functionalJ(u) := ∫ ‖gradu‖ 2 dx−u(y),y ∈ Ω
Ω
is not bounded from below onH0 1(Ω)!
Numerical
Methods
for PDEs
There is a product rule in higher dimensions, see [29, Sect. 7.2]
Numerica
Methods
for PDEs
Thus, it is clear that the attempt to minimize J will run into difficulties.
functional underlying the variational problem (2.3.17).
Yet, this is the quadratic
✬
Lemma 2.4.3 (General product rule).
For allj ∈ (C 1 (Ω)) d ,v ∈ C 1 (Ω) holds
✩
2.4 Equilibrium models: Boundary value problems
△
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✫
An important differential operator, see [29, Def. 8.8.1]:
div(jv) = v divj+j·gradv . (2.4.4)
divergence of aC 1 -vector fieldj = (f 1 ,...,f d ) T : Ω ↦→ R d
divj(x) := ∂f 1
∂x 1
(x)+···+ ∂f d
∂x d
(x) , x ∈ Ω .
R. Hiptm
C. Schwa
✪H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Recall the derivation of an ODE from a variational problem on a 1D domain (interval) in Sect. 1.3.3:
Tool: Integration by parts (1.3.22)
This section elucidates how to extend this approach to domainsΩ ⊂ R d ,d ≥ 1 (usuallyd = 2,3).
Crucial issue: Integration by parts in higher dimensions ?
2.4
p. 229
Numerical
Methods
for PDEs
A truly fundamental result from differential geometry provides a multidimensional analogue of the
fundamental theorem of calculus:
✬
Theorem 2.4.5 (Gauss’ theorem). → [29, Sect. 8.8]
With n : ∂Ω ↦→ R d denoting the exterior unit normal vectorfield on ∂Ω and dS indicating
integration over a surface, we have
∫ ∫
divj(x)dx = j(x)·n(x)dS(x) ∀j ∈ (Cpw(Ω)) 1 d . (2.4.6)
✫
Ω
∂Ω
✩
✪
2.4
p. 23
Numerica
Methods
for PDEs
Remember the origin of integration by parts: fundamental theorem of calculus [29, Satz 6.3.4]: for
F ∈ Cpw([a,b]),a,b 1 ∈ R,
∫ b
F ′ (x)dx = F(b)−F(a) , (2.4.1)
a
where ′ stands for differentiation w.r.tx. This formula is combined with the product rule [29, Satz 5.2.1
(ii)]
F(x) = f(x)·g(x) ⇒ F ′ (x) = f ′ (x)g(x)+f(x)g ′ (x) . (2.4.2)
∫ b
f ′ (x)g(x)+f(x)g ′ (x)dx = f(b)g(b)−f(a)g(a) ,
a
which amounts to (1.3.22).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.4
p. 230
Note: In (2.4.6) integration again allows to relax smoothness requirements, cf. Sect. 1.3.2.
✬
Theorem 2.4.7 (Green’s first formula).
For all vector fieldsj ∈ (Cpw(Ω)) 1 d and functionsv ∈ Cpw(Ω) 1 holds
∫ ∫ ∫
j·gradvdx = − divjvdx+ j·nvdS . (2.4.8)
Ω Ω ∂Ω
✫
DRAFT
✩
✪
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
2.4
p. 23

Note that the dependence on the integration variablexis suppressed in the formula (2.4.8) to achieve
a more compact notation. The first Green formula could also have been written as
∫
∫ ∫
j(x)·(gradv)(x)dx = − (divj)(x)v(x)dx+ j(x)·n(x)v(x)dS(x) . (2.4.8)
Ω
Ω
∂Ω
Proof. (of Thm. 2.4.7) Straightforward from Lemma 2.4.3 and Thm. 2.4.5. ✷
Now we apply Green’s first formula to the variational problem (2.3.3), which covers the membrane
model and electrostatics:
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Now we can invoke the multidimensional analogue of the fundamental lemma of the calculus of variations,
see Lemm 1.3.23
✬
Lemma 2.4.10 (Fundamental lemma of calculus of variations in higher dimensions).
Iff ∈ L 2 (Ω) satisfies
∫
f(x)v(x)dx = 0 ∀v ∈ C0 ∞ (Ω) ,
Ω
thenf ≡ 0 can be concluded.
✫
✩
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
✪H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
The role ofjin (2.4.8) is played by the vector field αgradu : Ω ↦→ R d .
(2.3.3)
αgradu∈C 1 pw(Ω)
Partial differential equations (PDE)
−div(α(x)gradu) = f inΩ .
(2.4.11)
∫
Ω
α(x)gradu(x) ·gradv(x)dx
} {{ }
=:j(x)
∫
∫
= − div(α(x)gradu(x))v(x)dx+ (α(x)gradu(x))·n(x)v(x)dS(x) .
Ω
∂Ω
2.4
p. 233
Numerical
Methods
for PDEs
Again, for the sake of brevity, dependence gradu = gradu(x), f = f(x) is not made explicit in
the PDE in (2.4.11).
2.4
p. 23
Numerica
Methods
for PDEs
∫
(2.3.3) −
Ω
div(α(x)gradu(x))v(x)dx
∫
+
(α(x)gradu(x))·n(x)v(x)dS(x)
∂Ω
(2.4.9)
} {{ }
=0 , since v |∂Ω =0
∫
= f(x)v(x)dx ∀v ∈ C0,pw 1 (Ω) ,
Ω
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 2.4.12 (Laplace operator).
Ifαagrees with a positive constant, by rescaling of (2.5.6) we can achieve
−∆u = f inΩ . (2.4.13)
∆ = div◦grad = ∂2
∂x 2 + ∂2
1 ∂x 2 + ∂2
2 ∂x 2 = Laplace operator
3
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
where we have to assume that u,αare sufficiently smooth: αgradu ∈ C 1 pw(Ω)
(2.4.13) is called Poisson equation, ∆u = 0 inΩ is called Laplace equation
∫
Ω
(div(α(x)gradu(x))+f(x)) v(x)dx = 0 ∀v ∈ C 1 0,pw (Ω) .
2.4
p. 234
△
2.4
p. 23

Finally: PDE (2.4.11) + boundary conditions
−div(α(x)gradu) = f inΩ , u = g on∂Ω . (2.4.14)
Numerical
Methods
for PDEs
test space in variational formulation V 0 := {u ∈ H 1 (Ω): u |Γ0 = 0}
(Remember: test space comprises “admissible perturbations” of configurations, cf. Sect. 1.3.1.)
Numerica
Methods
for PDEs
(2.4.14) = second-order elliptic BVP with Dirichlet boundary conditions
Short name for BVPs of the type (2.4.14): “Dirichlet problem”
Remark 2.4.15 (Extra smoothness requirement for PDE formulation).
Same situation as in Sect. 1.3.3, cf. Assumption (1.3.21):
Transition from variational equation to PDE requires
extra assumptions on smoothness of solution and coefficients.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Variational formulation, c.f. (2.3.2)
u ∈ H 1 ∫
∫
(Ω) ,
: σ(x)gradu(x)·gradv(x)dx =
u = g onΓ 0
Ω
Ω
☞ Application of Green’s first formula (2.4.8) to (2.4.17) leads to
∫
− (div(σ(x)gradu(x))+f(x)) v(x)dx
Ω
+
∫
f(x)v(x) ∀v ∈ V 0 . (2.4.17)
∂Ω\Γ 0
((σ(x)gradu(x))·n(x))v(x)dS(x) ∀v ∈ V 0 . (2.4.18)
Note that, unlike above, the boundary integral term cannot be dropped entirely, because v ≠ 0 on
∂Ω\Γ 0 .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
△
2.4
p. 237
Assumption (→ Rem. 2.4.15):
extra smootnessu ∈ C 2 pw(Ω),σ ∈ C 1 pw(Ω)
2.4
p. 23
Remark 2.4.16 (Membrane with free boundary values).
Numerical
Methods
for PDEs
How to deal with the boundary term ?
Numerica
Methods
for PDEs
(Graph description of membrane shape by u :
Ω ↦→ R, see Sect. 2.1.1 )
6
5
Idea:
➊ First restrict test functionv toC ∞ 0 (Ω)
➣ Boundary term vanishes !
Now: membrane clamped only on a part
Γ 0 ⊂ ∂Ω of its edge.
– – – : prescribed bondary values here (Γ 0 )
——-: “free boundary”
u(x)
4
3
2
1
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
Fig. 56
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Then, apply Lemma 2.4.10
➋ Then test with genericv ∈ V 0 , while making use of (2.4.19):
∫
div(σ(x)gradu(x))+f(x) = 0 inΩ . (2.4.19)
∂Ω\Γ 0
((σ(x)gradu(x))·n(x))v(x)dS(x) = 0 ∀v ∈ V 0 .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Configuration space V := {u ∈ H 1 (Ω): u |Γ0 = g} → Def. 2.2.12
Lemma 2.4.10 on∂Ω\Γ 0
=⇒ (σ(x)gradu(x))·n(x) = 0 on∂Ω\Γ 0 . (2.4.20)
When removing pinning conditions on ∂Ω \ Γ 0 the equilibrium conditions imply the (homoge-
Total potential energy as in (2.1.3):
neous) Neumann boundary conditions(σ(x)gradu(x))·n(x) = 0 on∂Ω\Γ 0 .
J M (u) := ∫ Ω
1
2 σ(x)‖gradu‖2 −f(x)u(x)dx . (2.1.3)
2.4
p. 238
2.4
p. 24

Boundary value problem for membrane clamped atΓ 0 ⊂ ∂Ω
u = g onΓ 0 ,
−div(σ(x)gradu) = f inΩ ,
(σ(x)gradu)·n = 0 on∂Ω\Γ 0 .
(2.4.21)
Numerical
Methods
for PDEs
Heat flux = power flux: [j] = W m 2
Numerica
Methods
for PDEs
(2.4.21) = Second-order elliptic BVP with Neumann boundary conditions on∂Ω\Γ 0
Short name for BVPs of the type (2.4.21):
“Mixed Neumann–Dirichlet problem”
Vector field j : Ω :=]0,1[ 2 ↦→ R 3 ✄
normal vectorn
Σ
1
0.9
0.8
2.5 Diffusion models (Stationary heat conduction)
△
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Total heat flux through oriented surfaceΣ ⊂ R 3
∫
Power P Σ = j·ndS . (2.5.1)
Σ
x 3
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.9
0.8
0.7
0.6
0.5
0.4
x 2
0.3
0.2
0.1
0
1
0.8
0.6
0.4
0.2
0
x 1
Fig. 57
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Now we look at a class of physical phenomena, for which models are based on two building blocks
P Σ ([P Σ ] = 1W): directed total power flowing through the oriented surfaceΣper unit time. Note that
the sign ofP Σ will change when flipping the normal ofΣ!
1. a conservation principle (of mass, energy, etc.),
2. a a potential driven flux of the conserved quantity.
Mathematical modelling for these phenomena naturally involves partial differential equations in the
first steps, which are supplemented with boudary conditions. Hence, second-order elliptic boundary
value problems arise first, while variational formulations are deduced from them, thus reversing the
order of steps followed for equilibrium models in Sects. 2.1–2.4.
2.5
p. 241
Numerical
Methods
for PDEs
✬
✫
Conservation of energy
∫ ∫
j·ndS = f dx for all “control volumes”V . (2.5.2)
∂V V
power flux through surface ofV
heat production insideV
f = heat source/sink ([f] = W m 3), f = f(x) andf can be discontinuous (f ∈ C0 pw(Ω))
✩
✪
2.5
p. 24
Numerica
Methods
for PDEs
In order to keep the presentation concrete, the discussion will target heat conduction, about which
R. Hiptmair
Fundamental concept: heat flux, modelled by vector field j : Ω ↦→ R 3
everybody should have a sound “intuitive grasp”.
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
Intuition: heat flows from hot zones to cold zones
✎ notation: Ω ⊂ R 3 : bounded open region occupied by solid object
(ˆ= Ω→computational domain)
DRAFT
the larger the temperature differece, the stronger the heat flow
Experimental evidence supports this intuition and, for many materials, yields the following quantitative
relationship:
✬
Fourier’s law
✩
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.5
p. 242
✫
j(x) = −κ(x)gradu(x) , x ∈ Ω . (2.5.3)
✪
2.5
p. 24

Meaning of quantities:
j = heat flux ([j] = 1 W m 2)
u = temperature ([u] = 1K)
κ = heat conductivity ([κ] = 1 W Km )
Numerical
Methods
for PDEs
} {{ }
Combine equations (2.5.5) & (2.5.3)
Numerica
Methods
for PDEs
j = −κ(x)gradu (2.5.3) + divj = f (2.5.5)
(2.5.3)⇒Heat flow from hot to cold regions linearly proportional to gradient of temperature
Some facts about the heat conductivity:
κ: •κ = κ(x) for non-homogeneous materials (spatially varying heat conductivity)
•κcan even be discontinuous for composite materials
•κmay be R 3,3 -valued (heat conductivity tensor)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
−div(κ(x)gradu) = f inΩ . (2.5.6)
Linear scalar second order elliptic PDE (for unknown temperatureu)
2.6 Boundary conditions
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
The most general form of the heat conductivity (tensor) enjoys the very same properties as the dielectric
tensor introduced in Sect. 2.1.2:
2.5
p. 245
2.6
p. 24
From thermodynamic principles, cf. (2.1.8):
∃κ − ,κ + > 0: 0 < κ − ≤ κ(x) ≤ κ + < ∞ for almost allx ∈ Ω . (2.5.4)
Numerical
Methods
for PDEs
In the examples from Sects. 2.1.1, 2.1.2 we fixed the value of the unknown function u : Ω ↦→ R on
the boundary∂Ω: Dirichlet boundary conditions in (2.4.14).
Numerica
Methods
for PDEs
Exception: free edge of taut membrane, see Rem. 2.4.16: Neumann boundary conditions in
(2.4.21).
Terminology: (2.5.4) ↔ κ is bounded and uniformly positive, see Def. 2.1.11.
From 2.5.2 by Gauss’ theorem Thm. 2.4.5
∫ ∫
divj(x)dx = f(x)dx for all “control volumes” V ⊂ Ω .
V
V
Now appeal to another version of the fundamental lemma of the calculus of variations, see
Lemma 2.4.10, this time sporting piecewise constant test functions.
local form of energy conservation:
divj = f inΩ . (2.5.5)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.5
p. 246
In this section we resume the discussion of boundary conditions and examine them for stationary heat
conduction, see previous section. This has the advantage that for this everyday physical phenomenon
boundary conditions have a very clear intuitive meaning.
Boundary conditions on surface/boundary∂Ω ofΩ:
(i) Temperatureuis fixed:
(ii) Heat fluxjthrough∂Ω is fixed:
vectorfield) on∂Ω
withg : ∂Ω ↦→ R prescribed
u = g on∂Ω . (2.6.1)
Dirichlet boundary conditions
with h : ∂Ω ↦→ R prescribed (n : ∂Ω ↦→ R 3 exterior unit normal
j·n = −h on ∂Ω . (2.6.2)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.6
p. 24

Neumann boundary conditions
(iii) Heat flux through∂Ω depends on (local) temperature:
Example 2.6.4 (Convective cooling (simple model)).
with increasing functionΨ : R ↦→ R
j·n = Ψ(u) on∂Ω (2.6.3)
radiation boundary conditions
Numerical
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for PDEs
Ω
Γ R
Γ N
Γ D
Example 2.6.7 (“Wrapped rock on a stove”).
• Non-homogeneous Dirichlet boundary conditions onΓ D ⊂ ∂Ω
• Homogeneous Neumann boundary condtions onΓ N ⊂ ∂Ω
• Convective cooling boundary conditions onΓ R ⊂ ∂Ω
Partition: ∂Ω = Γ D ∪Γ N ∪Γ R , Γ D ,Γ N ,Γ R mutually disjoint
} {{ }
✸
Numerica
Methods
for PDEs
Heat is carried away from the surface of the body by a fluid at bulk temperature u 0 . A crude model
assumes that the heat flux depends linearly on the temperature difference between the surface of Ω
and the bulk temperature of the fluid.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
−div(κ(x)gradu) = f + boundary conditions ⇒ elliptic boundary value problem (BVP)
★
For second order elliptic boundary value problems exactly one boundary condition is needed on
any part of∂Ω.
✧
✥
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
✦A. Chern
DRAFT
j·n = q(u−u 0 ) on∂Ω , where0 < q − ≤ q(x) ≤ q + < ∞ for almost allx ∈ ∂Ω .
Remark 2.6.8 (Linear BVP).
Observe that the solution mapping
(
f
g)
↦→ u for (2.5.6), (2.6.1) is linear.
Example 2.6.5 (Radiative cooling (simple model)).
✸
2.6
p. 249
Numerical
Methods
for PDEs
This means that ifu i solves the Dirichlet problem with source functionf i and Dirichlet datag i ,i = 1,2,
thenu 1 +u 2 solves (2.5.6) & (2.6.1) for sourcef 1 +f 2 and boundary valuesg 1 +g 2 .
△
2.6
p. 25
Numerica
Methods
for PDEs
A hot body emits electromagnetic radiation (blackbody emission), which drains thermal energy. The
radiative energy loss is roughly proportional to the 4th power of the temperatur difference between
the surface temperature of the body and the ambient temperature.
2.7 Characteristics of elliptic boundary value problems
Qualitative insights gained from heat conduction model:
→
j·n = α|u−u 0 |(u−u 0 ) 3 on∂Ω , withα > 0
Non-linear boundary condition
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
continuity: the temperatureumust be continuous (jump inu→j = ∞).
normal component ofjacross surfaces insideΩmust be continuous
(jump inj·n→heat sourcef of infinite intensity).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Terminology:
Ifg = 0 orh = 0→homogeneous Dirichlet or Neumann boundary conditions
interior smoothness ofu: u smooth wheref andD smooth.
Remark 2.6.6 (Mixed boundary conditions).
non-locality: local alterations inf,g,haffectueverywhere inΩ.
Different boundary conditions can be prescribed on different parts of∂Ω
(→ mixed boundary conditions, cf. Rem. 2.4.16)
△
2.6
p. 250
quasi-locality: If local changes inf,g,h confined toΩ ′ ⊂ Ω, their effects decay away fromΩ ′ .
2.7
p. 25

f
maximum principle: (in the absence of heat sources extremal temperatures are on the boundary)
✤
✣
iff ≡ 0, then
inf u(y) ≤ u(x) ≤ sup u(y) for allx ∈ Ω
y∈∂Ω y∈∂Ω
} {{ }
Typical features of solutions of elliptic boundary value problems
Example 2.7.1 (Scalar elliptic boundary value problem in one space dimension).
Poisson equation→(2.4.13) in 1D:
−u ′′ = f
➣ f discontinuous, piecewiseC 0 ⇒ u ∈ C 1 , piecewiseC 2 ✸
✜
✢
Numerical
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for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
⎧
⎨
δ −2 ∥
, if ∥x− ( f δ (x) =
1/2) ∥ ∥2 ≤ δ ,
, δ > 0 . (2.7.6)
⎩0 elsewhere.
u(x,0.5)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
Cross−section of solution u
δ = 0.400000
δ = 0.200000
δ = 0.100000
δ = 0.050000
δ = 0.025000
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Example 2.7.2 (Smoothness of solution of scalar elliptic boundary value problem).
−∆u = f(x) inΩ :=]0,1[ 2 , u = 0 on∂Ω , (2.7.3)
f(x) := sign(sin(2πk 1 x 1 )sin(2πk 2 x 2 )) , x ∈ Ω , k 1 ,k 2 ∈ N .
Fig. 60
0
0 0.2 0.4 0.6 0.8 1
x
Fig. 61
✸
Approximate solution computed by means of linear Lagrangian finite elements + lumping
(→ Ch. 3, details in Sect. 3.2, 3.5.4)
2.7
p. 253
Numerical
Methods
for PDEs
2.8 Second-order elliptic variational problems
2.8
p. 25
Numerica
Methods
for PDEs
1
0.5
0
In Ch. 1 and Sects. 2.1–2.4 we pursued the derivation:
−0.5
−1
1
0.8
0.6
0.4
0.6
0.4
0.2
0.2
0 0
y
x
Source termf(x),k 1 = k 2 = 2
0.8
1
Fig. 58
Solution of (2.7.3)
Fig. 59
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Minimization problem
(e.g., (2.1.4), (2.1.14))
Now we are proceeding in the opposite direction:
PDE
(e.g. (2.5.6))
+
➣
Variational problem
(e.g., (2.3.1), (2.3.2))
boundary conditions
(e.g., (2.6.1), (2.6.2), (2.6.3))
➣
➣
BVP for PDE
(e.g., (2.4.14), (2.4.21))
variational problem
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
➣ “Smooth”udespite “rough”f !
✸
Example 2.7.4 (Quasi-locality of solution of scalar elliptic boundary value problem).
Formal approach:
−∆u = f δ (x) inΩ :=]0,1[ 2 , u = 0 on∂Ω , (2.7.5)
2.7
p. 254
STEP 1:
test PDE with smooth functions
(do not test, where the solution is known, e.g., on the boundary)
2.8
p. 25

STEP 2:
STEP 3:
STEP 4:
integrate over domain
perform integration by parts
(e.g. by using Green’s first formula, Thm. 2.4.7)
[optional] incorporate boundary conditions into boundary terms
Example 2.8.1 (Variational formulation for heat conduction with Dirichlet boundary conditions).
STEP 1 & 2:
BVP: −div(κ(x)gradu) = f inΩ , u = g on∂Ω . (2.8.2)
test withv ∈ C0 ∞(Ω)
∫
∫
− div(κ(x)gradu)vdx = fvdx . (2.8.3)
Ω
Ω
Note: v |∂Ω = 0 for test function, becauseualready fixed on∂Ω.
STEP 3: use Green’s formula from Thm. 2.4.7 on Ω ⊂ R d (multidimensional integration by parts):
Apply (2.4.8) to (2.8.3) withj := κ(x)gradu:
∫
∫
∫
κ(x)gradu·gradvdx− κ(x)gradu·nvdS = fvdx ∀v ∈ C0 ∞ (Ω) .
Ω
∂Ω
Ω
} {{ }
=0,because v |∂Ω =0
This gives the variational formulation after we switch to “maximal admissible function spaces”
(Sobolev spaces, see Sect. 2.2, as spaces of functions with finite energy)
✬
Variational form of (2.8.2): seek
u ∈ H 1 ∫
(Ω)
u = g on∂Ω
∫Ω
: κ(x)gradu·gradvdx = fvdx ∀v ∈ H0 1 (Ω) .
Ω
✫
Numerical
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C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.8
p. 257
R. Hiptmair
✩C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
✪
Numerical
Methods
for PDEs
DRAFT
BVP: −div(κ(x)gradu) = f inΩ , −κ(x)gradu·n = Ψ(u) on∂Ω . (2.8.6)
STEP 1 & 2: u |∂Ω not fixed ⇒ test withv ∈ C ∞ (Ω)
∫
∫
− div(κ(x)gradu)vdx = fvdx ∀v ∈ C ∞ (Ω) .
Ω
Ω
STEP 3 & 4: apply Green’s first formula (2.4.8) and incorporate boundary conditions:
∫
∫
∫
κ(x)gradu·gradvdx− κ(x)gradu·n vdS = fvdx ∀v ∈ C ∞ (Ω) .
Ω
∂Ω} {{ } Ω
=−Ψ(u) (STEP 4)
✬
Variational formulation of (2.8.6): seek
∫
∫ ∫
u ∈ H 1 (Ω): κ(x)gradu·gradvdx+ Ψ(u)vdS = fvdx ∀v ∈ H 1 (Ω) .
Ω
∂Ω Ω
(2.8.7)
✫
✬
Theorem 2.8.8. If κ ∈ C 1 (Ω), classical solutions u ∈ C 2 (Ω) of the boundary value problems
(2.8.2) and (2.8.6) also solve the associated variational problems.
✫
Proof. Apply Theorem 2.4.7 as in the derivation of the weak formulations.
Example 2.8.9 (Variational formulation for Neumann problem).
2nd-order elliptic (inhomogeneous) Neumann problem
BVP:
−div(κ(x)gradu) = f inΩ ,
κ(x)gradu·n = h(x) on∂Ω .
DRAFT
✩
✪
✸
(2.8.10)
✩
✪
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
2.8
p. 25
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
(2.8.4)
✸
We confront Neumann boundary conditions (2.6.2) (prescribed heat flux) on the whole boundary.
Example 2.8.5 (Variational formulation: heat conduction with general radiation boundary conditions).
2.8
p. 258
Variational formulation derived as in Ex. 2.8.5, withΨ(u) = −h.
∫
∫ ∫
u ∈ H 1 (Ω): κ(x)gradu·gradvdx− hvdS = fvdx ∀v ∈ H 1 (Ω) . (2.8.11)
Ω
∂Ω Ω
2.8
p. 26

Observation: when we test (2.8.7) with v ≡ 1 − ∫ ∂Ω hdS = ∫ Ωf dx (2.8.12)
This is a compatibility condition for the existence of (variational) solutions of the Neumann problem!
Numerical
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Variational formulation of Neumann problem:
u ∈ H 1 ∗(Ω):
∫
Ω
∫ ∫
κ(x)gradu·gradvdx = fvdx+ hvdS ∀v ∈ H∗(Ω) 1 . (2.8.15)
Ω ∂Ω
Numerica
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for PDEs
Interpretation of (2.8.12) against the backdrop of the stationary heat conduction model:
△
conservation of energy → (2.5.2): Heat generated inside Ω (↔ f) must be offset by heat flux
through∂Ω (→h).
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.9 Essential and natural boundary conditions
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Remark 2.8.13 (Uniqueness of solutions of Neumann problem).
Synopsis:
2.8
p. 261
☛
2nd-order elliptic Dirichlet problem:
−div(α(x)gradu) = f inΩ , u = g on∂Ω . (2.4.14)
2.9
p. 26
Observation: if compatibility condition (2.8.12) holds true, then
v ∈ H 1 (Ω) solves (2.8.7) ⇐⇒ v +γ solves (2.8.7) ∀γ ∈ R ,
we say, “the solution is unique only up to constants”.
Numerical
Methods
for PDEs
with variational formulation
u ∈ H 1 ∫
∫
(Ω) ,
u = g on ∂Ω : (α(x)gradu(x))·gradv(x)dx =
Ω
Ω
☛ 2nd-order elliptic Neumann problem:
f(x)v(x)dx ∀v ∈ H0 1 (Ω) . (2.3.3)
Numerica
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for PDEs
Complementary observation: a(u,v) := ∫ Ωκ(x)gradu·gradvdx is not s.p.d (→ Def. 2.1.25) on
H 1 (Ω).
Idea:
Restore uniqueness of solutions by
enforcing average temperature to be zero
∫
u(x)dx = 0
Ω
This amounts to posing the variational problem (2.8.7) over the constrained function space
∫
H∗(Ω) 1 := {v ∈ H 1 (Ω): v(x)dx = 0} . (2.8.14)
Ω
The norm onH∗(Ω) 1 is the same as onH0 1 (Ω), see Def. 2.2.12. Obviously (why ?), the norm property
(N1) is satisfied. These arguments also show thatais s.p.d (→ Def. 2.1.25) onH∗(Ω), 1 cf. Thm. 2.9.6.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.8
p. 262
−div(α(x)gradu) = f inΩ , (α(x)gradu)·n = −h on∂Ω . (2.9.1)
with variational formulation
∫
∫
u ∈ H∗(Ω):
1 α(x)gradu·gradvdx =
Ω
Ω
☛ 2nd-order elliptic mixed Neumann-Dirichlet problem, see Rem. 2.4.16:
−div(α(x)gradu) = f inΩ ,
with variational formulation
u ∈ H 1 ∫
∫
(Ω) ,
: (α(x)gradu(x))·gradv(x)dx =
u = g on Γ 0
Ω
Ω
for allv ∈ H 1 (Ω) withv |Γ0 = 0.
∫
fvdx+ hvdS ∀v ∈ H∗(Ω) 1 . (2.8.15)
∂Ω
u =g onΓ 0 ⊂ ∂Ω ,
(α(x)gradu)·n = −h on∂Ω\Γ 0 .
(2.9.2)
∫
f(x)v(x)dx+ hvdS (2.9.3)
∂Ω\Γ 0
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.9
p. 26

In the variational formulations of 2nd-order elliptic BVPs of Sect. 2.8:
Numerical
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for PDEs
Remark 2.9.5 (Admissible Neumann data).
Numerica
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Dirichlet boundary conditions are directly imposed on trial space and (in homogeneous form)
on test space.
Terminology:
essential boundary conditions
Neumann boundary conditions are enforced only through the variational equation.
Terminology:
natural boundary conditions
The attribute “natural” has been coined, because Neumann boundary conditions “naturally” emerge
when removing constraints on the boundary, as we have seen for the partially free membrane of
Rem. 2.4.16.
Remark 2.9.4 (Admissible Dirichlet data).
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C. Schwab,
H.
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V.
Gradinaru
A. Chernov
DRAFT
In the variational problem (2.8.15) Neumann data h : ∂Ω ↦→ R enter through the linear form on the
right hand side
∫
l(v) :=
Ω
∫
f(x)v(x)dx+
∂Ω
h(x)v(x)dS(x) .
Remember the discussion in the beginning of Sect. 2.2, also Rem. 2.3.16: we have to establish that
l is continuous on H 1 ∗(Ω) defined in (2.8.14). This is sufficient, because the coefficient function κ is
uniformly positive and bounded, see (2.5.4). Thus, the enery‖·‖ a associated with the bilinear form
∫
a(u,v) = κ(x)gradu·gradvdx
Ω
can be bounded from above and below by|·| H 1 (Ω), cf. the estimate (2.3.14).
✬
Theorem 2.9.6. Second Poincaré-Friedrichs inequality]
IfΩ ⊂ R d ,d ∈ N, is bounded, then
✩
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
2.9
p. 265
✫
∃C = C(Ω) > 0: ‖u‖ 0 ≤ C diam(Ω)‖gradu‖ 0 ∀u ∈ H 1 ∗(Ω) .
✪
2.9
p. 26
Requirement for “Dirichlet data”g : ∂Ω ↦→ R in (2.4.14):
there isu ∈ H 1 (Ω) such that
u |∂Ω = g
Numerical
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✎ notation: C = C(Ω) indicates that the constantC may depend on the shape of the domain
Ω.
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Analoguous to Thm. 2.2.17:
★
If g : ∂Ω ↦→ R is piecewise continuously differentiable (and bounded with bounded piecewise
derivatives), then it can be extended to anu 0 ∈ H 1 (Ω), if and only if it is continuous on∂Ω.
✧
Bottom line:
Dirichlet boundary values have to be continuous
This is also stipulated by physical insight, e.g. in the case of the taut membrane model of Sect. 2.1.1:
discontinuous displacement on∂Ω would entail ripping apart the membrane.
✥
R. Hiptmair
C. Schwab,
H.
✦Harbrecht
V.
Gradinaru
A. Chernov
△
DRAFT
2.9
p. 266
Proof. (ford = 1,Ω = [0,1] only, technically difficult in higher dimensions, see [5, Thm. 1.6.6])
As in the proof of Thm. 2.2.16, we employ a density argument and assume that u is sufficiently
smooth,u ∈ C 1 ([0,1]).
By the fundamental theorem of calculus (2.4.1)
∫x
du
u(x) = u(y)+ (τ)dτ , 0 ≤ x,y ≤ 1 .
dx
y
∫1
u(x) =
∫1
u(x)dy =
∫1 ∫x
u(y)dy+
0 0
} {{ }
=0
Then use the Cauchy-Schwarz inequality (2.2.15)
0
y
du
dx
(τ)dτ dy .
∫1 ∫x ∫1 ∫x
∣
u(x) 2 ≤ 1dτ dy
du ∣∣∣ 2 ∫1
∣
∣dx (τ) dτ dy ≤
du ∣∣∣ 2
∣dx (τ) dτ .
0 y 0 y 0
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2.9
p. 26

Integrate overΩyields the estimate
∫1
‖u‖ 2 0 =
0
∫1
∣
u 2 (x)dx ≤
du ∣∣∣ 2
∣dx (τ) dτ = |u| 2 H 1 (Ω) .
0
By (2.2.15), Thm. 2.9.6 implies the continuity of the first term inl.
Continuity of the boundary contribution tolhinges on a trace theorem
✬
Theorem 2.9.7 (Multiplicative trace inequality).
✫
∃C = C(Ω) > 0: ‖u‖ 2 L 2 (∂Ω) ≤ C‖u‖ L 2 (Ω)·‖u‖ H 1 (Ω) ∀u ∈ H 1 (Ω) .
(✷)
R. Hiptmair
C. Schwab,
H.
✩Harbrecht
V.
Gradinaru
A. Chernov
✪
Numerical
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for PDEs
DRAFT
∫
∂Ω
✗
✖
the 2nd Poincaré-Friedrichs inequality of Thm. 2.9.6,
the multiplicative trace inequality of Thm. 2.9.7:
hvdS (2.2.15)
≤ ‖h‖ L 2 (∂Ω) ‖v‖ L 2 (∂Ω)
Thm. 2.9.7
≤ ‖h‖ L 2 (∂Ω) ‖v‖ H 1 (Ω)
Thm. 2.9.6
≤ ‖h‖ L 2 (∂Ω) |v| H 1 (Ω) ∀v ∈ H 1 ∗(Ω) .
h ∈ L 2 (∂Ω) provides valid Neumann data for the 2nd order elliptic BVP (2.9.1).
✔
✕
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
Proof.
(ford = 1,Ω = [0,1] only, technically difficult in higher dimensions)
In, particular Neumann datahcan be discontinuous.
As in the proof of Thms. 2.2.16, 2.9.6, we employ a density argument and assume thatuis sufficiently
smooth,u ∈ C 1 ([0,1]).
2.9
p. 269
△
2.9
p. 27
By the fundamental theorem of calculus (2.4.1):
∫1
u(1) 2 dw
=
dξ (x)dx , with w(ξ) := ξu2 (ξ) ,
0
∫1
u(1) 2 =
Then use the Cauchy-Schwarz inequality (2.2.15)
∫1
u(1) 2 ≤
0
∫1
u 2 (x)dx+2
A similar estimate holds foru(0) 2 .
0
0
u 2 (x)+2u(x) du
dx (x)xdx .
∣ |x||u(x)|
du ∣∣∣
∣dx (x) dx ≤ ‖u‖ 2 0 +2‖u‖ du
0∥dx∥ .
0
✷
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3 Finite Element Methods (FEM)
CORE CHAPTER
Relies heavily on Sects. 2.2, 2.8, 2.9
In this chapter:
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Now we can combine
Problem : linear scalar second-order elliptic boundary value problem → Ch. 2
Perspective : variational interpretation in Sobolev spaces → Sect. 2.8
Objective : algorithm for the computation of an approximate numerical solution
the Cauchy-Schwarz inequality (2.2.15) on∂Ω,
2.9
p. 270
3.0
p. 27

Numerical
Methods
for PDEs
with reaction coefficientc : Ω ↦→ R + ,c ∈ C 0 pw(Ω). Note that no compatibilty conditions is required in
this case.
Numerica
Methods
for PDEs
Preface
Sect. 1.5.1 introduced the fundamental ideas of the Galerkin discretization of variational problems,
or, equivalently, of minmization problems, posed over function spaces. A key ingredient are suitably
chosen finite-dimensional trial and test spaces, equipped with ordered bases.
In Sect. 1.5.1.2 the abstract approach was discussed for two-point boundary value problems and
the concrete case of piecewise linear trial and test spaces, built upon a partition (mesh/grid) of the
interval (domain). In this context the locally supported tent functions lent themselves as natural basis
functions.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Rem. 1.5.3 still applies: all functions (coefficientα, source functionf, Dirichlet datag) may be given
only in procedural form. Recall the discussion of the consequences in Rem. 1.5.36 and Sect. 1.5.1.2
3.1 Galerkin discretization
Recall the concept of “discretization”, see Sect. 1.5:
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
This chapter is devoted to extending the linear finite element method in 1D to
Not a moot point:
any computer can only handle a finite amount of information (reals)
2nd-order linear variational problems on bounded spatial domainsΩin two and three dimensions,
piecewise polynomial trial/test functions of higher degree.
The leap fromd = 1 tod = 2 will encounter additional difficulties and many new aspects will emerge.
This chapter will elaborate on them and present policies how to tackle them.
3.0
p. 273
Numerical
Methods
for PDEs
Variational boundary value
problem
Targetted: linear variational problem (1.4.6)
DISCRETIZATION
−−−−−−−−−−−−−−−→
System of a finite number of
equations for (real) unknowns
u ∈ V 0 : a(u,v) = l(v) ∀v ∈ V 0 , (3.1.1)
3.1
p. 27
Numerica
Methods
for PDEs
Throughout, we will restrict ourselves to linear 2nd-order elliptic variational problems on spatial
domainsΩ ∈ R d ,d = 2,3, with the properties listed in Rem. 2.1.1.
☛ 2nd-order elliptic Dirichlet problem:
u ∈ H 1 ∫
∫
(Ω) ,
u = g on ∂Ω : (α(x)gradu(x))·gradv(x)dx =
Ω
Ω
with continuous (→ Rem. 2.9.4) Dirichlet datag ∈ C 0 (∂Ω).
f(x)v(x)dx ∀v ∈ H0 1 (Ω) , (2.3.3)
☛ 2nd-order elliptic Neumann problems:
∫
∫ ∫
u ∈ H∗(Ω):
1 (α(x)gradu)·gradvdx = fvdx+ hvdS ∀v ∈ H∗(Ω) 1 , (2.8.15)
Ω
Ω ∂Ω
with piecewwise continuous (→ Rem. 2.9.5) Neumann data h ∈ C 0 pw(∂Ω) that satisfy the
compatibility condition (2.8.12).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
V 0 ˆ= vector space (Hilbert space) (usually a Sobolev space→Sect. 2.2) with norm‖·‖ V ,
a(·,·) ˆ= bilinear form, continuous onV 0 , which means
∃C > 0: |a(u,v)| ≤ C‖u‖ V ‖v‖ V ∀u,v ∈ V . (3.1.2)
l ˆ= continuous linear form in the sense of, cf. (2.2.1),
∃C > 0: |l(v)| ≤ C‖v‖ V ∀v ∈ V 0 . (3.1.3)
Ifais symmetric and positive definite (→ Def. 2.1.25), we may choose‖·‖ V := ‖·‖ a , “energy norm”,
see Def. 2.1.27. Continuity ofaw.r.t. ‖·‖ a is clear.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
A simpler version with homogeneous Neumann data and reaction term:
∫
∫
u ∈ H 1 (Ω): α(x)gradu·gradv +c(x)uvdx = fvdx∀v ∈ H 1 (Ω) , (3.0.1)
Ω
Ω
3.0
p. 274
3.1
p. 27

Recall from Sect. 1.5.1:
Numerical
Methods
for PDEs
of equations. It will turn out that under the assumptions of the theorem, the resulting system matrix
will be symmetric and positive definite in the sense of [18, Def. 2.7.9].
Numerica
Methods
for PDEs
Idea of Galerkin discretization
ReplaceV 0 in (3.1.1) with a finite dimensional subspace.
(V 0,N ⊂ V 0 called Galerkin (or discrete) trial space/test space)
The estimate (3.1.6) is immediate from settingv N := u N in (3.1.4)
|a(u N ,u N )| = |l(u N )| ≤ C l (a(u N ,u N )) 1/2 .
✷
Twofold nature of symbol “N”:
N = formal index, tagging “discrete entities” (→ “finite amount of information”)
N = dimV N,0 ˆ= dimension of Galerkin trial/test space
Discrete variational problem, cf. (1.5.7),
✬
u N ∈ V 0,N : a(u N ,v N ) = l(v N ) ∀v N ∈ V 0,N . (3.1.4)
Galerkin solution
✩
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.1
p. 277
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
R. Hiptm
Recall from Sect. 1.5.1:
DRAFT
2nd step of Galerkin discretization:
Introduce (ordered) basisB N ofV 0,N :
B N := {b 1 N ,...,bN N } ⊂ V N , V N = Span{B N } , N := dim(V N ) .
Unique basis representations:
u N = µ 1 b 1 N +···+µ Nb N N , µ i ∈ R
v N = ν 1 b 1 N +···+ν Nb N N , ν i ∈ R : plug into (3.1.4). 3.1
p. 27
Theorem 3.1.5 (Existence and uniqueness of solutions of discrete variational problems).
If the bilinear form a : V 0 ×V 0 ↦→ R is symmetric and positive definite (→ Def. 2.1.25) and the
linear forml : V 0 ↦→ R is continuous in the sense of
∃C l > 0: |l(u)| ≤ C l ‖u‖ a ∀u ∈ V 0 , (2.2.1)
then the discrete variational problem has a unique Galerkin solution u N ∈ V 0,N that satisfies
the stability estimate (→ Sect. 2.3.2)
Numerical
Methods
for PDEs
Of course, there are infinitely many ways to choose the basisB N . Below we will study the impact of
different choices.
What follows repeats the derivation of (1.5.18) and, in particular, (1.5.43).
Numerica
Methods
for PDEs
✫
Proof.
Uniqueness ofu N is clear:
‖u N ‖ a ≤ C l . (3.1.6)
a(u N ,v N ) = l(v N ) ∀v N ∈ V 0,N
a(w N ,v N ) = l(v N ) ∀v N ∈ V 0,N
⇒ a(u N −w N ,v N ) = 0 ∀v N ∈ V N,0
v N :=u N −w N ∈V 0,N
=⇒ ‖u N −w N ‖ a = 0 a s.p.d.
=⇒ u N −w N = 0 .
The discrete linear variational problem (3.1.4) is set in the finite-dimensional space V 0,N .
uniqueness of solutions is equivalent to existence of solutions (→ linear algebra).
Thus,
If you do not like this abstract argument, wait and see the equivalence of (3.1.4) with a linear system
R. Hiptmair
C. Schwab,
✪H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.1
p. 278
u N ∈ V 0,N : a(u N ,v N ) = l(v N ) ∀v N ∈ V 0,N . (3.1.4)
N∑ N∑
µ k ν j a(b k N ,bj N ) = ∑ N ν j l(b j N ) ∀ν 1,...,ν N ∈ R ,
k=1j=1
j=1
[ u N = µ 1 b 1 N +···+µ Nb N N ,µ i ∈ R
v N = ν 1 b 1 N +···+ν Nb N N ,ν i ∈ R ]
⎛
⎞
N∑ N∑
ν j
⎝ µ k a(b k N ,bj N )−l(bj N ) ⎠ = 0 ∀ν 1 ,...,ν N ∈ R ,
j=1 k=1
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.1
p. 28

✬
✫
N∑
µ k a(b k N ,bj N ) = l(bj N ) for j = 1,...,N .
k=1
A⃗µ = ⃗ϕ , with
A linear system of equations
Linear discrete variational problem
u N ∈ V 0,N : a(u N ,v N ) = l(v N ) ∀v N ∈ V 0,N
(
Galerkin matrix: A =
Right hand side vector: ⃗ϕ =
[⃗µ = (µ 1 ,...,µ N ) ⊤ ∈ R N ]
(
A = a(b k ) N
N ,bj N ) j,k=1 ∈ RN,N ,
(
⃗ϕ = l(b j ) N
N ) j=1 .
R. Hiptmair
✩C. Schwab,
H.
Harbrecht
V.
Linear system
Gradinaru
A. Chernov
Choosing basis B
−−−−−−−−−−−−→ N
of equations
DRAFT
✫
A⃗µ = ⃗ϕ
k=1 µ kb k N .
a(b k ) N
N ,bj N ) j,k=1 ∈ RN,N ,
(
l(b j ) N
N ) j=1 ∈ RN ,
Coefficient vector: ⃗µ = (µ 1 ,...,µ N ) ⊤ ∈ R N ,
Recovery of solution: u N = ∑ N
✪
Numerical
Methods
for PDEs
3.1
p. 281
✬
Lemma 3.1.9. Consider (3.1.4) and two bases ofV 0,N ,
related by
Proof.
B N := {b 1 N ,...,bN N } , B N := {b 1 N ,...,bN N } ,
b j N N = ∑
s jk b k N with
k=1
S = (s jk) N j,k=1 ∈ KN,N regular.
Galerkin matricesA,A ∈ K N,N , right hand side vectors ⃗ϕ,⃗ϕ ∈ K N ,
and coefficient vectors ⃗µ,⃗µ ∈ R N , respectively, satisfy
Make use of the bilinearity ofa:
A = SAS T , ⃗ϕ = S⃗ϕ , ⃗µ = S −T ⃗µ . (3.1.10)
A lm = a(b m N N ,bl N ) = ∑ N∑
N∑
s mk a(b k (
N ,bj N )s ∑ N )
lj = s lj A jk s mk = (SAS T ) lm ,
k=1j=1
k=1 j=1
} {{ }
(SA) lk
✩
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✪
3.1
p. 28
(Legacy) terminology for FEM: Galerkin matrix = stiffness matrix
Right hand side vector = load vector
Galerkin matrix for(u,v) ↦→ ∫ Ω uvdx = mass matrix
✤
Corollary 3.1.7. (3.1.4) has unique solution ⇔ A nonsingular
✜
Numerical
Methods
for PDEs
Reminder of linear algebra:
nn
Definition 3.1.11 (Congruent matrices).
Two matricesA ∈ K N,N ,B ∈ K N,N ,N ∈ N, are called congruent, if there is a regular matrix
S ∈ K N,N such thatB = SAS H .
Numerica
Methods
for PDEs
✣
Remark 3.1.8 (Impact of choice of basis).
Choice of B N in theory does not affectu N ⇒ No impact on discretization error !
✢
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✬
Lemma 3.1.12.
Equivalence relation on square matrices
Matrix property invariant under Property of Galerkin matrix invariant
⇔
congruence
under change of basisB N
R. Hiptm
C. Schwa
✩H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
But: Key properties (e.g., conditioning) of matrixAcrucially depend on basisB N !
✫
Matrix properties invariant under congruence:
• regularity → [18, Def. 2.0.5]
• symmetry
• positive definiteness → [18, Def. 2.7.9]
✪
3.1
p. 282
△
3.2
p. 28

3.2 Case study: Triangular linear FEM in two dimensions
Numerical
Methods
for PDEs
What is the 2D counterpart of mesh/gridMfrom Sect. (1.5.1.2) ?
Numerica
Methods
for PDEs
TriangulationMofΩ:
This section elaborates how to extend the linear finite element Galerkin discretization of Sect. 1.5.1.2
to two dimensions. Familiarity with the 1D setting is essential for understanding the current section.
Initial focus: well-posed 2nd-order linear variational problem posed onH 1 (Ω) (→ Def. 2.2.12)
Example: Neuman problem with homogeneous Neumann data and reaction term
∫
∫
u ∈ H 1 (Ω): α(x)gradu·gradv +c(x)uvdx = fvdx∀v ∈ H 1 (Ω) , (3.0.1)
Ω
Ω
⇕← see Sect. 2.4
−div(α(x)gradu)+c(x)u = f inΩ ,
BVP:
gradu·n = 0 on∂Ω .
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Ω
Fig. 62
(i) M = {K i } M i=1 ,M ∈ N,K i ˆ= open triangle
(ii) disjoint interiors: i ≠ j ⇒K i ∩K i = ∅
M⋃
(iii) tiling property: K i = Ω
i=1
(iv) intersectionK i ∩K j ,i ≠ j,
is – either∅
✎ notation: ˆ= a subset of R d together with its boundary (“closure”)
– or an edge of both triangles
– or a vertex of both triangles
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.2
p. 285
Parlance: vertices of triangles = nodes of mesh (= setV(M))
3.2
p. 28
Numerical
Methods
for PDEs
Ω
Numerica
Methods
for PDEs
Assumptions on domainΩ ⊂ R 2 ,
see Rem. 2.1.1:
Ω
Ω is a polygon
polygon with 10 corners
✄
A mesh that does not comply with the property (iv)
from above.
✄
By default, the domainΩis assumed to be an open
set, that is,x ∈ Ω impliesx ∉ ∂Ω!
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Fig. 63
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.2.1 Triangulations
3.2.2 Linear finite element space
3.2
p. 286
3.2
p. 28

Next goal: generalize the spline space S 0 1 (M) ⊂ H1 ([a,b]) of piecewise linear functions on a 1D
gridM, see Fig. 25, that was used as Galerkin trial/test space in 1D in Sect. 1.5.1.2
V 0,N = S 0 1,0 (M) := {
v ∈ C 0 ([0,1]): v |[xi−1 ,x i ] linear, i = 1,...,M,v(a) = v(b) = 0 }
.
d = 1 d = 2
Numerical
Methods
for PDEs
Remark 3.2.1 (Piecewise gradient).
Thm. 2.2.17
⇒ S 0 1 (M) ⊂ H1 (Ω)
⇒ for u N ∈ S1 0(M) the gradient gradu N can be computed on each triangle as
piecewise constant function, cf. Ex. 2.2.20.
(OnK ∈ M: grad(α K +β K ·x) = β K )
△
Numerica
Methods
for PDEs
Grid/mesh cells: intervals]x i−1 ,x i [,i = 1,...,M trianglesK i ,i = 1,...,M
Linear functions: x ∈ R ↦→ α+β ·x,α,β ∈ R x ∈ R 2 ↦→ α+β ·x,α ∈ R,β ∈ R 2
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.2.3 Nodal basis functions
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
V 0,N = S 0 1 (M) := {v ∈ C 0 (Ω): ∀K ∈ M:
}
v |K (x) = α K +β K ·x,
α K ∈ R,β K ∈ R 2 ⊂H 1 (Ω)
,x ∈ K
see Thm. 2.2.17
Next goal: generalization of “tent functions”, see (1.5.53).
3.2
p. 289
Recall condition (1.5.54), which defines a tent function in the space S1 0 (M). This approach carries
over to 2D.
3.2
p. 29
Functions of the formx ↦→ α K +β K ·x,α K ∈ R,β K ∈ R 2 are called (affine) linear.
✎ notation:
S 0 1 (M)continuous functions, cf. C0 (Ω)
locally 1st degree polynomials
scalar functions
Numerical
Methods
for PDEs
Idea: define (?) basis function
b x N ,x ∈ V(M), by
b x N (y) = {
1 , ify = x ,
0 , ify ∈ V(M)\{x} .
(3.2.2)
Numerica
Methods
for PDEs
Fig. 64
R.
✁ continuous piecewise affine linear function ∈
S1 0 (M) on a triangular meshM
Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.2
p. 290
Is this possible ?
R. Hiptm
C. Schwa
H.
Fig. 65
Harbrech
V.
Gradinar
A. Chern
DRAFT
there is exactly one plane through three non-collinear points inR 3 . The graph of a linear
v N ∈ S1 0(M) uniquely determined by{v N(x), x node ofM}!
dimS1 0 (M) = ♯V(M) (V(M) = set of nodes (= vertices of triangles) ofM) 3.2
Reasoning:
function R 2 ↦→ R is a plane.
➣
On a triangle K with vertices a 1 ,a 2 ,a 3 : (affine) linear q : K ↦→ R uniquely determined by
valuesq(a i ).
p. 29

Numerical
Methods
for PDEs
Numerica
Methods
for PDEs
Writing V(M) = {x 1 ,...,x N }, the nodal basisB N := {b 1 N ,...,bN N } ofS0 1 (M)
is defined by the conditions
{
b i 1 , ifi = j ,
N (xj ) = i,j ∈ {1,...,N} .
0 else,
(3.2.3)
Ω
✁ “Location” of nodal basis functions:
(meshM→Fig. 149)
•,•→nodal basis functions ofS1 0(M)
•→nodal basis functions ofS1,0 0 (M)
Ordering (↔ numbering) of nodes assumed !
Piecewise linear nodal basis function
(“hat function”/ “tent function”)
N∑
u N = µ i b i N ∈ S0 1 (M)
i=1
1
coefficient µ j = “nodal value” of u N at j-th
node ofM
u N (x j ) = µ j
Remark 3.2.4 (Linear finite element space for homogeneous Dirichlet problem).
Fig. 66
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.2
p. 293
Numerical
Methods
for PDEs
Fig. 67
Bottom line: the Galerkin trial/test space contained in H0 1 (Ω) is obtained by dropping all “tent functions”
that do not vanish on∂Ω from the basis.
3.2.4 Sparse Galerkin matrix
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.2
p. 29
Numerica
Methods
for PDEs
Recall that the Dirichlet problem with homogeneous boundary conditions u |∂Ω = 0 is posed on the
Sobolev spaceH0 1 (Ω) (→ Def. 2.2.10), see (2.3.3), Ex. 2.8.1.
Galerkin space for homogeneous Dirichlet b.c.:
V 0,N = S 0 1,0 (M) := S0 1 (M)∩H1 0 (Ω)
Notation: S 0 1,0 (M) zero on∂Ω, cf. H1 0 (Ω)
Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Now: a ˆ= any (symmetric) bilinear form occurring in a linear 2nd-order variational problem, most
general form
∫
a(u,v) :=
Ω
∫
(α(x)gradu)·gradv +c(x)uvdx+
∂Ω
hvdS , u,v ∈ H 1 (Ω) . (3.2.5)
b j N ˆ= nodal basis function assciated with vertexxj of triangulationMofΩ, see Sect. 3.2.3.
Note: a symmetric ⇒ symmetric Galerkin matrix
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
S1,0 0 {b (M) = Span j }
N : xj ∈ Ω (interior node !)
R.
dimS1,0 0 (M) = ♯{x ∈ V(M): x ∉ ∂Ω}
Now we study the sparsity (→ [18, Sect. 2.6]) of the Galerkin matrix A :=
R N,N ,N := dimS1 0 (M) = ♯V(M), see Sect. 3.1.
(
a(b j N ,bi N ) ) N
i,j=1 ∈
3.2
p. 294
The consideration are fairly parallel to those that made us understand that the Galerkin matrix for the
1D case was tridiagonal, see (1.5.58).
3.2
p. 29

Ω
Numerical
Methods
for PDEs
Recall from [18, Def. 2.6.1]:
Numerica
Methods
for PDEs
x j
x i
Ω
x j x i
Fig. 69
Notion 3.2.7 (Sparse matrix).A ∈ K m,n ,m,n ∈ N, is sparse, if
nnz(A) := #{(i,j) ∈ {1,...,m}×{1,...,n}: a ij ≠ 0} ≪ mn .
Fig. 68
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Sloppy parlance: matrix sparse :⇔ “almost all” entries= 0 /“only a few percent of” entries≠ 0
Galerkin discretization of a 2nd-order linear variational problems
utilizing the nodal basis ofS 0 1 (M)/S0 1,0 (M)
leads to sparse linear systems of equations.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Nodesx i ,x j ∈ V(M)
not connected by an edge
⇔ Vol(supp(b i N )∩supp(bj N )) = 0 ⇒ (A) ij = 0 .
Example 3.2.8 (Sparse Galerkin matrices).
✬
Lemma 3.2.6 (Sparsity of Galerkin matrix).
∃C = C(topology ofΩ): ♯{(i,j) ∈ {1,...,N} 2 : (A) ij ≠ 0} ≤ 7·N +C .
✫
✩
✪
3.2
p. 297
Numerical
Methods
for PDEs
M = triangular mesh, V 0,N = S1,0 0 (M), homogeneous Dirichlet boundary conditions, linear 2ndorder
scalar elliptic differential operator.
1.5
1
0.5
0
50
100
150
3.2
p. 29
Numerica
Methods
for PDEs
Proof. Euler’s formula (http://en.wikipedia.org/wiki/Euler_characteristic)
0
−0.5
200
250
♯M−♯E(M)+♯V(M) = χ Ω , χ Ω = Euler characteristic ofΩ .
Note thatχ Ω is a topological invariant (alternating sum of Betti numbers).
By combinatorial considerations (traverse edges and count triangles):
2·♯E I (M)+♯E B (M) = 3·♯M ,
−1
R. Hiptmair
C. Schwab,
H.
Harbrecht
−1.5
V.
Gradinaru
A. Chernov
Fig. 70 2.5
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2
DRAFT
Triangular meshM
300
350
400
0 50 100 150 200 250 300 350 400
nz = 2670 Fig. 71
Resulting sparsity pattern of Galerkin matrix
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
whereE I (M),E B (M) stand for the sets of interior and boundary edges ofM, respectively.
Then use
♯E I (M)+2♯E B (M) = 3(♯V(M)−χ Ω ) .
Recall: visualization of sparsity pattern by means of MATLAB spy-command.
N = ♯V(M) , nnz(A) ≤ N +2·♯E(M) ≤ 7·♯V(M)−6χ Ω .
✷
3.2
p. 298
✸
3.2
p. 30

3.2.5 Computation of Galerkin matrix
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Numerica
Methods
for PDEs
For sake of simplicity consider
∫
a(u,v) := gradu·gradvdx , u,v ∈ H0 1 (Ω) .
Ω
and Galerkin discretization based on
• triangular mesh, see Sect. 3.2.1,
• discrete trial/test space S1,0 0 (M) ⊂ H1 0
{
(Ω),
• nodal basis B N = b j }
N
according to (3.2.2).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Fig. 72
Restrictionsλ 1 ,λ 2 ,λ 3 of p.w. linear nodal basis functions ofS1 0 (M) to triangleK
The functionsλ 1 ,λ 2 ,λ 3 on the triangleK are also known as barycentric coordinate functions.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
∫
(A) i,j = a(b j N ,bi N ) = gradb j N ·gradbi N dx
Ω
Sect. 3.2.4: we need only study the cases, wherex i ,x j ∈ V(M)
They provide the nonzero restrictions of tent functions to triangles, see Fig. 87.
1. are connected by an edge of the triangulation,
2. coincide.
Idea:
∫
(A) ij =
“Assembly”
(add up cell contributions)
K 1
gradb j N|K 1
·gradb i N|K 1
dx+
∫
K 2
gradb j N|K 2
·gradb i N|K 2
dx
x i
K 1
K 2
Zero in on single triangleK ∈ M:
∫
a K (b j N ,bi N ) := gradb j N|K ·gradbi N|K dx , xi ,x j vertices ofK . (3.2.9)
K
Use analytic representation forb i N|K :
x j
ifa1 ,a 2 ,a 3 vertices ofK, λ i := b j N|K ,ai = x j
(i↔local vertex number,j ↔ global node number)
3.2
p. 301
Numerical
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for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
λ 1 (x) = 1 (
2)·( a 2
x−a 2 −a 3 )
2
2|K| a 3 1 −a2 1
λ 2 (x) = 1 (
3)·( a 3
x−a 2 −a 1 )
2
2|K| a 1 1 −a3 1
λ 3 (x) = 1 ( )
2|K|
x−a 1)·( a 1
2 −a 2 2
a 2 1 −a1 1
= − |e 1|
2|K| (x−a2 )·n 1 ,
= − |e 2|
2|K| (x−a3 )·n 2 ,
= − |e 3|
2|K| (x−a1 )·n 3 .
(e i = edge opposite vertexa i , see Figure for numbering
scheme ✄)
From the distance formula for a point w..r.t to a line given in Hesse normal form:
a 3 = ( a 3 ) T
1 ,a3 2
ω 3
n 1
n 2
n 3
Fig. 73
ω 1
ω 2
a 1 = ( a 1 ) T
1 ,a1 2
a 2 = ( a 2 1,a 2 2
(a i −a j )·n i = dist(a i ;e i ) = h i (h i ˆ= height) and2|K| = |e i |h i ⇒ λ i (a i ) = 1.
3.2
p. 30
Numerica
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for PDEs
) T
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.2
p. 302
This shows that the λ i really provide the restrictions of p.w. linear nodal basis functions
(tent functions) of S1 0 (M) to triangle K, because they are clearly (affine) linear and comply with
(3.2.2).
3.2
p. 30

gradλ 1 = 1
2|K|
( a 2 −a 3 )
2
a 2 1 −a3 , gradλ 2 = 1
1 2|K|
( a 3
2 −a 1 )
2
a 1 1 −a3 , gradλ 3 = 1
1 2|K|
( a 1 −a 2 )
2
a 2 1 −a1 .
1
Numerical
Methods
for PDEs
– the gradientsgradλ i are scaled byρ −1 (the barycentric coordinate functionsλ i become
steeper when the triangle shrinks in size.).
Both effects just offset ina K from (3.2.9) such thatA K remains invariant under scaling.
Numerica
Methods
for PDEs
(∫ ) 3
element (stiffness) matrixA
gradλ i·gradλ j dx =
K
K
i,j=1
⎛
= 1 cotω ⎞
3+cotω 2 −cotω 3 −cotω 2
⎝ −cotω
2 3 cotω 3 +cotω 1 −cotω 1
⎠ . (3.2.10)
−cotω 2 −cotω 1 cotω 2 +cotω 1
The local numbering and naming conventions are displayed in Fig. 110.
Derivation of (3.2.10), see also [19, Lemma 3.47]: obviously, because the gradients gradλ i are
constant onK,
∫
a(λ i ,λ j ) =
K
gradλ i·gradλ j dx = 1
4|K| |e i||e j |n i ·n j .
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.2
p. 305
“Assembly” of(A) ij starts from the sum
∫
∫
(A) ij = gradb j ·gradb N|K i 1 N|K dx+ 1
K 1
➣
K 2
gradb j N|K 2
·gradb i N|K 2
dx .
(A) ij can be obtained by summing respective (∗) entries of the elements matrices of the elements
adjacent to the edge connectingx i andx j
(∗): watch correspondence of local and global vertex numbers !
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.2
p. 30
Then use: n i·n j = cos(π −ω k ) = −cosω k , (i ≠ j)
|K| = 1 2 |e i||e j |sinω k , (i ≠ j).
3∑
3∑
Casei = j employs a trick: λ i = 1 ⇒ a(λ i ,λ j ) = 0.
i=1 i=1
Remark 3.2.11 (Scaling of entries of element matrix for−∆).
(3.2.10): A K does not depend on the “size” of triangleK!
(more precisely, element matrices are equal for similar triangles)
This can be seen by the following reasoning:
• Obviously translation and rotation ofK does not change. A K
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
ω 3
K 1 ω 2
⎛
cotω ⎞
3+cotω 2 −cotω 3 −cotω 2
⎝ −cotω 3 cotω 3 +cotω 1 −cotω 1
⎠
−cotω 2 −cotω 1 cotω 2 +cotω 1
+
⎛
ω 1
x i
α 2
K 2
α 1
α 3
cotα ⎞
3+cotα 2 −cotα 3 −cotα 2
⎝ −cotα 3 cotα 3 +cotα 1 −cotα 1
⎠
−cotα 2 −cotα 1 cotα 2 +cotα 1
(A) ij by summing entries of two element matrices
x j
Fig. 74
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• Scaling ofK by a factorρ > 0 has the following effect that
– the area|K| is scaled byρ 2 ,
3.2
p. 306
“Assembly” of diagonal entry (A) ii : summing corresponding diagonal entries of element matrices
belonging to triangles adjacent to nodex i .
3.2
p. 30

➀
➁
➀
➀
⎛
∗ ∗ ∗ ⎞
⎝∗ ∗ ∗⎠
∗ ∗ ∗
⎛
∗ ∗ ∗ ⎞
⎝∗ ∗ ∗⎠
∗ ∗ ∗
⎛
∗ ∗ ∗ ⎞
➁
➀
⎝∗ ∗ ∗⎠
∗ ∗ ∗
⎛
∗ ∗ ∗ ⎞
➂ ⎝∗ ∗ ∗⎠
➂ ➁ ∗ ∗ ∗
➁ ➀
⎛
∗ ∗ ∗ ⎞
⎝∗ ∗ ∗⎠
∗ ∗ ∗
➂ ➁
x i
➂
➂
(A) ii by summing diagonal entries of element matrices of adjacent triangles
Fig. 75
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.2.6 Computation of right hand side vector
We consider the linear form (right hand side of linear variational problem), see (2.3.3), (3.0.1):
∫
l(v) := f(x)v(x)dx , v ∈ H 1 (Ω) , f ∈ L 2 (Ω) .
Recall formula for right hand side vector
(⃗ϕ) j = l(b j N ) = ∫
Ω
Ω
f(x)b j N (x)dx , j = 1,...,N .
(3.2.15)
△
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Remark 3.2.12 (Assembly algorithm for linear Lagrangian finite elements).
Assume: numbering of nodal basis functions ↔ numbering of mesh vertices∈ V(M)
numbering of triangles (cells) of meshM = {K 1 ,...,K M },M := ♯M
Data structure: idx ∈ N ♯M,3 : local→global index mapping array
idx(k,l) = global number of vertexl ofk-th cell
x idx(k,l) = a l when a 1 ,a 2 ,a 3 are the vertices ofK k .
Code 3.2.14: Assembly of finite element Galerkin matrix for linear finite elementsS 0 1 (M)
1 A = zeros(N,N); % N = ♯V(M)
2 f o r i=1:M % M = ♯M
3 Ak = getElementMatrix(i); % Compute 3×3 element matrix, see (3.2.10)
4 A(idx(i,:),idx(i,:)) = A(idx(i,:),idx(i,:)) + Ak;
5 end
(3.2.13)
Note: ☞ Homogeneous Dirichlet boundary conditions not taken into account in Code 3.2.13
☞ Regard Code 3.2.13 as “MATLAB pseudo-code”: in actual implementation A must be
initialized as sparse matrix, see Rem. 3.5.18.
Computational effort
= O(♯M)
3.2
p. 309
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.2
p. 310
Idea: “Assembly”
K 5
N ∑j
∫
(⃗ϕ) j = f(x) b j (x)dx , N|K l
l=1K l
where K 1 ,...,K Nj ˆ= triangles
adjacent to nodex j .
(Integration confined tosupp(b j N )!) K 1
K 2
K 4
K 3
R. Hiptm
use of numerical quadrature for approximate evaluation ofl K (b j N ), cf. (1.5.59). Mandatory:
Zero in on single triangleK ∈ M:
∫
l K (b j N ) := f(x) b j N|K (x)dx , xj vertex ofK . (3.2.16)
DRAFT
K
Rem. 1.5.3: f : Ω ↦→ R given in procedural form
function y = f(x)
x j
Fig. 76
3.2
p. 31
Numerica
Methods
for PDEs
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
3.2
p. 31

f
1D setting of Sect. 1.5.1.2: use of composite quadrature rules based on low Gauss/Newton-Cotes
quadrature formulas on the cells[x j−1 ,x j ] of the grid, e.g. composite trapezoidal rule (1.5.59).
Numerical
Methods
for PDEs
Note: with index arrayidx from Rem. 3.2.12: idx(l,i(l,j)) = j
Numerica
Methods
for PDEs
What is the 2D counterpart of the composite trapezoidal rule ?
Recall:
trapezoidal rule [18, Eq. 10.2.7] integrates linear
interpolant of integrand based on endpoint values
3
2.5
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
2
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
1.5
000000000000000
111111111111111
000000000000000
111111111111111
1
X
000000000000000
000000000000000
000000000000000
111111111111111
111111111111111
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
0.5
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
0
000000000000000
111111111111111
0 0.5 1000000000000000
111111111111111
1.5 2 2.5 3 3.5 4
t
Fig. 77
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
➁
➀
➀
➀
⎛
∗ ⎞
⎝∗⎠
∗
⎛
∗ ⎞
⎝∗⎠
∗
⎛
∗ ⎞
⎝∗⎠
∗
➂
➂ ➁
➁ ➀
⎛
∗ ⎞
⎝∗⎠
∗
➁
⎛
∗ ⎞
⎝∗⎠
∗
➀
➂
➂
x i
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
➂
➁
Fig. 78
Idea:
2D trapezoidal rule
for triangleK with verticesa 1 ,a 2 ,a 3
∫
K
f(x)dx ≈ |K|
3 (f(a1 )+f(a 2 )+f(a 3 )) . (3.2.17)
ˆ= integration of linear interpolant ∑ 3
i=1 f(a i )λ i off.
⎛
(
element (load) vector: ⃗ϕ K := l K (b j(i) ) 3
N ) i=1 = |K| f(a 1 ⎞
)
⎜
⎝f(a 2 ⎟
) ⎠ ,
3
f(a 3 )
wherex j(i) = a i ,i = 1,2,3 (global node number↔local vertex number).
3.2
p. 313
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 3.2.19 (Assembly of right hand side vector for linear finite elements). → Rem. 3.2.12
Code 3.2.20: Assembly of right hand side vector for linear finite elements, see (3.2.18)
1 phi = zeros(N,1); % N = ♯V(M)
2 f o r i=1:M % M = ♯M
3 phiK = getElementVector(i); % Compute element (load) vector ∈ R 3
4 phi(idx(i,:)) = phi(idx(i,:)) + phiK; % idx according to (3.2.13)
5 end
3.3 Building blocks of general FEM
△
3.2
p. 31
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
As above in Fig. 75: “Assembly” of (⃗ϕ) j by summing up contributions from element vectors of triangles
adjacent tox j .
N j
∑
(⃗ϕ) j = l Kl (b j ) = N|K l
l=1
N j
∑
(⃗ϕ K ) i(l,j) = f(x j )· 13
l=1
N j ∑
l=1
wherei(l,j) is the local vertex index of the nodex j (global indexj) in the triangleK l .
|K l | , (3.2.18)
3.2
p. 314
The previous section explored the details of a simple finite element discretization of 2nd-order elliptic
variational problems. Yet, it already introduced key features and components that distinguish the
finite element approach to the discretization of linear boundary value problems for partial differential
equations:
3.3
p. 31

a focus on the variational formulation of a boundary value problem→Sect. 2.8,
a partitioning of the computational domainΩby means of a meshM(→ Sect. 3.2.1)
Numerical
Methods
for PDEs
Types of meshes:
Numerica
Methods
for PDEs
the use of Galerin trial and test spaces based on piecewise polynomials w.r.t. M (→ Sect. 3.2.2),
the use of locally supported basis functions for the assembly of the resulting linear system of
equations (→ Sect. 3.2.3).
In this section a more abstract point of view is adopted and the components of a finite element method
for scalar 2nd-order elliptic boundary value problems will be discussed in greater generality. However,
prior perusal of Sect. 3.2 is strongly recommended.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
Triangular mesh in 2D
Fig. 79
Quadrilateral mesh in 2D
✁ 2D hybrid mesh comprising
Fig. 80
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• triangles
DRAFT
3.3.1 Meshes
First main ingredient of FEM: triangulation/mesh ofΩ → Sect. 3.2.1
Fig. 81
Tetrahedral meshes in 3D (created with NETGEN):
• quadrilaterals
• curvilinear cells (at∂Ω)
3.3
3.3
p. 317
p. 31
Definition 3.3.1. A mesh (or triangulation) of Ω ⊂ R d is a finite collection{K i } M i=1 , M ∈ N, of
open non-degenerate (curvilinear) polygons (d = 2)/polyhedra (d = 3) such that
Numerical
Methods
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Numerica
Methods
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(A) Ω = ⋃ {K i , i = 1,...,M},
(B) K i ∩K j = ∅ ⇔ i ≠ j,
(C) for all i,j ∈ {1,...,M}, i ≠ j, the intersectionK i ∩K j is either empty or a vertex, edge,
or face of bothK i andK j .
“vertex”, ”edge”, ”face” of polygon/polyhedron: → geometric intuition
Terminology: Given meshM := {K i } M i=1 : K i called cell or element.
Vertices of a mesh→nodes (setV(M))
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Tensor product mesh = grid
in 2D: a = x 0 < x 1 < ... < x n = b ,
✄
d
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
c = y 0 < y 1 < ... < y m = d .
3.3
p. 318
M = {]x i−1 ,x i [×]y j−1 ,y j [: (3.3.2)
1 ≤ i ≤ n,1 ≤ j ≤ m} . c
☞ Restricted to tensor product domains a b
Fig. 82
3.3
p. 32

If (C) does not hold
Triangular non-conforming mesh
(with hanging nodes)
Numerical
Methods
for PDEs
Special case:
⎧
|α| = α 1 +α 2 +···+α d . (3.3.5)
⎫
Numerica
Methods
for PDEs
K i ∩ K j is only part of an edge/face for at most
one of the adjacent cells.
(However, conforming if degenerate quadrilaterals admitted)
Fig. 83
Example:
⎪⎨
d = 2: P p (R 2 ) =
⎪⎩
∑
α 1 ,α 2 ≥0
α 1 +α 2 ≤p
c α1 ,α 2
x α 1
1 xα 2
⎪⎬
2 , c α 1 ,α 2
∈ R .
⎪⎭
P 2 (R 2 ) = Span
{1,x 1 ,x 2 ,x 2 }
1 ,x2 2 ,x 1x 2
Terminology: Simplicial mesh =
triangular mesh in 2D
tetrahedral mesh in 3D
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✬
Lemma 3.3.6 (Dimension of spaces of( polynomials). )
dimP p (R d d+p
) =
for allp ∈ N
p 0 ,d ∈ N
R. Hiptm
C. Schwa
H.
✩Harbrech
V.
Gradinar
A. Chern
DRAFT
✫
✪
3.3.2 Polynomials
Proof. Distributep“powers” to thedindependent variables or discard them✄d+1 bins.
3.3
p. 321
Combinatorial ( ) model: number of different linear arrangements ofpidentical items anddseparators
d+p
= . ✷
p
3.3
p. 32
Second main ingredient of FEM:
In FEM:
Galerkin trial/test space comprise locally polynomial functions onΩ
Numerical
Methods
for PDEs
Leading order dimP p (R d ) = O(p d )
Numerica
Methods
for PDEs
Clear: polynomials of degree≤ p,p ∈ N 0 , in 1D (univariate polyomials), see (1.5.19)
P p (R) := {x ↦→ c 0 +c 1 x+c 2 x 2 +...c p x p } .
In higher dimensions this concept allows various generalizations, one given in the following definition,
one given in Def. 3.3.7.
Definition 3.3.3 (Multivariate polynomials).
Space of multivariate (d-variate) polynomials of (total) degreep ∈ N 0 :
P p (R d ) := {x ∈ R d ↦→ ∑ α∈N d 0 ,|α|≤pc αx α , c α ∈ R} .
Def. 3.3.3 relies on multi-index notation:
α = (α 1 ,...,α d ): x α := x α 1
1 ·····xα d
d ,
R. Hiptmair
C. Schwab,
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V.
Gradinaru
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DRAFT
✬
✫
(3.3.4)
3.3
p. 322
Definition 3.3.7 (Tensor product polynomials).
Space of tensor product polynomials of degreep ∈ N in each coordinate direction
Example:
Q p (R d ) := {x ↦→ p 1 (x 1 )·····p d (x d ), p i ∈ P p (R),i = 1,...,d} .
Q 2 (R 2 {
) = Span 1,x 1 ,x 2 ,x 1 x 2 ,x 2 1 ,x2 1 x 2,x 2 1 x2 2 ,x 1x 2 }
2 ,x2 2
Lemma 3.3.8 (Dimension of spaces of tensor product polynomials).
dimQ p (R d ) = (p+1) d for all p ∈ N 0 ,d ∈ N
✩
✪
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.3
p. 32

Terminology:
P p (R d )/Q p (R d ) = complete spaces of polynomials/tensor product polynomials
Numerical
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Numerica
Methods
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3.3.3 Basis functions
Fig. 84
Fig. 85
Fig. 86
Third main ingredient of FEM:
locally supported basis functions
(see Sect. 3.1 for role of bases in Galerkin discretization)
Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
Support of node-associated
basis function, cf. Fig. 87
Support of edge-associated
basis function
Support of cell-associated basis
function
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
DRAFT
Basis functionsb 1 N ,...,bN N for a finite element trial/test spaceV 0,N built on a meshMsatisfy:
(a) B N := {b 1 N ,...,bN N } is basis ofV 0,N ➣ N = dimV 0,N ,
Requirement (c) implies that
(b) eachb i N
is associated with a single cell/edge/face/vertex ofM,
global finite element basis functions are locally supported.
R.
(c) supp(b i N ) = ⋃ {K: K ∈ M,p ⊂ K}, ifb i N
associated with cell/edge/face/vertexp.
3.3
p. 325
What is the rationale for this requirement ?
3.3
p. 32
Finite element terminology: b i N
= global shape functions/global basis functions
MeshM+global shape functions ➨ complete description of finite element space
Numerical
Methods
for PDEs
Consider a generic bilinear form a arising from a linear scalar 2nd-order elliptic BVP, see (3.2.5): it
involves integration over Ω/∂Ω of products of (derivatives of) basis functions. Thus the integrand for
a(b j N ,bi N ) vanishes outside the overlap of the supports ofbj N andbi N .
Numerica
Methods
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Example 3.3.9 (Supports of global shape functions in 1D). → Sect. 1.5.1.2
Ω =]a,b[ ˆ= interval
Equidistant mesh
M := {]x j−1 ,x j [, j = 1,...,M} ,
x j := a+hj,h := (b−a)/M,M ∈ N.
Support (→ Def. 1.5.57) of global shape function
associated withx 7
Example 3.3.10 (Supports of global shape functions on triangular mesh).
0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✗
✖
Galerkin matrixA ∈ R N,N with (A) ij := a(b j N ,bi N
), i,j = 1,...,N satisfies
a ij ≠ 0
only if
b i N andbj N
associated with
vertices/faces/edges(cells) adjacent to common
cell
Finite element stiffness matrices are sparse (→ Notion 3.2.7)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
✔A. Chern
✕
DRAFT
Global shape functions
Restriction to element
−−−−−−−−−−−−−→ local shape functions (3.3.11)
3.3
p. 326
3.3
p. 32

Definition 3.3.12 (Local shape functions).
Given finite element function space on meshMwith global shape functionsb i N ,i = 1,...,N:
{b j N|K , K ⊂ supp(bj N )} = set of local shape functions onK ∈ M .
Numerical
Methods
for PDEs
Taken for granted: finite element meshMaccording to Def. 3.3.1.
Goal:
construction of finite element spaces and global shape functions of higher polynomials degrees.
Numerica
Methods
for PDEs
Local shape functions b 1 K ,...,bQ K
, Q = Q(K) ∈ N also associated with vertices/edges/-
faces/interior ofK
Example 3.3.13 (Local shape functions forS1 0 (M) in 2D). → Sect. 3.2.3
Global basis function forS 0 1 (M)
On “unit triangle”K with vertices
( (
0 1
0)
0)
a 1 =
, a 2 =
Local shape functions:
, a 3 =
(
0
1)
✄
b 1 K (x) = 1−x 1−x 2 ,
b 2 K (x) = x 1 ,
b 3 K (x) = x 2 .
1
Fig. 87
R. Hiptmair
C. Schwab,
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Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.3
p. 329
Lagrangian finite element spaces provide spacesV 0,N ofM-piecewise polynomials that fulfill
Notation:
(Lagrangian FE spaces)
V N,0 ⊂ C 0 (Ω)
3.4.1 Simplicial Lagrangian FEM
Thm. 2.2.17
=⇒ V N,0 ⊂ H 1 (Ω) .
S 0 p(M) continuous functions, cf. C0 (Ω)
locally polynomials of degreep,e.g. P p (R d )
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DRAFT
3.4
p. 33
Numerical
Methods
for PDEs
M: Simplicial mesh, consisting of triangles in 2D, tetrahedra in 3D.
Numerica
Methods
for PDEs
Now we generalizeS 0 1 (M)/S0 1,0 (M) from Sect. 3.2 to higher polynomial degreep ∈ N 0.
These are the barycentric coordinate functionsλ 1 ,λ 2 ,λ 3 introduced in Sect. 3.2.5
Fig. 88
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C. Schwab,
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Harbrecht
V.
Gradinaru
A. Chernov
Definition 3.4.1 (Simplicial Lagrangian finite element spaces).
Space ofp-th degree Lagrangian finite element functions on simplicial meshM
S 0 p(M) := {v ∈ C 0 (Ω): v |K ∈ P p (K) ∀K ∈ M} .
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DRAFT
DRAFT
3.4 Lagrangian FEM
✸
3.4
p. 330
Def. 3.4.1 merely describes the space of trial/test functions used in a Lagrangian finite element
method on a Simplicial mesh. A crucial ingredient is still missing (→ Sect. 3.3.3): the global shape
functions still need to be specified. This is done by generalizing (3.2.2) based on sets of special
interpolation nodes.
Example 3.4.2 (Triangular quadratic Lagrangian finite elements).
3.4
p. 33

interpolation nodes
N := V(M)∪{midpoints of edges} ,
N = {p 1 ,...,p N } .
Numerical
Methods
for PDEs
Selected local shape functions:
Numerica
Methods
for PDEs
Nodal basis functions b j N
, j = 1,...,N defined
by, cf. (3.2.2)
Fig. 91
b j N (p i) =
{
1 , ifi = j ,
0 else.
(3.4.3)
A “definition” like (3.4.3) is cheap, but it may be pointless, in case no such functions b j N
exist. To
establish their existence, we first study the case of a single triangleK.
We have to show that there is a basis ofP 2 (R 2 ) that satisfies (3.4.3) in the case of a mesh consisting
of a single triangleM = {K}.
Fig. 89
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C. Schwab,
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Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
So far we have seen that local shape functions can be found that satisfy (3.4.3).
Issue: can the local shape functions from (3.4.4) be “stiched together” across interelement edges
such that they yield a continuous gobal basis function? (Remember that Thm. 2.2.17 demands
global continuity in order to obtain a subspace ofH 1 (Ω).)
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DRAFT
A first simple consistency check: does the number of interpolation nodes ♯N for M = {K} agree
withdimP 2 (R 2 ) = 6? Yes, it does!
Local shape functions (barycentric coordinate representation)
b 1 K = (2λ 1 −1)λ 1 ,
b 2 K = (2λ 2 −1)λ 2 ,
b 3 K = (2λ 3 −1)λ 3 ,
b 4 K = 4λ 1λ 2 ,
b 5 K = 4λ 2λ 3 ,
b 6 K = 4λ 1λ 3 .
(3.4.4)
To see the validity of the formulas (3.4.4), note that
➀
➄
K
➃
➂
➅
➁
Fig. 90
3.4
p. 333
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
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Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Fig. 92
The restriction of a quadratic polynomial to an edge is an univariate
quadratic polynomial.
Fixing its value in three points, the midpoint of the edge and
the endpoints, uniquely fixes this polynomial.
The local shape functions associated with the same
interpolation node “from left and right” agree on the edge.
➣ continuity !
3.4
p. 33
Numerica
Methods
for PDEs
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C. Schwa
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Gradinar
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DRAFT
•λ i (a i ) = 1 andλ i (a j ) = 0, ifi ≠ j, wherea 1 ,a 2 ,a 3 are the vertices of the triangleK,
•λ 1 (m 12 ) = λ 1 (m 13 ) = 1 2 , wheremij = 1 2 (ai +a j ) denotes the midpoint of the edge connecting
a i anda j ,
• each barycentric coordinate functionλ i is affine linear such thatλ i λ j ∈ P 2 (R 2 ).
3.4
p. 334
Fig. 93
Fig. 94
3.4
p. 33

✁ Global basis function for S2 0 (M) associated
with a vertex
Numerical
Methods
for PDEs
Now we consider tensor product meshes (grids), see (3.3.2), Fig. 82, for a 2D example.
Example 3.4.6 (Bilinear Lagrangian finite elements).
Numerica
Methods
for PDEs
Fig. 95
(3.4.3): this function attaints value = 1 at a vertex
(•) and vanishes at the midpoints (•) of the edges
Example 3.4.5 (Interpolation nodes for cubic and quartic Lagrangian FE in 2D).
of adjacent triangles, as well as at any other vertex.
✸
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C. Schwab,
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Harbrecht
V.
Gradinaru
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DRAFT
Sought: generalization of 1D piecewise linear finite element functions from Sect. 1.5.1.2, see Fig. 25,
to 2D tensor product gridM.
Tensor product structure ofM ➤
tensor product construction of FE space
This is best elucidated by a tensor product construction of basis functions:
b j N,x (x) : 1D tent function onM x = {[x j−1 ,x j ], j = 1,...,n}
b l N,y (y) : 1D tent function onM y = {[y j−1 ,y j ], j = 1,...,n}
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DRAFT
2D tensor product “tent function” associated with nodep:
3.4
p. 337
b p N (x) = bj N,x (x 1)·b l N,y (x 2) , where p = (x j ,y l ) T . (3.4.7)
3.4
p. 33
Numerical
Methods
for PDEs
1.2
1
1.2
1
1.2
1
Numerica
Methods
for PDEs
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.2
0
−0.2
2.5
2
1.5
1
0.5
0.5
0
0
−0.5 −0.5
b j N,x (x)
1
1.5
2
⊕
0.4
0.2
0
−0.2
2.5
2.5
2
1.5
1
0.5
0.4
0.2
= 0
−0.2
2.5
2.5
2
1.5
1
0.5
0
0
−0.5 −0.5
b l N,y (y)
2
1.5
1
0.5
0.5
0
0
−0.5 −0.5
b p N (x)
1
2.5
2
1.5
Fig. 98
Fig. 96
(local) interpolation nodes forS 0 3 (M) Fig. 97
(local) interpolation nodes forS 0 4 (M)
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✁ 2D tensor product tent function
No pyramid !
Basis functions associated (→ Sect. 3.3.3,
condition (c)) with nodes ofM,
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C. Schwa
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Gradinar
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DRAFT
3.4.2 Tensor-product Lagrangian FEM
3.4
p. 338
Fig. 99
3.4
p. 34

Tensor product construction ➤ bilinear local shape functions, e.g. onK =]0,1[ 2
Numerical
Methods
for PDEs
The following is a natural generalization of (3.4.9) to higher degree local tensor product polynomials,
see Def. 3.3.7:
Numerica
Methods
for PDEs
b 1 K (x) = (1−x 1)(1−x 2 ) ,
b 2 K (x) = x 1(1−x 2 ) ,
b 3 K (x) = x 1x 2 ,
b 4 K (x) = (1−x 1)x 2 .
(3.4.8)
➃
➀
K
x 2
Fig. 100
➂
➁
x 1
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C. Schwab,
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Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Definition 3.4.10 (Tensor product Langrangian finite element spaces).
Space ofp-th degree Lagrangian finite element functions on tensor product meshM
Terminology:
S 0 p(M) := {v ∈ C 0 (Ω): v |K ∈ Q p (K) ∀K ∈ M} .
S1 0 (M) = multilinear finite elements (p = 1,d = 2 = bilinear finite elements)
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C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Remaining issue:
definition of global basis functions (global shape functions)
Policy: use of interpolation nodes as in Sect. 3.4.1, see Ex. 3.4.2.
3.4
p. 341
Numerical
Methods
for PDEs
Example 3.4.11 (Quadratic tensor product Lagrangian finite elements).
Consider casep = 2,d = 2 of Def. 3.4.10:
Interpolation nodes forS2 0(M)
N = V(M)∪{midpoints of edges} .
d
3.4
p. 34
Numerica
Methods
for PDEs
Fig. 101
Note: number of interpolation nodes belonging to
one cell is
Bilinear local shape functions on unit squareK
{
Span b 1 }
K ,b2 K ,b3 K ,b4 K = Q 1 (R 2 ) .
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Gradinaru
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9 = dimQ 2 (R 2 ) .
c
a
Fig.
b
102
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V.
Gradinar
A. Chern
Bilinear Lagrangian finite element space on 2D tensor product meshM:
DRAFT
Global basis functions defined analoguously to (3.4.3).
DRAFT
S 0 1 (M) := {v ∈ C0 (Ω): v |K ∈ Q 1 (R 2 ) ∀K ∈ M} . (3.4.9)
✸
✸
3.4
p. 342
Choice of interpolation nodes for tensor product Lagrangian finite elements:
3.4
p. 34

Numerical
Methods
for PDEs
2. Does the space from (3.4.15) allow for locally supported basis functions associated with nodes of
the mesh?
Numerica
Methods
for PDEs
p = 1 p = 2
Remark 3.4.12 (Imposing homogeneous Dirichlet boundary conditions).
p = 3
Fig. 103
We wil give a positive answer to both question by constructing the basis functions:
Define global shape functionsb j N
according to (3.2.3)
What is a global basis forSp(M)∩H 0 0 1 (Ω), whereMis either a simplicial mesh or a tensor product
mesh?
We proceed analoguous to Rem. 3.2.4:
interpolation nodesp j ,j = 1,...,N, see (3.4.3).
recall that global basis functions are defined via
Sp,0 0 (M) := S0 p(M)∩H0 {b 1(Ω) = Span j }
N : pj ∈ Ω (interior node)
. (3.4.13)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
This makes sense, because
linear/bi-linear functions onK are uniquely determined by their values in the vertices,
the restrictions to an edge of K of the local linear and bi-linear shape functions are both linear
univariate functions, see Figs. 72, 101.
Fixing vertex values for v N ∈ S1 0 (M) uniquely determines v on all edges of M already, thus,
ensuring global continuity, which is necessary due to Thm. 2.2.17.
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Remark 3.4.14 ((Bi)-linear Lagrangian finite elements on hybrid meshes).
△
3.4
p. 345
Numerical
Methods
for PDEs
Exercise 3.4.1. Consider a tensor product meshMofΩ :=]0,1[ 2 and the space
{
V 0,N := v ∈ H0 1 (Ω): v |K ∈ P 1(R 2 }
)∀K ∈ M .
What is the dimension of this space?
3.4
p. 34
Numerica
Methods
for PDEs
M: 2D hybrid mesh comprising triangles & rectangles
✄
Two issues arise:
Idea:
use
linear functions (→ Def. 3.3.3,
p = 1) on triangular cells,
bi-linear functions (→
Def. 3.4.10, p = 1) on rectangles.
S 0 1 (M) = {v ∈ H 1 (Ω): v |K ∈
{
P1 (R 2 ) , ifK ∈ M is triangle,
Q 1 (R 2 ) , ifK ∈ M is rectangle
}
Fig. 104
. (3.4.15)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 3.4.16 (Lagrangian finite elements on hybrid
meshes).
☞
M: 2D hybrid mesh comprising triangles & rectangles
Matching interpolation nodes on edges of triangles
and rectangles
Glueing of local shape functions on triangles
and rectangles possible
gobal interpolation nodes forp = 2✄
△
Fig. 105
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1. Does the prescription (3.4.15) yield a large enough space? (Note that v ∈ H 1 (Ω) ⇒ S 0 1 (M) ⊂
C 0 (Ω), see Thm. 2.2.17, but continuity might enforce too many constraints.)
3.4
p. 346
3.5
p. 34

3.5 Implementation of FEM
Numerical
Methods
for PDEs
File format for storing triangular mesh (of polygonal domain):
# Two-dimensional simplicial mesh
Numerica
Methods
for PDEs
1 ξ 1 η 1 # Coordinates of first node
This section discusses algorithmic details of Galerkin finite element discretization of 2nd-order elliptic
variational problems for spatial dimension d = 2,3 on bounded polygonal/polyhedral domains Ω ⊂
R d .
The presentation matches the LehrFEM finite element MATLAB library, parts of which will be made
available for participants of the course. A detailed documentation is available from [6].
The guiding principle behind the implementation of finite element codes is
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DRAFT
2 ξ 2 η 2 # Coordinates of second node
.
N ξ N η N
# Coordinates of N-th node
1 n 1 1 n 1 2 n 1 3 X 1 # Indices of nodes of first triangle
2 n 2 1 n 2 2 n 2 3 X 2 # Indices of nodes of second triangle
.
M n M 1 n M 2 n M 3 X M # Indices of nodes of M-th triangle
X i ,i = 1,...,M → extra information (e.g. material properties in triangle#i).
(3.5.2)
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DRAFT
✓
✒
to rely on local computations as much as possible!
✏
✑
This is made possible by the local supports of the global basis functions, see Sect. 3.3.3, Ex. 3.3.10.
3.5
3.5
p. 349
p. 35
3.5.1 Mesh file format
Numerical
Methods
for PDEs
Optional:
additional information about edges (on∂Ω):
K ∈ N # Number of edges on ∂Ω
Numerica
Methods
for PDEs
Data flow in (most) finite element software packages:
n 1 1 n1 2 Y 1
# Indices of endpoints of first edge
n 2 1 n2 2 Y 2
# Indices of endpoints of second edge
(3.5.3)
CAD data
Parameters
.
n K 1 nK 2
Y K # Indices of endpoints of K-th edge
Mesh generator
Finite element solver
(computational kernel)
Post-processor
(e.g. visualization)
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Gradinaru
A. Chernov
Y k ,k = 1,...,K → extra information
Example 3.5.4 (Mesh file format for MATLAB code “LehrFEM”).
✸
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C. Schwa
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DRAFT
Vertex coordinate file:
Cell information file:
DRAFT
Here “ ” designates passing of information, which is usually done by writing and reading files to
and from hard disk. This requires particular file formats.
Example 3.5.1 (Triangular mesh: file format).
% List of vertices
1 +0.000000e+00 -1.000000e+00
2 +1.000000e+00 +0.000000e+00
3 +0.000000e+00 +1.000000e+00
4 -1.000000e+00 +0.000000e+00
5 +0.000000e+00 +0.000000e+00
% List of elements
1 1 2 5
2 2 3 5
3 3 4 5
4 4 1 5
3.5
3.5
p. 350
p. 35

Loading a mesh
m = load_Mesh(’Coord_Circ.dat’,...
’Elem_Circ.dat’);
plot_Mesh(m,’apts’);
Option flags:
’a’: with axes
’p’: vertex labels on
’t’: cell labels on
’s’: caption/title on
For details see [6, Sect. 1.3.1], [6, Sect. 1.3.2].
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
4 5
3
4
2D triangular mesh
3
1
−1 −0.5 0 0.5 1
# Vertices : 5, # Elements : 4, # Edges : 8
Fig. 106
✸
2
1
2
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
MATLAB-CODE: mesh generation for circular domain
BBOX = [-1 -1; 1 1];
H0 = 0.1;
DHD = @(x) sqrt(x(:,1).^2+x(:,2).^2)-1;
HHANDLE = @(x) ones(size(x,1),1);
Mesh = init_Mesh(BBOX,H0,DHD,...
HHANDLE,[],1);
save_Mesh(Mesh,’Coordinates.dat’,...
’Elements.dat’);
3.5.2 Mesh data structures [6, Sect. 1.1]
Bounding box
Largest reasonable edge length
Signed distance functionϕ(x):
(distance from ∂Ω, ϕ(x) < 0 ⇔
x ∈ Ω)
Element size function
(determines local edge length)
✸
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
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V.
Gradinar
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DRAFT
3.5
p. 353
Issue:
internal representation of mesh (→ Def. 3.3.1) in computer code
3.5
p. 35
How to create a mesh ?
→
Mesh generation (beyond scope of this course)
→ http://www.andrew.cmu.edu/user/sowen/mesh.html
Free software: • DistMesh (MATLAB, used in “LehrFEM”, see [6, Sect. 1.2])
• NETGEN (industrial strength open source mesh generator)
• Triangle (easy to use 2D mesh generator)
• TETGEN (Tetrahedral mesh generation)
Example 3.5.5 (Mesh generation in LehrFEM).
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
mesh data structure must provide:
1. offer unique identification of cells/(faces)/(edges)/vertices
2. represent mesh topology (= incidence relationships of cells/faces/edges/vertices)
3. describe mesh geometry (= location/shape of cells/faces/edges/vertices)
4. allow sequential access to edges/faces of a cell
(→ traversal of local shape functions/degrees of freedom)
5. make possible traversal of cells of the mesh (→ global numbering)
Focus: array oriented data layout (→ MATLAB, FORTRAN)
Numerica
Methods
for PDEs
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C. Schwa
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V.
Gradinar
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DRAFT
Algorithm & details → [24], more explanations in [6, Sect. 1.2].
Notation:
M = mesh (set of elements),V(M) = set of nodes (vertices) inM,E(M) = set of edges inM
Case: d-dimensional simplicial triangulationM, minimal data structure (cf. Sect. 3.5.1)
3.5
p. 354
→ Coordinates of verticesV(M) : ♯V(M)×d-array Coordinates of reals
→ Vertex indices for cells: ♯M×(d+1)-array Elements of integers.
3.5
p. 35

(−1,1)
1
1
2
5
3
(−1,−1)
2
6
4
9
5
3
7
8
4
(1,1)
8
7
6
(1,−1)
Fig. 107
Example 3.5.6 (Arrays storing 2D triangular mesh).
i Coordinates
1 -1 1
2 -1 0
3 -1 -1
4 0 -1
5 0 0
6 1 -1
7 1 0
8 1 1
9 0 1
Array Coordinates
K j Vertex indices
1 1 2 9
2 2 5 9
3 5 8 9
4 5 7 8
5 3 4 2
6 4 5 2
7 4 7 5
8 4 6 7
Array Elements
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
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Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
mesh =
Coordinates: [5x2 double]
Elements: [4x3 double]
Edges: [8x2 double]
Vert2Edge: [5x5 double]
Edge2Elem: [8x2 double]
EdgeLoc: [8x2 double]
Notation:
How to number↔order
E(M) ˆ= edges of 2D mesh
vertex coordinates, see Ex. 3.5.4
vertex indices of triangles, see Ex. 3.5.4
indices of endpoints in Coordinates array
♯V(M)×♯V(M) sparse integer matrix:
entry(i,j) = edge index, if≠ 0
♯E(M)×2 integer array:
indices of adjacent cells in Elements array
♯E(M)×2 integer array: local indices of edges w.r.t. adjacent cells
local shape functions
global shape functions
?
✸
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Already offers complete description of the mesh topology and geometry !
This information is enough for efficient assembly of finite element Galerkin matrices/right hand side
vectors for (bi-)linear Lagrangian finite elements, see Rem. 3.2.12, 3.2.19.
Note: Global shape functions associated with edges/faces ➣ extra information required !
Optional extra information:
3.5
p. 357
Numerical
Methods
for PDEs
✬
Elements, Edges arrays ➣ ordering of vertices of cells/endpoints of edges
Arrays (of vertices,cells,edges) ➣ array indices ➣ numbering of global shape functions
✫
Remark 3.5.8. Second option: C++/JAVA-style object oriented data layout
Nodes, cells ofM ←→ dynamically allocated objects (instances of classes Node, Cell)
✩
✪
3.5
p. 35
Numerica
Methods
for PDEs
→ Edge connecting vertices: ♯V(M)×♯V(M) symmetric sparse integer matrixI E
{
0 , if vertex♯inot linked to♯j
(I E ) ij :=
e ij , if edge connecting♯iand♯j
heree ij is the unique edge number∈ {1,2.....♯E(M)}
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C. Schwab,
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Harbrecht
V.
Gradinaru
A. Chernov
class Node {
private:
double x,y;
ID id;
public:
Node(double x,double y,ID id=0);
Point getCoords(void) const;
ID getId(void) const;
};
class Cell {
private:
const vector vertices;
ID id;
public:
Cell(const vector &vertices,ID id=0);
int NoNodes(void) const;
const Node &getNode(int) const;
ID getId(void) const;
};
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C. Schwa
H.
Harbrech
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Gradinar
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→ End points of the edges: ♯E(M)×2 array of integer (= vertex indices of end points).
→ Cell adjacent to edges: ♯E(M)×2 array of integers (=cell indices)
(one cell index =0 if edge is on∂Ω)
Example 3.5.7 (Extended MATLAB mesh data structure). → [6, Sect. 1.1]
mesh = add_Edge2Elem(add_Edges(init_Mesh(BBOX,H0,DHD,HHANDLE,[],1)))
(init_Mesh → Ex. 3.5.5)
DRAFT
3.5
p. 358
class BdFace {
private:
const vector vertices;
BdCond bdcond;
public:
BdFace(const vector &vertices);
int NoNodes(void) const;
const Node &getNode(int) const;
BdCond getBdCond(void) const;
};
class Mesh {
private:
list nodes;
list cells;
list bdfaces;
public:
Mesh(istream &file);
virtual Mesh(void);
const list &Nodes(void) const;
const list &Cells(void) const;
const list &BdFaces(void) const;
};
DRAFT
3.5
p. 36

ID getId()→provides unique identifier for each node/cell.
Distinguish: − local objects (→ classes Node, Cell, BdFace)
− global objects (“mesh management” class Mesh, see below)
3.5.3 Assembly [6, Sect. 5]
△
Numerical
Methods
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R. Hiptmair
C. Schwab,
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Harbrecht
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Gradinaru
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DRAFT
Example: bilinear forms/linear forms arising from 2nd-order elliptic BVPs, e.g, (2.9.1), (2.9.2),
(2.9.3), can be localized in straightforward fashion by restricting integration to mesh
cells (→ Rem. 3.2.1): foru,v ∈ H 1 (Ω)
∫
a(u,v) := α(x)gradu·gradvdx = ∑ ∫
α(x)gradu·gradvdx , (3.5.10)
Ω
K∈M
K
} {{ }
=:a K (u |K ,v |K )
∫
l(v) := fvdx = ∑
Ω
K∈M
∫
fvdx . (3.5.11)
K
} {{ }
=:l K (v |K )
Numerica
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DRAFT
“Assembly” = term used for computing entries of stiffness matrix/right hand side vector (load vector)
in a finite element context.
From the dictionary: “Assemble” = to fit together all the separate parts of sth.
3.5
p. 361
Start with discussion for linear finite elements. Jumping right to quadratic finite elements has not
worked out well.
✗
✔
Recall (3.3.11): Restrictions of global shape functions to cells = local shape functions
✖
✕
3.5
p. 36
Aspects of assembly for linear Lagrangian finite elements (V 0,N = S1,0 0 (M)) were discussed in
Sects. 3.2.5, 3.2.6. (Refresh yourself on these sections in case you cannot remember the main ideas
behind building the Galerkin matrix and right hand side vector.)
We consider a discrete variational problem (V 0,N = FE space,dimV 0,N = N ∈ N, see (3.1.4))
Numerical
Methods
for PDEs
Definition 3.5.12 (Element (stiffness) matrix and element (load) vector).
Given local shape functions{b 1 K ,...,bQ K },Q ∈ N, we call
(
element (stiffness) matrix A K :=: a K (b j ) Q
K ,bi K ) i,j=1 ∈ RQ,Q ,
(
element (load) vector ⃗ϕ K := l K (b i ) Q
K ) i=1 ∈ RQ .
Numerica
Methods
for PDEs
u N ∈ V 0,N : a(u N ,v N ) = l(v N ) ∀v N ∈ V 0,N . (3.1.4)
To be computed (see also Sect. 3.2.5, Sect. 3.2.6):
(
• Galerkin matrix (stiffness matrix): A = a(b j ) N
N ,bi N ) i,j=1 ∈ RN,N
(
• r.h.s. vector (load vector): ⃗ϕ := l(b i ) N
N ) i=1 ∈ RN
} {{ }
both can be written in terms of local cell contributions, since usually
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
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DRAFT
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
R. Hiptm
Note: Here Q, the number of local shape functions on element K ∈ M, is independent of K.
In general, we could also have Q = Q K when we blend several element types in one mesh, see
DRAFT
Rem. 3.4.14.
Type of FE space
Q
degreepLagrangian FE on triangular mesh dimP p (R 2 ) = 2 1(p+1)(p+2)
degreepLagrangian FE on tetrahedral mesh dimP p (R 3 ) = 1 6 (p+1)(p+2)(p+3)
degreepLagrangian FE on tensor product mesh in 2D dimQ p (R 2 ) = (p+1) 2
a(u,v) = ∑
a K (u |K ,v |K ) , l(v) = ∑
l K (v |K ) . (3.5.9)
K∈M
K∈M
3.5
p. 362
3.5
p. 36

Again scrutinize Figs. 74, 75 and the accompanying remarks in Sect. 3.2.5. We learn that in the
special setting of this section
Numerical
Methods
for PDEs
➣
By (3.5.15), the indicesl(i) encode the T-matrix according to
(T K ) l(i),i = 1 , i = 1,...,N ,
Numerica
Methods
for PDEs
the entries of the finite element Galerkin matrix can be obtained by summing corresponding
entries of some element matrices,
this corresponding entry of an element matrices is determined by the unique association
of a local basis function to a global basis function.
These insights are formalized in the next theorem.
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C. Schwab,
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Gradinaru
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DRAFT
where all other entries ofT K are understood to vanish.
⇒ (A) ij =
∑
K∈M,supp(b j N )∩K≠∅, l=1 n=1
supp(b i N )∩K≠∅
Q ∑
Q∑
(T K ) li (A K ) ln (T K ) nj . ✷
Example 3.5.16 (Assembly for linear Lagrangian finite elements on triangular mesh).
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C. Schwa
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Gradinar
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DRAFT
✬
Theorem 3.5.13. The stiffness matrix and load vector can be obtained from their cell counterparts
by
✫
A = ∑ K
T ⊤ K A KT K , ⃗ϕ = ∑ K
with the index mapping matrices (“T-matrices”)
{
j 1 , if (b
(T K ) ij := N ) |K = bi K ,
0 , otherwise.
T ⊤ K ⃗ϕ K , (3.5.14)
T K ∈ R Q,N , defined by
Note: Every T-matrix has exactly one non-vanishing entry per row.
1 ≤ i ≤ Q,1 ≤ j ≤ N . (3.5.15)
✩
3.5
p. 365
Numerical
Methods
for PDEs
✪R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
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DRAFT
Using the local/global numbering indicated beside
⎛
0 1 0 0 0 0 0 0 0 0 0 0 ⎞
→ T K ∗ = ⎝0 0 0 0 0 0 1 0 0 0 0 0⎠
0 0 0 0 0 0 0 0 1 0 0 0
11
10
8
9
3
2 7
6
12 K ∗
1
5
1 2
4
3
Fig. 108
✸
3.5
p. 36
Numerica
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R. Hiptm
C. Schwa
H.
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DRAFT
Proof. (of Thm. 3.5.13)
(A) ij = a(b j N ,bi N ) = ∑
a K (b j N|K ,bi N|K ) = ∑
a K (b l(j)
K∈M K∈M,supp(b j N )∩K≠∅,
supp(b i N )∩K≠∅
K ,bl(i) K ) = ∑
(A K ) l(i),l(j) .
K∈M,supp(b j N )∩K≠∅,
supp(b i N )∩K≠∅
l(i) ∈ {1,...,Q},1 ≤ i ≤ N ˆ= index of the local shape function corresponding to the global shape
functionb i N onK.
3.5
p. 366
★
✧
Cell oriented assembly ↔ (3.5.14)↔A = ∑ K
T ⊤ K A KT K
⎧
A = ∑ ⎨
T ⊤ K A KT K :=
K ⎩
enddo
✥
✦
⇕
foreachK ∈ M do
local operations onK (→A K ) andA = A+T ⊤ K A KT K
⎫
⎬
⎭
3.5
p. 36

Notion: local operations ˆ= required only data from fixed “neighbourhood” ofK
computational effort “O(1)”: independent of♯M
Computational cost(Assembly of Galerkin matrixA)= O(♯M)
Cell oriented assembly in LehrFEM [6, Sect. 5.1]
function A = assemble(Mesh)
for k = Mesh.Elements’
idx = ❶
Aloc = ❷
A(idx,idx) = A(idx,idx)+Aloc;
end
❶ row vector of index numbers of
global shape functions b i 1
N ,...,b i Q
N ∈ V N
corresponding to local shape functions
b 1 K ,...,bQ K : idx= (i 1 ,...,i Q )
(encodes index mapping matrixT K )
❷ Q×Q element stiffness matrix
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
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DRAFT
3.5
p. 369
MATLAB-CODE: assembly for quadratic Lagrangian FE
function A = assemMat_QFE(Mesh,EHa 1 ndle,varargin)
nV = size(Mesh.Coordinates,1);
nE = size(Mesh.Elements,1)
I 2 = zeros(36*nE,1); J = I; a = I; offset = 0;
for k =1:nE
vidx = Mesh.Elements(k,:)
idx 3 = [vidx,...
Mesh.Vert2Edge(vidx(1),vidx(2))+nV,...
Mesh.Vert2Edge(vidx(2),vidx(3))+nV,...
Mesh.Vert2Edge(vidx(3),vidx(1))+nV];
Aloc 4 = transpose(EHandle(Mesh.Coordinates(vidx,:),...
Mesh. 5 ElemFlag(k),varargin{:}));
Qsq = prod(size(Aloc)); range = offset + 1:Qsq;
t = idx(ones(length(idx),1),:)’; I(range) = t(:);
t = idx(ones(1,length(idx)),:); J(range) = t(:);
a(range) = Aloc(:);
offset = offset + Qsq;
end
A 6 = sparse(I,J,a);
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
3.5
p. 37
For Lagrangian FEM of fixed degreep (→ Sect. 3.4):
Numerical
Methods
for PDEs
1: EHandle (function handle)→provides element stiffness matrixA K ∈ R 6,6
Numerica
Methods
for PDEs
the total computational effort is of the orderO(♯M) = O(N),N := dimS 0 p(M).
2: I,J,a ˆ= linear arrays storing(i,j,(A) ij ) for stiffness matrixA.
Initialized with 0 for the sake of efficiency→Ex. 3.5.18
Example 3.5.17 (Assembly for quadratic Lagrangian FE in MATLAB code).
3: idx ˆ= index mapping vector, see ❶ above
Setting: FE spaceS 0 2 (M) on triangular meshMof polygonΩ ⊂ R2 , see Ex. 3.4.2
Recall: 6 local shape functions: 3 vertex-associated, 3 edge-associated → (3.4.4)
Convention:
vertex-associated global shape functions→ b 1 N ,...,b♯V(M) N
edge-associated global shape functions→ b ♯V(M)+1
N ,...,b ♯V(M)+♯E(M)
N
3
Local numbering
→
1
6
K
4
5
2
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C. Schwab,
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Gradinaru
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DRAFT
3.5
p. 370
4: Aloc =A K ∈ R 6,6 (element stiffness matrix→Def. 3.5.12)
5: Mesh.ElemFlag(k) marks groups of elements (e.g. to select local coefficient functionα(x)
in (2.8.4))
6: Build sparse MATLAB-matrix (→ Def. 3.2.7) from index-entry arrays, see manual entry for MAT-
LAB function sparse and [18, Sect. 2.6.2].
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.5
p. 37

Remark 3.5.18 (Efficient implementation of assembly). → [18, Sect. 2.6.2]
tic-toc-timing (min of 4v runs), MATLAB V7, Intel Pentium 4 Mobile CPU 1.80GHz, Linux
Computation of element stiffness matrices skipped !
• Sparse assembly:
A(idx,idx) = A(idx,idx) + Aloc;
• Array assembly I: “growing arrays”
I = []; J = []; a = [];
...
t = idx(:,ones(length(idx),1))’;
I = [I;t(:)];
t = idx(:,ones(1,length(idx)));
J = [J;t(:)];
a = [a; Aloc(:)];
• Array assembly III
→ see code fragment above
Time [s]
Timing for different assembly routines
10 2
10 1
10 0
10 −1
10 −2
Sparse assembly
Array assembly, type 1
Array assembly, type 2
Array assembly, type 3
10 −3
10 1 10 2 10 3 4
# Dofs
Fig. 109
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
whereλ i are the affine linear barycentric coordinate functions (linear shape functions), see Fig. 72.
For the barycentric coordinate representation of the quadratic local shape functions see (3.4.4), for a
justification of (3.5.19) consult Rem. 3.6.9.
⇒ gradb i K =
∑ (
κ α
α∈N 3 0 ,|α|≤p
To evaluate
∫
K
α 1 λ α 1−1
1 λ α 2
2 λα 3
3 gradλ 1 +α 2 λ α 1
1 λα 2−1
2 λ α 3
3 gradλ 2+
α 3 λ α 1
1 λα 2
2 λα 3−1
3 gradλ 3
)
.
(3.5.20)
λ β 1
1 λβ 2
2 λβ 3
3 gradλ i·gradλ j dx , i,j ∈ {1,2,3}, β k ∈ N . (3.5.21)
Numerica
Methods
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C. Schwa
H.
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V.
Gradinar
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DRAFT
More detailed discussion→[31] and [18, Sect. 2.6.2].
3.5.4 Local computations and quadrature
We have seen that the (global) Galerkin matrix and right hand side vector are conveniently generated
by “assembling” entries of element (stiffness) matrices and element (load) vectors.
Now we study the computation of these local quantities, see also Sect. 3.2.5, 3.2.6.
First option:
analytic evaluations
△
3.5
p. 373
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
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DRAFT
Ifa 1 ,a 2 ,a 3 vertices ofK (counterclockwise ordering):
λ 1 (x) = 1 ( a 2
))
x−(
1 a 2
·(
2 −a 3 )
2|K| a 2 2
2 a 3 1 −a2 ,
1
λ 2 (x) = 1 ( a 3
))
x−(
1 a 3
·(
2 −a 1 )
2|K| a 3 2
2 a 1 1 −a3 ,
1
λ 3 (x) = 1 ( a 1
))
x−( a 1
·( −a 2 )
2|K| a 1 2
2 a 2 1 −a1 .
1
a 3 = ( a 3 1,a 3 ) T
2
ω 3
n 1
n 2
n 3
Fig. 110
ω 1
ω 2
a 1 = ( a 1 1,a 1 2) T
a 2 = ( a 2 1 ,a2 2
) T
3.5
p. 37
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
We discuss bilinear form related to−∆, triangular Lagrangian finite elements of degreep, Sect. 3.4.1,
Def. 3.4.1:
∫
K triangle: a K (u,v) := gradu·gradvdx element stiffness matrix .
K
Use barycentric coordinate representations of local shape functions, in 2D
b i K =
∑
κ α λ α 1
1 λα 2
2 λα 3
3 , κ α ∈ R , (3.5.19)
α∈N 3 0 ,|α|≤p
3.5
p. 374
gradλ 1 = 1
2|K|
( a 2
2 −a 3 )
2
a 3 1 −a2 , gradλ 2 = 1
1 2|K|
( a 3
2 −a 1 )
2
a 1 1 −a3 , gradλ 3 = 1
1 2|K|
( a 1 −a 2 )
2
a 2 1 −a1 .
1
(3.5.22)
3.5
p. 37

✬
Lemma 3.5.23 (Integration of powers of barycentric coordinate functions).
For any non-degenerated-simplexK andα j ∈ N,j = 1,...,d+1,
∫
λ α 1
1 ·····λα d+1
d+1 dx = d!|K| α 1 !α 2 !·····α d+1 !
K
(α 1 +α 2 +···+α d+1 +d)!
✫
∀α ∈ N d+1
0 . (3.5.24)
✩
✪
Numerical
Methods
for PDEs
Local quadrature formula, cf. (3.2.17)
Terminology:
∫
f(x)dx ≈ ∑ P∑
|K| ωl K f(ζ K l ) , ζ K l ∈ K, ωl K ∈ R , P ∈ N . (3.5.25)
Ω
K∈M l=1
ω K l → weights , ζ K l → quadrature nodes
(3.5.25) =P -point local quadrature rule
Numerica
Methods
for PDEs
Proof ford = 2
Step #1: transformationK → “unit triangle” ̂K := convex{ (00 ) ,
( 10 ) ,
( 01 ) } ,
⇒
∫
K
∫1
λ β 1
1 λβ 2
2 λβ 3
3 dx = 2|K|
(∗)
= 2|K|
0
∫1
0
∫1
= 2|K|
0
1−ξ ∫ 1
ξ β 1
1 ξβ 2
2 (1−ξ 1−ξ 2 ) β 3 dξ 2 dξ 1
0
∫1
ξ β 1
1 (1−ξ 1 ) β 2+β 3 +1 s β 2(1−s) β 3dsdξ 1
0
ξ β 1
1 (1−ξ 1) β 2+β 3 +1 dξ1·B(β 2 +1,β 3 +1)
R. Hiptmair
C. Schwab,
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Gradinaru
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DRAFT
3.5
p. 377
Mandatory • for computation of load vector (f complicated/only available in procedural form,
Rem. 1.5.3,)
• for computation of stiffness matrix, ifα = α(x) does not permit analytic integration.
Example for local quadrature rule: 2D trapezoidal rule from (3.2.17)
Guideline [18, Sect. 10.2]:
only quadrature rules with positive weights are numerically stable.
R. Hiptm
C. Schwa
H.
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V.
Gradinar
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DRAFT
3.5
p. 37
(∗) ˆ= substitutions(1−ξ 1 ) = ξ 2 ,
∫1
B(α,β) :=
= 2|K| B(β 1 +1,β 2 +β 3 +2)·B(β 2 +1,β 3 +1) ,
0
B(·,·) ˆ= Euler’s beta function
t α−1 (1−t) β−1 dt , 0 < α,β < ∞ .
UsingΓ(α+β)B(α,β) = Γ(α)Γ(β),Γ ˆ= Gamma function,Γ(n) = (n−1)!,
∫
⇒ λ β 1
1 λβ 2
2 λβ 3
3 dx = 2|K|· Γ(β 1+1)Γ(β 2 +1)Γ(β 3 +1)
K
Γ(β 1 +β 2 +β 3 +3)
Remark. Alternative: symbolic computing (MAPLE, Mathematica) for local computations
Second option:
cell-based quadrature
At this point turn the pages back to (1.5.61) and remember the use of numerical quadrature for
computing the Galerkin matrix for the linear finite element method in 1D.
✷ .
Numerical
Methods
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
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DRAFT
How to gauge the quality of parametric local quadrature rules ? → [18, Sect. 10.3]
✬
Quality of a parametric local quadrature rule on K ∼ maximal degree of polynomials (multivariate
→ Def. 3.3.3, or tensor product → Def. 3.3.7) on K integrated exactly by the corresponding
quadrature rule onK.
✫
Parlance:
Quadrature rule exact forP p (R d ) ⇒
quadrature rule of orderp+1
degree of exactnessp
How are quadrature rules specified for the many different cells of a finite element mesh ?
Remark 3.5.26 (Affine transformation of triangles).
✩
✪
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Reminder: numerical quadrature mandatory in the presence of coefficients/source terms in procedural
form → Rem. 1.5.3.
3.5
p. 378
3.5
p. 38

Definition 3.5.27 (Affine (linear) transformation).
MappingΦ : R d ↦→ R d is affine (linear), ifΦ(x) = Fx+τ with someF ∈ R d,d ,τ ∈ R d .
Numerical
Methods
for PDEs
Remark 3.5.30 (Transformation of local quadrature rules on triangles).
△
Numerica
Methods
for PDEs
{( ( (
✎ notation: ‘unit triangle” ̂K 0 1 0
:= convex , ,
0)
0)
1)}
✬
✩
Φ K (̂x) := F K̂x + τ K ˆ= affine transformation (→ Def. 3.5.27) mapping ̂K to triangle K, see
Lemma 3.5.28.
Lemma 3.5.28 (Affine transformation of triangles).
For any non-degenerate triangleK ⊂ R 2 (|K| > 0) there is a unique affine transformationΦ K ,
Φ K (̂x) = F K̂x+τ K (→ Def. 3.5.27), withK = Φ(̂K).
✫
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C. Schwab,
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V.
Gradinaru
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DRAFT
✪
By transformation formula for integrals [29, Satz 8.5.2]
∫ ∫
f(x)dx =
K
f(Φ K (̂x))|detF K |d̂x . (3.5.31)
̂K
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
P -point quadrature formula on ̂K
P -point quadrature formula onK
( 01 )
̂K
Φ K (̂x) =
( ) 1 2
̂x
3 4
( 24 )
K
( 13 )
3.5
p. 381
Numerical
Methods
for PDEs
➣
∫
f(̂x)d̂x ≈ |̂K|
̂K
P∑
̂ω l f(̂ζ l )
l=1
∫
f(x)dx ≈ ∑ P∑
|K| ωl K f(ζ K l )
Ω
K∈M l=1
with ωl K = ̂ω l , ζ K l = Φ K (̂ζ l ) .
Only quadrature formula (3.5.25) on unit triangle ̂K needs to be specified!
(The same applies to tetrahedra, where affine mappings ford = 3 are used.)
Since the spaceP p (R d ) is invariant under affine mappings,
(3.5.32)
3.5
p. 38
Numerica
Methods
for PDEs
q ∈ P p (R d ) ⇒ ̂x ↦→ q(Φ(̂x)) ∈ P p (R d ) for any affine transformationΦ , (3.5.33)
( 00 ) ( 10 )
Formula:
{( ) ( ) ( )}
a 1
K = convex 1 a 2
a 1 , 1 a 3
2 a 2 , 1
2 a 3 2
Note that |K| = 1 2 |detF K| .
( 00 )
⎛
Fig. 111
⇒ Φ K (̂x) = ⎝ a2 1 −a1 1 a3 1 −a1 1 ⎠̂x+ ⎝ a1 1
⎠
a 2 2 −a1 2 a3 2 −a1 2 a 1 .
2
(3.5.29)
⎞
⎛
⎞
R. Hiptmair
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Harbrecht
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Gradinaru
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DRAFT
3.5
p. 382
the orders of the quadrature rules on the left and right hand side of (3.5.31) agree.
Example 3.5.34 (Useful quadrature rules on triangles). → [6, Sect. 3.3.2]
Specification of quadrature rule for “unit triangle” ̂K := convex{ (00 ) ,
( 10 ) ,
( 01 ) } .
Quadrature rules described by pairs(̂ω 1 ,̂ζ 1 ),...,(̂ω P ,̂ζ P ),P ∈ N.
△
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C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.5
p. 38

Quadrature rule of order 2 (exact forP 1 ( ̂K))
{( ( ( ( ( (
1 0 1 0 1 1
3 0))
, 3 1))
, 3 0))}
,
,
,
. (3.5.35)
Numerical
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Quadrature rule P1O2 Quadrature rule P3O3 Quadrature rule P6O4 Quadrature rule P7O6
Numerica
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Quadrature rule of order 3 (exact forP 2 ( ̂K))
{( ( )) ( ( )) ( ( ))}
1
3 , 1/2 1 0 1
,
0 3 , ,
1/2 3 , 1/2
. (3.5.36)
1/2
One-point quadrature rule of order 2 (exact forP 1 (̂K))
{( ( ))}
1/3
1, . (3.5.37)
1/3
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Example 3.5.40 (Local quadrature rules on quadrilaterals).
IfK quadrilateral ⇒ ̂K := convex{ (00 ) ,
( 10 ) ,
( 01 ) ,
( 11 ) } (unit square).
△
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Quadrature rule of order 6 (exact forP 5 ( ̂K))
{( ( )) ( √ (
9
40 , 1/3 155+ 15 6+ √ )) ( √ (
15/21 155+ 15
, ,
1/3 1200 6+ √ 9−2 √ ))
15/21
, ,
15/21 1200 6+ √ ,
15/21
( √ (
155+ 15 6+ √ )) ( √ (
15/21 155− 15
,
1200 9−2 √ 6− √ ))
15/21
, ,
15/21 1200 9+2 √ ,
15/21
( √ (
155− 15 9+2 √ )) ( √ (
15/21 155− 15 6− √ ))}
15/21
In [10]: quadrature rules up to orderp = 21 with P ≤ 1/6p(p+1)+5
Remark 3.5.39 (Numerical quadrature in LehrFEM). → [6, Sect. 3]
Routines returnP -point quadrature formulas for
⎧ (00 ) (
⎨unit triangle convex{
,
10 ) ( , 01 ) } for triangular cell,
̂K =
(00 ) (
⎩unit square convex{
,
10 ) ( , 11 ) ( , 01 ) } for rectangular cell,
in MATLAB structure QuadRule with fields
QuadRule.w: weights ̂ω l of quadrature rule on ̂K,
QuadRule.x: coordinates of nodes ̂ζ l ∈ ̂K of quadrature rule on ̂K
(3.5.38)
✸
3.5
p. 385
Numerical
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R. Hiptmair
C. Schwab,
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V.
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DRAFT
On ̂K:
tensor product construction:
If{(ω 1 ,ζ 1 ),...,(ω P ,ζ P )},P ∈ N, quadrature rule on the interval]0,1[, exact forP p ]0,1[, then
{
(ω1 2,( ζ 1
)
ζ ) ··· (ω1 ω 1 P , ( ζ 1
)
ζ )
P
.
.
(ω 1 ω P , ( ζ ) P
ζ1
) ··· (ω 2
P , ( ζ ) } P
ζP
)
provides a quadrature rule on the unit square ̂K, exact forQ p ( ̂K).
Quadrature rules on]0,1[ (→ [18, Ch. 10]):
• classical Newton-Cotes formulas (equidistant
quadrature nodes).
• Gauss-Legendre quadrature rules, exact for
P 2P (]0,1[) using onlyP nodes.
• Gauss-Lobatto quadrature rules: P nodes including{0,1},
exact forP 2P−1 (]0,1[).
Number P of quadrature nodes
20
18
16
14
12
10
8
6
4
Gauss−Legendre nodes in [−1,1]
2
−1 −0.5 0 0.5 1
t
Fig. 112
✸
3.5
p. 38
Numerica
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R. Hiptm
C. Schwa
H.
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V.
Gradinar
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DRAFT
For triangles: QuadRule = PnOq(), ˆ= n-point quadrature of orderq
Location of quadrature nodes ̂ζ l in unit triangle ̂K:
3.5
p. 386
3.5
p. 38

3.5.5 Incorporation of essential boundary conditions
Numerical
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Numerica
Methods
for PDEs
Recall variational formulation of non-homogeneous Dirichlet boundary value problem from Ex. 2.8.1:
u ∈ H 1 (Ω)
u = g on∂Ω : ∫Ω
with (admissible→Rem. 2.9.4) Dirichlet data
∫
κ(x)gradu·gradvdx = fvdx ∀v ∈ H0 1 (Ω) . (2.8.4)
Ω
⇓
−div(κ(x)gradu) = f inΩ , u = g on∂Ω ,
g ∈ C 0 (∂Ω).
Recall from Sect. 2.9: Dirichlet b.c. = essential boundary conditions
(built into trial space)
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Ω
Fig. 113
Example 3.5.43 (offset functions for linear Lagrangian FE).
✁ Maximal support ofu 0 on triangular mesh.
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Remember offset function technique, see (1.3.19) and Sect. 2.1.3:
∫
w ∈ H0 1 (Ω): κ(x)gradw·gradvdx
(2.8.4) ⇔ u = u 0 +w , ∫ Ω
= −κ(x)gradu 0·gradv +fvdx ∀v ∈ H 1 (3.5.41)
0 (Ω) , Ω
3.5
p. 389
3.5
p. 39
where
u 0 = g on∂Ω
Adapt this to finite element Galerkin discretization by generalizing the 1D example Rem. 1.5.64 to
d = 2,3:
Numerical
Methods
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For Dirichlet datag ∈ C 0 (∂Ω)
∑
u 0 = g(x)b x N (3.5.44)
x∈V(M)∩∂Ω
Numerica
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Remember:
Rem. 3.4.12.
we already know finite element subspaces V 0,N := Sp,0 0 (M) ⊂ H1 0 (Ω), see
Idea from Rem. 1.5.64:
use offset functionu 0 ∈ V N := S 0 p(M)
locally supported near the boundary:
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b x N ˆ= tent function associated with node x ∈
V(M), cf. Sect. 3.2.3. (3.5.44) generalizes
(1.5.65) to 2D.
Fig. 114
✸
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C. Schwa
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V.
Gradinar
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DRAFT
supp(u 0 ) ⊂ ⋃ {K ∈ M: K ∩∂Ω ≠ ∅} . (3.5.42)
Remark 3.5.45 (Approximate Dirichlet boundary conditions).
Be aware that for the choice (3.5.44)
3.5
p. 390
u 0 ≠ g on∂Ω .
Rather, u 0 is a piecewise linear interpolant of the Dirichlet data g ∈ C 0 (∂Ω). Therefore, another
approximation comes into play when enforcing Dirichlet boundary conditions by means of piecewise
3.5
p. 39

polynomial offset functions.
Example 3.5.46 (Implementation of non-homogeneous Dirichlet b.c. for linear FE).
Consider (2.8.4) and assume the following ordering of the nodal basis functions, see Fig. 67
△
Numerical
Methods
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from the discretization ofaon the (larger) FE spaceS1 0(M):
( )( ) ( )
A0 A 0∂ ⃗µ0 ⃗ϕ
A T 0∂ A = . (3.5.49)
∂∂ ⃗µ ∂ ⃗ϕ ∂
Here, ⃗µ 0 ˆ= coefficients for interior basis functionsb 1 N ,...,bN N
⃗µ ∂ ˆ= coefficient for basis functions b N+1
N ,...,bM N
for basis functions associated with nodex
∈ ∂Ω.
➋ We realize that the coefficient vector of (3.5.49) is that of a FE approximation ofu
Numerica
Methods
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B 0 := {b 1 N ,...,bN N } ˆ= nodal basis ofS0 1,0 (M),
(tent functions associated with interior nodes)
B := B 0 ∪{b N+1
N ,...,bM N } ˆ= nodal basis ofS0 1 (M)
(extra basis functions associated with nodes∈ ∂Ω).
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⃗µ ∂ known = values ofg at boundary nodes: ⃗µ ∂ = ⃗γ
➌ Moving known quantities in (3.5.49) to the right hand side yields (3.5.48).
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DRAFT
Note:
M = ♯V(M),N = ♯{x ∈ V(M), x ∉ ∂Ω} (no. of interior nodes)
Example 3.5.50 (Non-homogeneous Dirichlet boundary conditions in LehrFEM).
✸
A 0 ∈ R N,N ˆ= Galerkin matrix for discrete trial/test spaceS 0 1,0 (M),
A ∈ R M,M ˆ= Galerkin matrix for discrete trial/test spaceS 0 1 (M).
( ) A0 A
A = 0∂
A T 0∂ A , A 0∂ ∈ R N,M−N , A ∂∂ ∈ R M−N,M−N . (3.5.47)
∂∂
Ifu 0 ∈ S1 0 (M) is chosen according to (3.5.44), then
{
u 0 ∈ Span b N+1 }
N ,...,bM N
M∑
⇔ u 0 = γ j−N b j N ,
j=N+1
which means that the coefficient vector ⃗ν of the finite element approximation w N ∈ S1,0 0 (M) of
w ∈ H0 1 (Ω) from (3.5.41) solves the linear system of equations
➣
A 0 ⃗ν = ⃗ϕ−A 0∂ ⃗γ . (3.5.48)
Non-homogeneous Dirichlet boundary data are taken into account through a modified right
hand side vector.
Alternative consideration leading to (3.5.48):
➊ First ignore essential boundary conditions and assemble the linear system of equations arising
3.5
p. 393
Numerical
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for PDEs
R. Hiptmair
C. Schwab,
H.
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V.
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DRAFT
3.5
p. 394
Code 3.5.51: Solving 2nd-order Dirichlet BVP with linear FE in LehrFEM
1 % Initialize constants anf functions
2 NREFS = 5; % Number of red refinement steps
3 F_HANDLE = @f_LShap; % Right hand side source term
4 GD_HANDLE = @g_D_LShap; % Dirichlet boundary data
5 % Load mesh
6 Mesh = load_Mesh(’meshvert.dat’,’meshel.dat’);
7 Mesh.ElemFlag = ones( s i z e(Mesh.Elements,1),1);
8 Mesh = add_Edges(Mesh);
9 Loc = get_BdEdges(Mesh);
10 Mesh.BdFlags = zeros( s i z e(Mesh.Edges,1),1);
11 Mesh.BdFlags(Loc) = 1;
12
13 % Assemble stiffness matrix and load vector for linear Lagrangian FE
14 A = assemMat_LFE(Mesh,@STIMA_Lapl_LFE);
15 phi = assemLoad_LFE(Mesh,P7O6(),F_HANDLE,0,1,2);
16 p
17 % Incorporate Dirichlet boundary data for vertices adjacent to edges
18 % with carryingflag 1. U contains nodal values for Dirichlet
19 % boundary data, FreeDofs contains numbers of interior nodes.
20 % See Ex. 3.5.46 for further explanations.
21 [U,FreeDofs] = assemDir_LFE(Mesh,-7,GD_HANDLE);
22 L = L - A*U; %
23
24 % Solve the linear system
25 U(FreeDofs) = A(FreeDofs,FreeDofs)\L(FreeDofs);
26
27 % Plot solution
28 plot_LFE(U,Mesh); c o l o r b a r;
3.5
p. 39
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C. Schwa
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DRAFT
3.5
p. 39

29 plotLine_LFE(U,Mesh,[0 0] ,[1 1]);
Numerical
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Methods
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Line 22: see (3.5.48)
Remark 3.6.1 (Pullback of functions).
3.6 Parametric finite elements
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
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DRAFT
In a natural way, a transformation of domains induces a transformation of the functions defined on
them:
Definition 3.6.2 (Pullback).
Given domainsΩ,̂Ω ⊂ R d and a bijective mapping Φ : ̂Ω ↦→ Ω, the pullbackΦ ∗ u : ̂Ω ↦→ R of
a functionu : Ω ↦→ R is a function on ̂Ω defined by
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Gradinar
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DRAFT
(Φ ∗ u)(̂x) := u(Φ(̂x)) , ̂x ∈ ̂Ω .
CORE SECTION
Parametric mapping techniques essential for understanding the local assembly functions of the MAT-
LAB finite element library.
3.6
p. 397
Implicitly, we used the pullback of integrands when defining quadrature rules through transformation,
see (3.5.31).
3.6
p. 39
?
? ? ?
?
?
?
?
Fig. 115
✁ 2D hybrid mesh M with (curvilinear) triangles
and quadrilaterals
How to buildS 0 1 (M)?
Numerical
Methods
for PDEs
Obviously, the pullbackΦ ∗ induces a linear mapping between spaces of functions on Ω and ̂Ω,
respectively.
Φ ∗ u defined here
Φ
u defined here
Numerica
Methods
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3.6.1 Affine equivalence
Ω
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DRAFT
Recall Lemma 3.5.28: affine transformation of triangles (3.5.29)
All cells of a triangular mesh are affine images of “unit triangle” ̂K
“Unit triangle”: ̂K
〈 (00 ) (
= , 10 ) ( , 01 ) 〉 ̂x 2
a 3
1 Φ
{
K (̂x) = F K̂x+τ K
ForK = convex a 1 ,a 2 ,a 3} :
a 1
( a 2
F K = 1 −a 1 )
1 a3 1 −a1 1
a 2 2 −a1 2 a3 2 −a1 , τ K = a 1 .
2
̂K
K
3.6
p. 398
̂Ω
In the context of numerical quadrature we made the observation, cf. (3.5.33):
Φ ∗
Fig. 117
△
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C. Schwa
H.
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V.
Gradinar
A. Chern
DRAFT
3.6
p. 40

✬
Lemma 3.6.3 (Preservation of polynomials under affine pullback).
IfΦ : R d ↦→ R d is an affine (linear) transformation (→ Def. 3.5.27), then
✫
Φ ∗ (P p (R d )) = P p (R d ) and Φ ∗ (Q p (R d )) = Q p (R d ) .
✩
✪
Numerical
Methods
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Proof. (of (3.6.4)) Recall the definition of global shape functions and also local shape functions for
S 0 p(M),p ∈ N, by means of the conditions (3.4.3) at interpolation nodes, see Ex. 3.4.2 forp = 2. ✷
Numerica
Methods
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In fact, Lemma 3.5.28 reveals another reason for the preference for polynomials in building discrete
Galerkin spaces.
Proof. (of Lemma 3.5.28)
Since the pullback is linear, we only need to study its action on the (monomial) basis x ↦→ x α ,
α ∈ N d 0 ofP p(R d ), see Def. 3.3.3 and the explanations on multi-index notation (3.3.4).
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Note: we already used the definition of basis functions through basis functions on the “reference cell”
[0,1] and affine pullback in 1D, see Rem. 1.5.33
Now write
p i K ˆ= (local) interpolation nodes on triangleK,
̂p i ˆ= (local) interpolation nodes on unit triangle ̂K.
Observe: Assuming a matching numbering p i K = Φ K(̂p i ). where Φ K : ̂K ↦→ K is the unique
affine transformation mapping ̂K ontoK, see (3.5.29).
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̂x 2
a 1 a 2
Then resort to induction w.r.t. degreep.
Φ ∗ K (xα ) = Φ ∗ K (x 1)·Φ ∗ K ( xα′ }{{}
∈P p−1 (R d )
d∑
) = ( (F) 1l̂x l +τ 1 )
l=1
} {{ }
∈P 1 (R d )
·Φ ∗ K (xα′ ) ∈ P p (R d ) ,
} {{ }
∈P p−1 (R d )
3.6
p. 401
This is clear for p = 2, because affine transformations
take midpoints of edges to midpoints
of edges. The same applies to the interpolation
nodes for higher degree Lagrangian finite elements
defined in Ex. 3.4.5.
1
̂K
Φ K̂x 1
1
K
a 3
Fig. 118
3.6
p. 40
with α ′ := (α 1 − 1,α 2 ,...,α d ), where we assumed α 1 > 0. Here, we have used the induction
hypothesis to concludeΦ ∗ K (xα′ )inP p−1 (R d ).
✷
Numerical
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The local shape functions b i K ∈ P p(R d ), ̂b i ∈ P p (R d ), i = 1,...,Q, are uniquely defined by the
interpolation conditions
Numerica
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b i K (pj K ) = δ ij , ̂b i (̂p j ) = δ ij . (3.6.5)
A simple observation:
ConsiderS 0 1 (M), triangleK ∈ M, unit triangle ̂K, affine mappingΦ K : ̂K ↦→ K
Together with p i K = Φ K(̂p i ) this shows that Φ ∗ K bi K
satisfies the interpolation conditions (3.6.5) on
̂K and, thus, has to agree witĥb i .
✷
• b 1 K ,b2 K ,b3 K
(standard) local shape functions onK,
• ̂b 1 ,̂b 2 ,̂b 3 (standard) local shape functions on ̂K,
→ Ex. 3.3.13
̂b i = Φ ∗ K bi K ⇔ ̂b i (̂x) = b i K (x) , x = Φ K(̂x) (3.6.4)
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Terminology:
finite element spaces satisfying (3.6.4) are called affine equivalent
Remark 3.6.6 (Evaluation of local shape functions at quadrature points).
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Of course, we assume thatΦ K respects the local numbering of the vertices of ̂K andK.
We consider Lagrangian finite element spaces on a simplicial meshM.
The proof of (3.6.4) is straightforward: both Φ ∗ K bi K (by Lemma 3.6.3) and̂b i are (affine) linear functions
that attain the same values at the vertices of ̂K. Hence, they have to agree.
Recall from Sect. 3.5.4:
definition (3.5.32) of local quadrature formulas via “unit simplex”.
In particular: quadrature nodes onK: ζ K l = Φ K (̂ζ l )
Note:
(3.6.4) holds true for all simplicial Lagrangian finite element spaces
3.6
p. 402
b i K (ζK Def. 3.6.2
l ) = Φ ∗ K (bi K )(̂ζ l ) (3.6.4)
= ̂b i (̂ζ l ) independent ofK ! . (3.6.7)
3.6
p. 40

∫
F(b i P K ,bj K )dx ≈ |K| ∑
ω l F(̂b i (ζ l ),̂b j (ζ l )) , (3.6.8)
K
for any functionF : R 2 ↦→ R.
➣
Precomputêb i (ζ l ),i = 1,...,Q,l = 1,...,P and store the values in a table!
l=1
Numerical
Methods
for PDEs
This is very interesting, because
the values ∂̂b i
∂̂x 1
(̂ζ l ) can be precomputed,
simple expressions forgradλ i·gradλ j are available, see Sect. 3.2.5.
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△
△
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DRAFT
3.6.2 Example: Quadrilaterial Lagrangian finite elements
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DRAFT
Remark 3.6.9 (Barycentric representation of local shape functions).
We consider Lagrangian finite element spaces on a simplicial meshM.
(3.4.4): formulas for local shape functions for S2 0 (M) (d = 2) in terms of
barycentric coordinate functions λ i , i = 1,2,3. Is this coincidence? NO! Does (3.5.19) hold
for any (simplicial) Lagrangian finite element space?
YES!
where
b i (3.6.4)
K (x) = (Φ −1
K )∗(̂x ↦→̂b i )
(̂x 1 ,̂x 2 )
,
=̂b i ((Φ −1
K )∗ (̂λ 2 )(x),(Φ −1
K )∗ (̂λ 3 )(x)) =̂b i (λ 2 (x),λ 3 (x))
λ 2 (̂x) = ̂x 1 ,λ 3 (̂x) = ̂x 2 ,λ 1 (̂x) = 1− ̂x 1 − ̂x 2 ˆ= barycentric coordinate functions on ̂K,
see Ex. 3.3.13,
λ i ˆ= barycentric coordinate functions on triangleK, see Fig. 72,
Φ K ˆ= affine transformation (→ Def. 3.5.27),Φ K (̂K) = K, see (3.5.29).
3.6
p. 405
Numerical
Methods
for PDEs
So far, see Sect. 3.3.3 and (3.3.11), we have adopted the perspective
global shape functions
Now we reverse this construction
local shape functions
Restriction to element
−−−−−−−−−−−−−→
local shape functions
“glueing”
−−−−−→ global shape functions (3.6.10)
In fact, when building the global basis functions for quadratic Lagrangian finite elements we already
proceeded this way, see Ex. 3.4.2. Fig. 94 lucidly conveys what is meant by “glueing”.
Be aware that the possibility to achieve a continuous global basis function by glueing together local
shape function on adjacent cells, entails a judicious choice of the local shape functions.
This section will demonstrate how the policy (3.6.10) together with the formula (3.6.4) will enable us
to extend Lagrangian finite element beyond the meshes discussed in Sect. 3.4.
3.6
p. 40
Numerica
Methods
for PDEs
➣
By the chain rule:
gradb i ∂̂b i
K (x) = (̂x)gradλ
∂̂x 2 + ∂̂b i
(̂x)gradλ
1 ∂̂x 3 , x = Φ K (̂x) .
2
This formula is convenient, becausegradλ i ≡ const, see (3.5.22).
This facilitates the computation of element (stiffness) matrices for 2nd-order elliptic problems in variational
form: when using a quadrature formula according to (3.5.32)
∫
(α(x)gradb i K )·gradbj K dx
K
P K∑
≈ |K|
l=1
⎛⎛
⎞
∂̂b i
⎜
ω l ⎝⎝∂̂x (̂ζ
1 l )
⎠T ( ) ⎛ ⎞⎞
∂̂b gradλ1 ·gradλ 1 gradλ 1 ·gradλ j
2 ⎝∂̂x (̂ζ
1 l )
⎠⎟
∂̂b i
∂̂x (̂ζ l ) gradλ 1 ·gradλ 2 gradλ 2 ·gradλ 2 ∂̂b j
∂̂x (̂ζ ⎠
l )
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DRAFT
3.6
p. 406
✁ quadrilateral meshMin 2D
What is “S 0 1 (M)”?
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C. Schwa
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DRAFT
3.6
p. 40

Clear: If K is a rectangle, ̂K the unit square, then there is a unique affine transformation Φ K (→
Def. 3.5.27) withK = Φ K ( ̂K).
In this case (3.6.4) holds for the local shape functions of bilinear Lagrangian finite elements from
Ex. 3.4.6 (and all tensor product Lagrangian finite elements introduced in Sect. 3.4.2)
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⇓
( )
α1 +β
Φ K (̂x) = 1̂x 1 +γ 1̂x 2 +δ 1̂x 1̂x 2 , α
α 2 +β 2̂x 1 +γ 2̂x 2 +δ i ,β i ,γ i ,δ i ∈ R .
2̂x 1̂x 2
Numerica
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Idea:
local shape functions
“glueing”
−−−−−→
Build local shape functions by “inverse pullback”
global shape functions
b i K = (Φ−1 K )∗̂b i , (3.6.11)
where
{̂b i} Q ˆ= set of shape functions on reference element ̂K.
i=1
➣ What isΦ K for a general quadrilateral ?
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
The mapping property Φ K (â i ) = a i is evident. In order to see Φ K ( ̂K) = K ( ̂K ˆ= unit square) for
(3.6.12), verify thatΦ K maps all parallels to the coordinate axes to straight lines.
Moreover, a simple computation establishes:
If ̂K is the unit square,Φ K : ̂K ↦→ K a bilinear transformation, and̂b i the bilinear local shape
functions (3.4.8) on ̂K,
then(Φ −1
K )∗̂b i are linear on the edges ofK.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.6
p. 409
“Glueing” of local shape functions possible
3.6
p. 41
̂x
unit square 2
â 4 â 3
parallelogram
Numerical
Methods
for PDEs
Explanation:
Numerica
Methods
for PDEs
â 1 â 2
Affine
mapping
̂x 1
Φ K
Fig. 119
Affine transformations fail to produce
general quadrilaterals from a square.
They only give parallelograms.
̂x
unit square 2
a 4 quadrilateral
â 4 â 3
a 3
a 1 a 2
Φ K
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
y
˜K
K
e
j
i
x
Fig. 121
➊ Pick a vertex x ∈ V(M) and consider an adjacent
quadrilateralK, on which there is a local
shape functionb i K such thatbi K (x) = 1 andbi K
vanishes on all other vertices of K. This local
shape function is obtained by inverse pullback
of thêb i associated withΦ −1
K (x).
➋ The same construction can be carried out for
another quadrilateral ˜K that shares the vertex
x and an edge e with K. On that quadrilateral
we find the local shape functionb j˜K
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
It takes bilinear transformations to obtain
a generic quadrilateral from the unit
square.
â 1 â 2
̂x 1
bilinear
mapping
Fig. 120
➌ Both b i K|e
and are linear and attain the same values, that is 0 and 1 at the endpointsxand
bj˜K|e
y ofe, respectively.
Bilinear transformation of unit square to quadrilateral with verticesa i ,i = 1,2,3,4:
Φ K (̂x) = (1− ̂x 1 )(1− ̂x 2 )a 1 + ̂x 1 (1− ̂x 2 )a 2 + ̂x 1̂x 2 a 3 +(1− ̂x 1 )̂x 2 a 4 . (3.6.12)
3.6
p. 410
b i K|e = bj˜K|e
Continuity of global shape function (defined by interpolation conditions at nodes)
3.6
p. 41

Remark 3.6.13 (Non-polynomial “bilinear” local shape functions).
Note that the components ofΦ −1
K are not polynomial even ifΦ K is a bilinear transformation (3.6.12).
Numerical
Methods
for PDEs
Definition 3.6.14 (Parametric finite elements).
A finite element space on a mesh M is called parametric, if there exists a reference element
̂K,Q ∈ N, and functionŝb i ∈ C 0 (̂K),i = 1,...,Q, such that
∀K ∈ M: ∃ bijectionΦ K : ̂K ↦→ K: ̂b i = Φ ∗ K bi K , i = 1,...,Q ,
where{b 1 K ,...,bQ K } = set of local shape functions onK.
Numerica
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for PDEs
✬
The local shape functionsb i K defined by (3.6.11), whereΦ K is a bilinear transformation and̂b i
are the bilinear local shape functions on the unit square, are not polynomial in general.
✫
Visualization of local shape functions on trapezoidal cellK := convex{ (00 ) ,
( 30 ) ,
( 21 ) ,
( 11 ) } :
✩
R. Hiptmair
C. Schwab,
H.
✪Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.6
p. 413
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
R. Hiptm
This definition takes the possibility of “glueing” for granted: the concept of a local shape function, see
(3.3.11), implies the existence of a global shape function with the right continuity properties.
DRAFT
How to implement parametric finite elements ?
We consider a generic elliptic 2nd-order variational Dirichlet problem
u ∈ H 1 ∫
∫
(Ω) ,
u = g on ∂Ω : (α(x)gradu(x))·gradv(x)dx = f(x)v(x)dx ∀v ∈ H0 1 (Ω) . (2.3.3) 3.6
Ω
Ω
p. 41
Numerical
Methods
for PDEs
Issue: computation of element (stiffness) matrices and element (load) vectors (→ Def. 3.5.12).
Challenge:
local shape functionsb 1 K ,...,bQ K ,K ∈ M, only known implicitly
b i K = (Φ−1 K )∗̂b i
➣ Known: transformationΦ : ̂K ↦→ K, ̂K reference element, functionŝb 1 ,...,̂b Q
̂b i = Φ ∗ b i K , i = 1,...,Q (→ pullback, Def. 3.6.2)
Numerica
Methods
for PDEs
△
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Idea:
Use transformation to ̂K to compute element stiffness matrix A K ,
element load vector ⃗ϕ K :
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.6.3 Transformation techniques
✗
✖
“Bilinear” Lagrangian finite elements = a specimen of parametric finite elements
✔
✕
3.6
p. 414
∫
(A K ) ij =
∫K
=
̂K
α(x)gradb j K (x)·gradbj K (x)dx
(Φ ∗ α)(̂x)(Φ ∗ (gradb j K
} {{ }
) )(̂x)·(Φ ∗ (gradb i K ) )(̂x)|detDΦ(̂x)|d̂x ,
} {{ }
= ? = ?
∫ ∫
(⃗ϕ K ) i = f(x)b i K (x)dx = (Φ ∗ K
K
̂K
f)(̂x)̂b i (̂x)|detDΦ(̂x)|d̂x ,
3.6
p. 41

y transformation formula (for multidimensional integrals, see also (3.5.31)):
∫ ∫
f(x)dx =
K
f(̂x)|detDΦ(̂x)|d̂x for f : K ↦→ R , (3.6.15)
̂K
All integrals have been transformed to the reference element ̂K, where we apply a
quadrature formula (3.5.32).
Needed: values of determinant of Jacobi matrixDΦ at quadrature nodes ̂ζ l .
Also needed: gradientsΦ ∗ (gradb i K ) at quadrature nodes ̂ζ l !?
(Seems to be a problem asb i K
may be elusive, cf. Rem. 3.6.13!)
✬
Lemma 3.6.16 (Transformation formula for gradients).
For differentiableu : K ↦→ R and any diffeomorphismΦ : ̂K ↦→ K we have
✫
Proof :
(grad̂x (Φ ∗ u))(̂x) = (DΦ(̂x)) T (grad x u)(Φ(̂x))
} {{ }
=Φ ∗ (gradu)(x)
use chain rule for components of the gradient
∀̂x ∈ ̂K . (3.6.17)
∂Φ ∗ u
(̂x) = ∂ d∑ ∂u
u(Φ(̂x)) = (Φ(̂x)) ∂Φ j
(̂x) .
∂̂x i ∂̂x i ∂x
j=1 j ∂̂x i
⎛
∂Φ ∗ ⎞
⎛ ⎞
u
∂̂x
(̂x)
∂u
1 ⎜
⎝
. ⎟
∂Φ ∗ ⎠ = (grad̂x Φ∗ u)(̂x) = DΦ(̂x) T ∂x (Φ(̂x))
⎜ 1
⎟
⎝ . ⎠ = DΦ(̂x) T (grad x u)(Φ(̂x)) .
u
∂u
∂̂x
(̂x)
d ∂x (Φ(̂x))
d
Here,DΦ(̂x) ∈ R d,d is the Jacobian ofΦat ̂x ∈ ̂K, see [29, Bem. 7.6.1].
Using Lemma 3.6.16 we arrive at:
∫
(A K ) ij = (α(Φ(̂x))(DΦ) −T grad̂b i )·((DΦ) −T grad̂b j )|detDΦ|d̂x . (3.6.18)
̂K
Note that the argument ̂x is suppressed for some terms in the integrand.
✩DRAFT
✪
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
3.6
p. 417
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Example 3.6.19 (Transformation techniques for bilinear transformations).
( )
α1 +β
Φ(̂x) = 1̂x 1 +γ 1̂x 2 +δ 1̂x 1̂x 2 , α
α 2 +β 2̂x 1 +γ 2̂x 2 +δ i ,β i ,γ i ,δ i ∈ R ,
( 2̂x 1̂x 2
)
β1 +δ
⇒ DΦ(̂x) = 1̂x 2 γ 1 +δ 1̂x 1 ,
β 2 +δ 2̂x 2 γ 2 +δ 2̂x 1
⇒ det(DΦ(̂x)) = β 1 γ 2 −β 2 γ 1 +(β 1 δ 2 −β 2 δ 1 )̂x 1 +(δ 1 γ 2 −δ 2 γ 1 )̂x 2 .
BothDΦ(̂x) anddet(DΦ(̂x)) are (componentwise) linear inx.
If Φ = Φ K for a generic quadrilateralK as in (3.6.12), then the coefficientsα i ,β i ,γ 1 ,δ i depend on
the shape ofK in a straightforward fashion:
( ) ( ) ( )
α1
= a
α 1 β1
, = a 2 β 2 −a 1 γ1
, = a 2 γ 4 −a 1 , 2
3.6.4 Boundary approximation
Intuition:
(
δ1
δ 2
)
= a 3 −a 2 −a 4 +a 1 .
Approximating a (smooth) curved boundary∂Ω by a polygon/polyhedron will introduce a
(sort of) discretization error.
Parametric finite elment constructions provide a tool for avoiding polygonal/polyhedral approximation
of boundaries.
Here we discuss this for a very simple case of triangular meshes in 2D (more details → [5,
Sect, 10.2]).
✸
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.6
p. 41
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✎ notation: M −T :=
(M −1) T =
(M T) −1
3.6
p. 418
3.6
p. 42

Idea: Piecewise polynomial approximation of
boundary (boundary fitting)
(∂Ω locally considered as function over
straight edge of an element)
a 2
˜K
E Γ
n
δ
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OPTIONAL SECTION
Contents of this section will not be needed later on.
Numerica
Methods
for PDEs
Example: Piecewise quadratic boundary
approximation
(Part of∂Ω betweena 1 anda 2
approximated by parabola)
a 3
∂Ω
a 1
Fig. 122
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
So far we have discussed the finite elements for linear second-order variational boundary value problems
only.
However, as we have learned in Ex. 1.5.66, in 1D the Galerkin approach based on linear finite elements
was perfectly capable of dealing with non-linear two-point boundary value problems. Indeed
the abstract discussion of the Galerkin approach in Sect. 1.5.1 was aimed at general and possibly
non-linear variational problems, see (1.5.7), (1.5.18).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Mapping ˜K → “curved element”K:
˜Φ K (˜x) := ˜x+4δλ 1 (˜x)λ 2 (˜x)n . (3.6.20)
(λ i barycentric coordinate functions on ˜K,nnormal toE Γ , see Fig. 122)
Note: Essential: δ sufficiently small =⇒ Φ bijective
The complete transformation Φ K : ̂K ↦→ K is obtained by joining an affine transformation (→
Def. 3.5.27)Φ a K : ̂K ↦→ ˜K,Φ a K (̂x) := F K̂x+τ K, and ˜Φ K :
Φ K = ˜Φ K ◦Φ a K .
3.6
p. 421
Numerical
Methods
for PDEs
It goes without saying that the abstract (and formal) discussion of Sect. 1.5.1 remains true for nonlinear
second-order boundary value problems in variational form.
Difficult: Characterization of “spaces of functions with finite energy” (→ Sobolev spaces, Sect. 2.2)
(Relief!)
for non-linear variational problems.
Recall (→ Rem. 1.3.12):
In this course we do not worry that much about function spaces.
Non-linear variational problem
u ∈ V: a(u;v) = l(v) ∀v ∈ V 0 , (1.3.14)
3.7
p. 42
Numerica
Methods
for PDEs
For parabolic boundary fitting:
D˜Φ K = I+4δn·grad(λ 1 λ 2 ) ⊤ ∈ R 2,2 , det(D˜Φ K ) = 1+4δn·grad(λ 1 λ 2 ) .
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
V 0 ˆ= test space, (real) vector space (usually a function space, “Sobolev-type” space→Sect. 2.2)
V ˆ= trial space, affine space: usuallyV = u 0 +V 0 , with offset functionu 0 ∈ V ,
f ˆ= a linear mappingV 0 ↦→ R, a linear form,
a ˆ= a mappingV ×V 0 ↦→ R, linear in the second argument, that is
a(u;αv +βw) = αa(u;v)+βa(u;w) ∀u ∈ V , v,w ∈ V 0 , α,β ∈ R . (3.7.1)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.7 Linearization
3.7
p. 422
Example 3.7.2 (Heat conduction with radiation boundary conditons).
3.7
p. 42

➣ 2nd-order elliptic boundary value problem, cf. (2.5.6) & (2.6.3)
−div(κ(x)gradu) = f inΩ ,
κ(x)gradu·n(x)+Ψ(u) = 0 on∂Ω .
Variational formulation from Ex. 2.8.5
∫
∫ ∫
u ∈ H 1 (Ω): κ(x)gradu·gradvdx+ Ψ(u)vdS = fvdx ∀v ∈ H 1 (Ω) . (2.8.7)
Ω
∂Ω Ω
IfΨ : R ↦→ R is not an affine linear function, then (2.8.7) represents a non-linear variational problem
(1.3.14) with
trial/test spaceV = V 0 = H 1 (Ω) (→ Def. 2.2.12),
∫
right hand side linear forml(v) := fvdx,
∫
Ω
∫
a(u;v) := κ(x)gradu·gradvdx+ Ψ(u)vdS.
Ω
∂Ω
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Idea:
Given ⃗ ξ (k) ∈ D ➣
Idea:
Newton iteration:
local linearization:
⃗ ξ (k+1) as zero of affine linear model function
F( ⃗ ξ) ≈ ˜F( ⃗ ξ) := F( ⃗ ξ (k) )+DF( ⃗ ξ (k) )( ⃗ ξ − ⃗ ξ (k) ) .
⃗ ξ (k+1) := ⃗ ξ (k) −DF( ⃗ ξ (k) ) −1 F( ⃗ ξ (k) ) , [ ifDF( ⃗ ξ (k) ) regular ] (3.7.3)
Givenu (k) ∈ V ➣
u (k+1) from
← apply idea to (1.3.14)
local linearization:
w ∈ V 0 : a(u (k) ;v)+D u a(u (k) ;v)w = l(v) ∀v ∈ V 0 ,
u (k+1) := u (k) +w .
(3.7.4)
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Note that the non-linearity enters only through the boundary term.
✸
The meaning ofDF( ⃗ ξ (k) ) in (3.7.3) is clear: it stands for the Jacobian ofF evaluated at ⃗ ξ (k)
3.7
p. 425
But what is the meaning ofD u a(u (k) ;v)w in (3.7.4)?
3.7
p. 42
Pursuing the policy of Galerkin discretization (choice of discrete spaces and corresponding bases,
→ Sect. 1.5.1) we can convert (1.3.14) into a non-linear system of equations
N∑
a(u 0 + µ j b j N ;bk N ) = f(bk N ) ∀k = 1,...,N . (1.5.18)
j=1
If the left hand side depends smoothly on the unkowns (the corefficients µ j of ⃗µ), then the classical
Newton method (→ [18, Sect. 4.4]) to solve it iteratively.
Numerical
Methods
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Remember the “definition” of the Jacobian (for sufficiently smoothF )
➣
try the “definition”
DF( ⃗ F(
ξ)⃗µ = lim
⃗ ξ +t⃗µ)−F( ⃗ ξ)
, ⃗ ξ ∈ D, ⃗µ ∈ R N . (3.7.5)
t→0 t
D u a(u (k) a(u+tw;v)−a(u;v)
;v)w = lim , u (k) ∈ V , v,w ∈ V t→0 t
0 . (3.7.6)
Numerica
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Here, we focus on a different approach that reverses the order of the steps:
1. Linearization of problem (“Newton in function space”),
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
If(u,v) ↦→ a(u;v) depends smoothly onu, then
(v,w) ↦→ D u a(u (k) ;v)w is a bilinear formV 0 ×V 0 ↦→ R.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2. Galerkin discretization of linearized problems.
“Newton in function space”:
Example 3.7.7 (Derivative of non-linearu ↦→ a(u;·)).
Recall idea of Newton’s method [18, Sect. 4.4] for the iterative solution of F(x) = 0, F : D ⊂
R N ↦→ R N smooth:
3.7
p. 426
Apply formula (3.7.6) to the non-linear boundary term in (2.8.7), that is, here
∫
a(u;v) := Ψ(u)vdS , u,v ∈ H 1 (Ω) .
∂Ω
3.7
p. 42

∫
a(u+tw;v)−a(u;v) = (Ψ(u+tw)−Ψ(u))vdS , u,v ∈ H 1 (Ω) .
∂Ω
Assume Ψ : R ↦→ R is smooth with derivative Ψ ′ and employ Taylor expansion for fixed w ∈ H 1 (Ω)
andt → 0
∫
a(u+tw;v)−a(u;v) = tΨ ′ (u)wvdS +O(t 2 ) .
∂Ω ∫
D u a(u (k) a(u+tw;v)−a(u;v)
;v)w = lim = Ψ ′ (u)wvdS .
t→0 t ∂Ω
= a bilinear form inv,w onH 1 (Ω)×H 1 (Ω)!
This example also demonstrates how to actually computeD u a(u (k) ;v)w!
✸
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Any of the Lagrangian finite element spaces introduced in Sect. 3.4 will supply validV N /V 0,N . Offset
functions can be chosen according to the recipes from Sect. 3.5.5.
Important aspect: termination of iteration, see [18, Thm. 4.4.3].
Option:
termination based on relative size of Newton update, withw,u (k+1)
N
from (3.7.8)
∥
STOP, if ‖w‖ ≤ τ ∥u (k+1)
N ∥ , (3.7.9)
where‖·‖ is a relevant norm (e.g., energy norm) onV (k+1)
N
andτ > 0 a prescribed relative tolerance.
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Idea: Galerkin discretization of the linear variational problem from (3.7.4)
w ∈ V 0 : c(w,v) = g(v) ∀v ∈ V 0 ,
c(w,v) = D u a(u (k) ;v)w , g(v) := l(v)−a(u (k) ;v) .
3.7
p. 429
3.7
p. 43
Newton-Galerkin iteration for (1.3.14)
Givenu (k)
N ∈ V (k)
N ➣ u (k+1)
N
∈ V (k+1)
N
from
Numerical
Methods
for PDEs
4
Finite Differences (FD) and Finite
Volume Methods (FV)
Numerica
Methods
for PDEs
w N ∈ V (k+1)
0,N : D ua(u (k)
N ;v N)w N = l(v N )−a(u (k)
N ;v N) ∀v N ∈ V (k+1)
0,N ,
u (k+1)
N
:= P (k+1)
N u (k)
(3.7.8)
N +w .
Newton update
Note: different Galerkin trial/test spacesV (k)
N ,V(k) 0,N
may be used in different steps of the iteration!
(It may enhance efficiency to use Galerkin trial/test spaces of a rather small dimension in the beginning
and switch to larger when the iteration is about to converge.)
Warning! IfV (k)
N ≠ V (k+1)
N
➣
Linear projection operatorP (k+1)
N
you cannot simply addu (k)
N andw
: V (k)
N
(k+1)
↦→ V N
required in (3.7.8)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
3.7
p. 430
CORE CHAPTER
Now we examine two approaches to the discretization of scalar linear 2nd-order elliptic BVPs that
offer an alternative to finite element Galerkin methods discussed in Ch. 3.
What these methods have in common with (low degree) Lagrangian finite element methods is
that they rely on meshes (→ Sect. 3.3.1) tiling the computational domainΩ,
they lead to sparse linear systems of equations.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
4.0
p. 43

Remark 4.0.1 (Collocation approach on “complicated” domains).
Sect. 1.5.2.2 taught us spline collocation methods. A crucial insight was that collocation methods
(see beginning of Sect. 1.5.2 for a presentation of the idea), which target the boundary value problem
in ODE/PDE form, have to employ discrete trial spaces comprised of continuously differentiable
functions, see Rem. 1.5.82.
It is very difficult to contruct spaces of piecewise polynomial C 1 -functions on non-tensor product
domains ford = 2,3 and find suitable collocation nodes, cf. (1.5.78).
Therefore we skip the discussion of collocation methods for 2nd-order elliptic BVPs on Ω ⊂ R d ,
d = 2,3.
△
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Methods
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2D model problem:
Homogeneous Dirichlet BVP for Laplacian:
−∆u = − ∂2 u
∂x 2 − ∂2 u
1 ∂x 2 = f inΩ :=]0,1[ 2 ,
2
u = 0 on∂Ω .
Discretization based on
M = (triangular) tensor-product grid
(meshwidthh = (1+N) −1 ,N ∈ N)
lexikographic (line-by-line) ordering of nodes ofM
✄
➊
N(N−1)+1
N+1
1
N−1
N*N
N+2 N+3 2N
2 3
N
Fig. 123
finite difference approach to−∆: approximation of derivatives by symmetric difference quotients
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
4.1 Finite differences
A finite difference scheme for a 2-point boundary value problem was presented in Sect. 1.5.3, which
you are advised to browse again. Its gist was
4.1
p. 433
Numerical
Methods
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This is nothing new: we did this in (1.5.97).
∂ 2
∂x 2 1
∂ 2
∂x 2 2
u(ξ −h,η)−2u(ξ,η)+u(ξ +h,η)
u ≈
h
|x=(ξ,η)
2 ,
u ≈
|x=(ξ,η)
u(ξ,η −h)−2u(ξ,η)+u(ξ,η +h)
h 2 .
−∆u |x=(ξ,η) ≈ 1 h 2 ( 4u(ξ,η)−u(ξ −h,η)−u(ξ +h,η)−u(ξ,η −h)−u(ξ,η +h)
) .
4.1
p. 43
Numerica
Methods
for PDEs
to replace the derivatives in the differential equation with difference quotients connecting approximate
values of the solutions at the nodes of a grid/mesh.
Recall: Finite differences target the “ODE/PDE-formulation” of the boundary value problem.
Our goal: extension to higher dimensions
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
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DRAFT
Use this approximation at grid point p = (ih,jh). This will connect the five point values u(ih,jh),
u((i−1)h,jh),u((i+1)h,jh),u(ih,(j −1)h),u(ih,(j +1)h).
Approximationsµ i,j to the point values u(ih,jh)
will be the unknowns of the finite difference method.
Centering the above difference quotients at grid points yields linear relationships between the unknowns:
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DRAFT
4.1
p. 434
4.1
p. 43

1
h 2 ( 4u(ih,jh)−u(ih−h,jh)−u(ih+h,jh)−u(ih,jh−h)−u(ih,jh+h)
) = f(ih,jh) ,
1
h 2 (
4µi,j −µ i−1,j −µ i+1,j −µ i,j−1 −µ i,j+1
) = f(ih,jh) . (4.1.1)
Numerical
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Already in Sect. 1.5.3 we saw that the linear system of equations popping out of the finite difference
discretization of the linear two-point BVP (1.5.81) was the same as that obtained via the linear finite
Galerkin approach on the same mesh.
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Also this is familiar from the discussion in 1D. Yet, in 1D the association of the point values and of
components of the vector ⃗µ of unknowns was straightforward and suggested by the linear ordering
of the nodes of the grid. In 2D we have much more freedom.
One option on tensor-product grids is the line-by-line ordering (lexikographic ordering) depicted in
Fig. 123. This allows a simple indexing scheme:
u(p) ↔ µ i,j ↔ µ (j−1)N+i
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C. Schwab,
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V.
Gradinaru
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DRAFT
In two dimensions we will also come to this conclusion! So, let us derive the Galerkin matrix and
right hand side vector for the 2D model problem on the tensor product mesh depicted in Fig. 123. To
begin with we convert it into a triangular meshMby splitting each square into two equal triangles by
inserting a diagonal (green lines in Fig. 123). On this mesh we use linear Lagrangian finite elements
as in Sect. 3.2.
Then we repeat the considerations of Sect. 3.2.
➋
Linear Lagrangian finite element Galerkin discretization→Sect. 3.2: V 0,N = S 0 1,0 (M)
(global shape functions ˆ= “tent functions”,→Fig. 87)
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−µ (j−2)N+i −µ (j−1)N+i−1 +4µ (j−1)N+i −µ (j−1)N+i+1 −µ jN+i
h 2
= f(ih,jh)
} {{ }
=ϕ (j−1)N+i
. (4.1.2)
➤ linear system ofN 2 equationsA⃗µ = ⃗ϕ withN 2 ×N 2 block-tridiagonal Poisson matrix
⎛
⎞ ⎛
⎞
T −I 0 ··· ··· 0 4 −1 0 0
A := 1 −I T −I .
−1 4 −1 .
h 2 0 −I T −I .
⎜ . . .. ... . .. 0
, T :=
0 −1 4 −1 .
⎟ ⎜ . . .. ... ...
∈ R N,N (4.1.3)
⎟
⎝ . −I T −I⎠
⎝ . −1 4 −1⎠
0 ··· ··· 0 −I T 0 ··· ··· 0 −1 4
⎛
A =
⎜
⎝
0
0
⎞
⎟
⎠
✁ band structure of Poisson matrix
The MATLAB command
A = gallery(’poisson’,n)
creates a sparsen 2 ×n 2 Poisson matrix.
4.1
p. 437
Numerical
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R. Hiptmair
C. Schwab,
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Gradinaru
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DRAFT
4.1
p. 438
h
a 3
K
a 1 a 2
h
Element stiffness matrix from (3.2.10):
⎛ ⎞
2 −1 −1
A K = 1 ⎝
2 −1 1 0 ⎠ .
−1 0 1
(← numbering of local shape functions)
Element load vector: use three-point quadrature formula (3.5.35)
⎛
f(a 1 ⎞
)
⃗ϕ K = 6 1h2 ⎜
⎝f(a 2 ⎟
) ⎠ .
f(a 3 )
4.1
p. 43
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4.1
p. 44

➂ ➂ K 5
K 6 K 4
➀
➀
➁ ➀
p
➂
➁ ➂
➂
K 1 K 3
K 2
➀
➀
➁
➀
➁
➂
➁
➁
Local assembly:
← green: local vertex numbers
Contributions to load vector component
associated with nodep:
FromK 1 : (⃗ϕ K1 ) 2
FromK 2 : (⃗ϕ K2 ) 3
FromK 3 : (⃗ϕ K3 ) 3
FromK 4 : (⃗ϕ K4 ) 1
FromK 5 : (⃗ϕ K5 ) 1
FromK 6 : (⃗ϕ K6 ) 2
⃗ϕ p = h 2 f(p) .
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➂
⎛
⎝ 1 −1 0
⎞
−1 2 −1⎠
0 −1 1
➀
➁
−1
➁
⎛
⎝ 1 0 ⎞ ➂
0 1 −1⎠
−1 −1 2
⎛
⎝ 1 −1 0
⎞
−1 2 −1⎠
0 −1 1
➂
➁
➂
➂
⎛
⎝ 1 0 ⎞
0 1 −1⎠
−1 −1 2
−1
➀
➀
4 −1
−1
➂
⎛
⎝ 1 0 ⎞
0 1 −1⎠
−1 −1 2
⎛
⎝ 1 −1 0
⎞
−1 2 −1⎠
0 −1 1
➁
➁
Numerica
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Assembly of finite element Galerkin matrix from element (stiffness) matrices (→ Sect. 3.5.3):
➀
➀
➁
➀
4.1
p. 441
Fig. 125
4.1
p. 44
➂
➀
1
2
➀
➀
1
2
⎛
⎝ 1 0 ⎞
0 1 −1⎠
−1 −1 2
1
2
⎛
⎝ 1 −1 0
⎞
−1 2 −1⎠
0 −1 1
1
2
⎛
⎝ 1 −1 0
⎞
−1 2 −1⎠
0 −1 1
➀
➁
➁
➂
➂
➁
➂
➂
➀
⎛
⎝ 1 0 ⎞
0 1 −1⎠
−1 −1 2
1
2
➀
1
2
⎛
⎝ 1 0 ⎞
0 1 −1⎠
−1 −1 2
➁
➂
⎛
⎝ 1 −1 0
⎞
−1 2 −1⎠
0 −1 1
➁
➁
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DRAFT
4.1
p. 442
Factor 1/2 is missing in Figure 125
➤ N 2 ×N 2 linear system of equationsh 2 A⃗µ = h 2 ⃗ϕ,A ˆ= Poisson matrix (4.1.3)
✬
✫
Discussion:
(Most) finite difference schemes
↔
finite element Galerkin schemes
with numerical quadrature
on structured meshes
finite differences vs. finite element Galerkin methods
(here focused on 2nd-order linear scalar problems)
Finite element methods can be used on general triangulations and structured (tensor-product)
meshes alike, which delivers superior flexibility in terms of geometry resolution (advantage FEM).
The correct treatment of all kinds of boundary conditions (→ Sect. 2.6). naturally emerges from
the variational formulations in the finite element method (advantage FEM).
✩
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✪C. Schwa
H.
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V.
Gradinar
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DRAFT
4.1
p. 44

Finite element methods have built-in “safety rails” because there are clear criteria for choosing
viable finite element spaces and once this is done, there is no freedom left to go astray (advantage
FEM).
Finite element methods are harder to understand (advantage FD, but only with students who have
not attended this course!)
Then, why are “finite difference methods” ubiquitous in scientific and engineering simulations ?
When people talk “finite differences” they have in mind structured meshes (translation invariant, tensor
product structure) and use the term as synonym for “discretization on structured meshes”. The
popularity of structured meshes is justified:
• structured meshes allow regular data layout and vectorization, which boost the performance of
algorithms on high performance computing hardware.
→ course “Parallel Computing for Scientific Simulations”
• translation invariant PDE operators give rise to simple Galerkin matrices that need not be assembled
and stored (recall the 5-point-stencil for −∆) and support very efficient matrix×vector
operations.
Numerical
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C j
First ingredient:
C k
Γ ik
n ik
p k
p j
.
C i
p i
Fig. 126
(finitely many) control volumes
Concrete choice:
Control volumes =
(polygonal) cells of a mesh ˜M = {C i } i
covering computational domainΩ.
Associate
Meaning:
cellC i ↔ nodal valueµ i
µ i ≈ u(p i ),p i = “center” ofC i
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✓
✒
Use structured meshes whenever possible!
4.2 Finite volume methods (FVM)
✏
✑
4.2
p. 445
Numerical
Methods
for PDEs
The conservation law (2.5.2) had to be linked to the flux law (2.5.3) in order to give rise to a 2nd-order
scalar PDE see (2.5.5)–(2.5.6).
Correspondingly, “heat conservation in control volumes” has to be supplemented by a rule that furnishes
the heat flux between two adjacent control volumes.
4.2
p. 44
Numerica
Methods
for PDEs
4.2.1 Gist of FVM
Second ingredient:
local numerical fluxes
For two adjacent cellsC k ,C i with common edgeΓ ik := C i ∩C k .
Focus:
linear scalar 2nd-order elliptic boundary value problem in 2D (→ Sect. 2.5), homogeneous
Dirichlet boundary conditions (→ Sect. 2.6), uniformly positive scalar heat conductivityκ =
κ(x)
−div(κ(x)gradu) = f inΩ , u = 0 on∂Ω .
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Numerical flux J ik = Ψ(µ i ,µ k ) ≈ ∫ Γ j·n ik ik dS
(Ψ = numerical flux function, j = (heat) flux, see (2.5.1),n ik ˆ= edge normal. )
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Gradinar
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Finite volume methods for 2nd-order elliptic BVP are inspired by the conservation principle (2.5.2).
∫ ∫
j·ndS = f dx for all “control volumes”V . (2.5.2)
∂V V
Physics requires that this holds for all (infinitely many) “control volumes”V ⊂ Ω.
Idea:
consider balance law on (fintely many !) control volumesC i
∫ ∫
j·n i dS = f dx ⇒ ∑ ∫
J ik = f dx .
∂C i C i k∈U
C i i
Since discretization has to lead to a finite number of equations, the idea is to demand that (2.5.2)
holds for only a finite number of special control volumes.
4.2
p. 446
✎ notation: U i := {j: C i and C j share edge,C j ∈ ˜M}, p i = node associated with control
volumeC i .
4.2
p. 44

System of equations (˜M := ♯ ˜M equations, unknownsµ i ):
∑
∫
Ψ(µ i ,µ k ) = f dx ∀i = 1,..., ˜M . (4.2.1)
k∈U
C i i
Further approximation: 1-point quadrature for approximate evaluation of integral overC i ,
∑
k∈U i
Ψ(µ i ,µ k ) = |C i |f(p i )dx ∀i = 1,..., ˜M . (4.2.2)
Note: homogeneous Dirichlet problem ➢ only “interior” control volumes in (4.2.2)
Numerical
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C. Schwab,
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V.
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Construction of Voronoi dual cells:
edges→perpendicular bisectors
nodes→circumcenters of triangles
straightforward generalization to 3D
Remark 4.2.4 (Geometric obstruction to Voronoi dual meshes).
C i
p i
Fig. 128
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4.2.2 Dual meshes
Dual meshes are a commonly used technique for the construction of control volumes for FVM, based
on conventional FE triangulationMofΩ(→ Sect. 3.3.1).
4.2
p. 449
4.2
p. 45
Focus:
dual mesh for triangular meshMin 2D,Ωpolygon
Numerical
Methods
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ω
.
⇐ Obtuse angleω:
Numerica
Methods
for PDEs
Popular choice:
Voronoi dual mesh
. .
➢ circumcenter∉triangle
⇏ ➢ C i ∩C j ≠ ∅ nodesi,j connected
by edge
V(M) = {p 1 ,...,p M } = nodes ofM
Define Voronoi cells
C i := {x ∈ Ω: |x−p i | < |x−p j | ∀j ≠ i} .
Voronoi dual mesh ˜M := {C i } M i=1
(4.2.3)
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Fig. 129
➢ geometric construction breaks down
➢ connectivity of unknowns hard to determine
△
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Fig. 127
4.2
p. 450
Angle condition to ensure C i ∩C j ≠ ∅ ⇔ nodesi,j connected by edge ofM:
(i) sum of angles facing interior edge≤ π,
(ii) angles facing boundary edges≤ π/2 (for non-Dirichlet boundary conditions).
(i), (ii) characterize Delaunay triangulations
4.2
p. 45

Popular choice:
Barycentric dual mesh
Numerical
Methods
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Idea:
obtain numerical flux from
Fourier’s law (2.5.3) applied to a (sufficiently smooth)u N : Ω ↦→ R
reconstructed from dual cell valuesµ i .
Numerica
Methods
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Fig. 130
Dual cells:
edges→union of lines connecting
barycenters and midpoints of
edges ofM
nodes→barycenters of triangles
No geometric obstructions
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Natural approach, since µ i is read as approximation of u(p i ), where the “center” p i of the dual cell
C i coincides with a node x i ∈ V(M) of the triangular meshM:
where
N∑
u N = I 1 ⃗µ := µ i b i N , (4.2.6)
i=1
N = ♯V(M) = number of dual cells, size of vector⃗µ,
b i N ˆ= nodal basis function (“tent function”) ofS0 1,0 (M) belonging to the node insideC i.
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u N ˆ= piecewise linear interpolant of vertex valuesµ i
4.2.3 Relationship of finite elements and finite volume methods
4.2
p. 453
Note that u N is not smooth across inner edges of M. However, we do not care when computing
j := κ(x)gradu N , because this flux is only needed at edges of the dual mesh, which lie inside
triangles ofM(with the exception of single points that are irrelevant for the flux integrals).
4.2
p. 45
Hardly surprising, finite volume methods and finite element Galerkin discretizations are closely related.
This will be explored in this section for a model problem.
Numerical
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Illustration of point evaluation
Numerica
Methods
for PDEs
Setting:
1
We consider the homogeneous Dirichlet problem for the Laplacian∆
1
0.8
0.6
0.4
0.6
0.8
−∆u = f inΩ , u = 0 on∂Ω . (4.2.5)
Discretization by finite volume method based on a triangular mesh M and on Voronoi dual cells
→ Fig. 127:
Assumption:
M = Delaunay triangulation ofΩ ⇔ angle condition
Number of control volumes = number of interior nodes ofM
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0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
vertex valuesµ i onV(M)
0.7
0.8
Choice of numerical flux:
0.9
1
0
0.2
0.4
Fig. 131
Fig. 132
p.w. linear interpolantu N := I 1 ⃗µ ∈ S 0 1,0 (M)
∫
J ik := − gradI 1 ⃗µ·n ik dS (4.2.7)
Γ ik
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Still missing:
specification of numerical flux functionΨ : R 2 ↦→ R for each dual edge
4.2
p. 454
(4.2.7) ➥ (4.2.2)↔one row of finite volume discretization matrix from
∑
∫
gradI 1 ⃗µ·n ik dS = ∑ (∑
∫
µ j gradb j ) ∫
N ·n ikdS = f(x)dx .
k∈U
Γ ik i j∈U i k∈U
Γ ik C i i
} {{ }
= matrix entry (A) ij
4.2
p. 45

n i ˆ= exterior unit normal vector to∂C i .
p i
∫
⇒ (A) ij = gradb j N ·n idS .
∂C i
(4.2.8)
Γ K i
K 1
p j
Part of the boundary of the control
volumeC i :
Γ K i := ∂C i ∩K .
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K 2 p j
K 1
e 2
Γ
e ik ij
p i e 1
Fig. 134
Now apply Gauss’ theorem
Thm. 2.4.5 to the domainsC i ∩K 1
andC i ∩K 2 (shaded in figure).
Also use again that gradb j N ≡
const onK 1 andK 2 .
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K 2
Fig. 133
4.2
p. 457
(A) ij = 1 ∫
gradb j ·n
2 N|K e1 dS + 1 ∫
gradb j ·n
1 2 N|K 1 1
e ij
dS
e 1 e ij
+ 1 ∫
gradb j ·n
2 N|K 2 2 e ij
dS + 1 ∫
gradb j ·n
2 N|K e2 dS . (4.2.10)
1
e ij e 2
4.2
p. 45
Now, consideri ≠ j ↔ off-diagonal entries ofA:
First, we recall that the intersection of the support of the “tent function”b j N with ∂C i is located inside
K 1 ∪K 2 , see Fig. 133.
∫ ∫
(A) ij = gradb j N ·n idS + gradb j N ·n idS .
Γ K 1
i
Γ K 2
i
Next observe thatgradb j N
is piecewise constant, which implies
divgradb j N = 0 inK 1 , divgradb j N = 0 inK 2 . (4.2.9)
Numerical
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DRAFT
On the other hand, an entry of finite element Galerkin matrix à based on linear Lagrangian finite
element spaceS1 0 (M) can be computed as, see Sect. 3.2.5:
∫
∫
(Ã) ij = gradb j N ·gradbi N dx+ gradb j N ·gradbi N dx .
K 1 K 2
Conduct local integration by parts using Green’s first formula from Thm. 2.4.7 and taking into account
(4.2.9) and the linearity of the local shape functions
∫
∫
(Ã) ij = (gradb j ·n N|K 1 1 )b i N dS +
∂K 1
∂K 2
(gradb j N|K 2
·n 2 )b i N dS
= 1 2 |e 1|gradb j N|K 1
·n e1 + 1 2 |e ij|gradb j N|K 1
·n 1 e ij
+
1
2 |e 2|gradb j N|K 2
·n e2 + 1 2 |e ij|gradb j N|K 2
·n 2 e ij
.
This is the same value as for(A) ij from (4.2.10)! Similar considerations apply to the diagonal entries
(A) ii and(Ã) ii.
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4.2
p. 458
The finite volume discretization and the finite element Galerkin discretization spawn the same
system matrix for the model problem (4.2.5).
4.2
p. 46

5 Convergence and Accuracy
Numerical
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a supplies an inner product onV 0
a induces energy norm‖·‖ a onV 0 (→ Def. 2.1.27)
☞ We want (3.1.1) to be well posed, see Rem. 2.3.11
Numerica
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In this chapter we resume the discussion of Sect. 1.6 of accuracy of a Galerkin solution u N of a
variational boundary value problem.
More precisely, we are going to study convergence, see Rem. 1.6.2
Focus: finite element Galerkin discretization of linear scalar 2nd-order elliptic boundary value problems
in 2D, 3D
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Assumption 5.1.2. The right hand side functionall : V 0 ↦→ R from (3.1.1) is continous w.r.t. to
the energy norm (→ Def. 2.1.27) induced bya:
☞ An assumption to appease fastidious mathematicians:
∃C > 0: |l(u)| ≤ C‖u‖ a ∀u ∈ V 0 . (2.2.1)
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Prerequisites (what you should know by now):
➣ Boundary value problems (from equilibrium models, diffusion models): Sects. 2.4, 2.6,
Assumption 5.1.3.V 0 equipped with the energy norm‖·‖ a is a Hilbert space, that is, complete.
➣ Variational formulation: Sect. 2.8, see also (2.3.3), (2.8.15), (3.0.1),
➣ Some Sobolev spaces and their norms: Sect. 2.2
➣ Abstract Galerkin discretization: Sect. 3.1,
➣ Lagrangian finite elements: Sects. 3.4, 3.2.
5.1 Galerkin error estimates
5.1
p. 461
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✬
Theorem 5.1.4 (Existence and uniqueness of solution of linear variational problem).
Under Assumptions 5.1.1–5.1.3 the linear variational problem has a unique solutionu ∈ V 0 .
✫
This theorem is also known as Riesz representation theorem for continuous linear functionals.
✩
✪
5.1
p. 46
Numerica
Methods
for PDEs
Setting:
linear variational problem (1.4.6) in the form
u ∈ V 0 : a(u,v) = l(v) ∀v ∈ V 0 , (3.1.1)
V 0 ˆ= (real) vector space, a space of functions Ω ↦→ R for scalar 2nd-order elliptic variational
problems,
a : V 0 ×V 0 ↦→ R ˆ= a bilinear form, see Def. 1.3.13,
l : V 0 ↦→ R ˆ= a linear form, see Def. 1.3.13,
☞ We want (3.1.1) to be related to a quadratic minimization problem (→ Def. 2.1.21):
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 5.1.5 (Well-posed 2nd-order linear elliptic variational problems).
For instance, Assumption 5.1.1 is satisfied for the bilinear form
∫
a(u,v) := (α(x)gradu)·gradvdx , u,v ∈ H0 1 (Ω) , (5.1.6)
Ω
and uniformly positive definite (→ Def. 2.1.11) coefficient tensorα : Ω ↦→ R d,d , see Sect. 2.1.3.
For the right hand side functional
∫ ∫
l(v) := f(x)v(x)dx+ h(x)v(x)dS , v ∈ H 1 (Ω) ,
Ω ∂Ω
we found in Sect. 2.2, see (2.2.15), and Rem. 2.9.5 that f ∈ L 2 (Ω) and h ∈ L 2 (∂Ω) ensures
Assumption 5.1.2.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Assumption 5.1.1. The bilinear form a : V 0 × V 0 ↦→ R in (3.1.1) is symmetric and positive
definite (→ Def. 2.1.25).
5.1
p. 462
Assumption 5.1.3 for a from (5.1.6) is a deep result in the theory of Sobolev spaces [12, Sect. 5.2.3,
Thm. 2].
△
5.1
p. 46

Now consider Galerkin discretization of (3.1.1) based on Galerkin trial/test space V 0,N ⊂ V 0 , N :=
dimV 0,N < ∞:
u N ∈ V 0,N : a(u N ,v N ) = l(v N ) ∀v N ∈ V 0,N . (3.1.4)
Numerical
Methods
for PDEs
Discretization errore N := u−u N “a(·,·)-orthogonal” to discrete trial/test spaceV N
u
Remark 5.1.8. If a(·,·) is inner product on V :
“Phythagoras’ theorem”→Fig. 136
Numerica
Methods
for PDEs
Thm. 3.1.5: existence and uniqueness of Galerkin solutionu N ∈ V 0,N
V 0
e N := u−u N
V 0,N
‖u−v N ‖ 2 a = ‖u−u N‖ 2 a −‖u N −v N ‖ 2 a .
(5.1.9)
Goal:
bound relevant norm of discretization erroru−u N
Here:
relevant norm = energy norm‖·‖ a
Why is the energy norm a “relevant norm” ?
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
u N
v N
Fig. 136
(5.1.9) ➣ (v N = 0) simple formula for computation
of energy norm of Galerkin
discretization error in numerical experiments
with knownu.
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
➣ Bounds of‖u−u N ‖ a provide bounds for the error in energy, see Rem. 1.6.7, (1.6.10)
✬
✩
|J(u)−J(u N )| = 1 2 |a(u,u)−a(u N,u N )|=| 1 2 a(u+u N,u−u N )|
(2.1.28)
≤ ‖u−u N ‖ a·‖u+u N ‖ a .
(No doubt, energy is a key quantity for the solution of an equilibrium problem, which is defined as the
minimizer of a potential energy functional.)
5.1
p. 465
Theorem 5.1.10 (Cea’s lemma).
Under Assumptions 5.1.1–5.1.3 the energy norm of the Galerkin discretization error satisfies
✫
‖u−u N ‖ a = inf
v N ∈V 0,N
‖u−v N ‖ a .
✪
5.1
p. 46
Other “relevant norms” were discussed in Sects. 1.6.1, 2.2:
• the mean square norm orL 2 (Ω)-norm, see Def. 2.2.5,
• the supremum norm orL ∞ (Ω)-norm, see Def. 1.6.4.
Numerical
Methods
for PDEs
Proof. Use bilinearity ofaand Galerkin orthogonality (5.1.7): for any v N ∈ V 0,N
‖u−u N ‖ 2 a = a(u−u N,u−u N ) = a(u−v N ,u−u N )+a(v N −u N ,u−u N ) .
} {{ }
=0
Next, use the Cauchy-Schwarz inequality for the inner producta:
Numerica
Methods
for PDEs
a(u,v) ≤ ‖u‖ a ‖v‖ a ∀u,v ∈ V 0 .
‖u−u N ‖ 2 a ≤ ‖u−v N‖ a·‖u−u N ‖ a ,
and cancel one factor‖u−u
R. Hiptmair
N ‖ a .
✷
The Galerkin approach allows a remarkably simple bound of the energy norm of the C. Schwab,
H.
discretization erroru−u N :
Harbrecht
V.
Gradinaru ✬
A. Chernov
a(u,v) = l(v) ∀v ∈ V 0 , V 0,N ⊂V 0
DRAFT
=⇒ a(u−u
a(u N ,v N ) = l(v) ∀v N ∈ V N ,v N ) = 0 ∀v N ∈ V 0,N .
Optimality of Galerkin solutions:
0,N
‖u−u N ‖ a = inf ‖u−v N ‖ a , (5.1.11)
v N ∈V 0,N
} {{ } } {{ }
↑ ↑
u
(norm of) discretization error best approximation error
Galerkin orthogonality
V 0
✫
e N := u−u N
a(u−u N ,v N ) = 0 ∀v N ∈ V 0,N . (5.1.7)
V 0,N
[Geometric meaning for inner producta(·,·)→]
5.1
u N Fig. 135
p. 466
R. Hiptm
C. Schwa
H.
Harbrech
V.
✩Gradinar
A. Chern
✪
DRAFT
5.1
p. 46

☞ To assess accuracy of Galerkin solution: study capability ofV 0,N to approximateu!
Numerical
Methods
for PDEs
• p-refinement: replaceV 0,N := S 0 p(M),p ∈ N withV ′
0,N := S0 p+1 (M) ⇒ V 0,N ⊂ V ′
0,N
Numerica
Methods
for PDEs
“Monotonicity” of best approximation: consider different trial/test spaces
V 0,N ,V ′
0,N ⊂ V 0 ,
V 0,N ⊂ V ′
0,N
⇒
inf ‖u−v N ‖
v N ∈V 0,N
′ a ≤ inf ‖u−v N ‖ a .
v N ∈V 0,N
The extreme case of p-refinement amounts to the use of global polynomials on Ω as trial and test
functions ➣ (polynomial) spectral Galerkin method, see Sect. 1.5.1.1.
Enhance accuracy by enlarging (“refining”) trial space.
Now return to finite element Galerkin discretization of linear 2nd-order elliptic variational problems.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Combination of h-refinement and p-refinement ? OF COURSE (hp-refinement, [26])
5.2 Empirical Convergence of FEM
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
How to achieve refinement of FE space ?
• h-refinement: replace M (underlying V 0,N ) → M ′ (underlying larger discrete trial space
V
0,N ′ )
5.1
p. 469
5.2
p. 47
Example 5.1.12 (regular refinement of triangular mesh in 2D).
1
2D triangular mesh
1
2D triangular mesh
1
2D triangular mesh
Numerical
Methods
for PDEs
Recall from Sect. 1.6.2:
✬
Crucial: convergence is an asymptotic notion !
Numerica
Methods
✩for PDEs
0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
✫
sequence of discrete models ⇒ sequence of approximate solutions(u (i)
N ) i∈N
⇒ study sequence(‖u (i)
N −u‖) i∈N
✪
−0.2
−0.2
−0.2
created by variation of a discretization parameter:
−0.4
−0.4
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
# Vertices : 45, # Elements : 64, # Edges : 108
−0.6
−0.6
−0.8
−0.8
−1
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
# Vertices : 153, # Elements : 256, # Edges : 408
# Vertices : 561, # Elements : 1024, # Edges : 1584
K
T 3
T 4
T
T 2 1
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
In this section some numerical experiments will demonstrate
meaningful notions of “discretization parameters”,
qualitative behaviors of the sequence(‖u (i)
N −u‖) i∈N we may expect,
for Lagrangian finite element discretization of linear scalar 2nd-order elliptic variational problems (→
Sect. 2.8).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Regular refinement of triangleK into four congruent trianglesT 1 ,T 2 ,T 3 ,T 4
Sequences of discrete models will be generated by eitherh-refinement orp-refinement.
✸
5.1
p. 470
5.2
p. 47

Model problem:
Dirichlet problem for Poisson equation:
−∆u = f ∈ L 2 (Ω) inΩ, u = g ∈ C 0 (∂Ω) on∂Ω . (5.2.1)
Numerical
Methods
for PDEs
The discretization parameter must be linked to the resolution (“capability to approximate generic
solution”) of the Galerkin trial/test spaceV 0,N . Possible choices are
Numerica
Methods
for PDEs
Example 5.2.2 (Convergence of linear and quadratic Lagrangian finite elements in energy norm).
Setting: Ω =]0,1[ 2 ,f(x 1 ,x 2 ) = 2π 2 sin(πx 1 )sin(πx 2 ),x ∈ Ω,g = 0
N := dimV 0,N as a measure of the “cost” of a discretization, see Sect. 1.6.2,
the maximum “size” of mesh cells, expressed by the mesh widthh M (→ Def. 5.2.3), see below.
➢ Smooth solution u(x,y) = sin(πx)sin(πy).
• Galerkin finite element discretization based on triangulas meshes and
– linear Lagrangian finite elements,V 0,N = S1,0 0 (M) ⊂ H1 0 (Ω) (→ Sect. 3.2),
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Definition 5.2.3 (Mesh width).
Given a meshM = {K}, its mesh widthh M is defined as
h M := max{diamK: K ∈ M} , diamK := max{|p−q|: p,q ∈ K} .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
– quadratic Lagrangian finite elements,V 0,N = S2,0 0 (M) ⊂ H1 0 (Ω) (→ Ex. 3.4.2),
• quadrature rule (3.5.38) for assembly of local load vectors (→ Sect. 3.5.4),
Monitored: H 1 (Ω)-semi-norm (→ Def. 2.2.10) of the Galerkin discretization erroru−u N
This generalizes the concept of “mesh width” introduced in Sect. 1.5.1.2.
Approximate(∗) computation of|u−u N | H 1 (Ω) on a sequence of meshes (created by
sucessive regular refinement (→ Ex. 5.1.12) of coarse initial mesh)
(∗): use of local quadrature rule (3.5.38) (on current FE mesh)
5.2
p. 473
5.2
p. 47
1
2D triangular mesh
1
2D triangular mesh
Numerical
Methods
for PDEs
10 0 Discretization errors with respect to H 1 semi−norm
10 0 Discretization errors with respect to H 1 semi−norm
p = −0.5
Quadratic FE
Linear FE
Numerica
Methods
for PDEs
0.9
0.9
10 −1
p = 1.0
10 −1
0.8
0.7
0.6
0.5
0.8
0.7
0.6
0.5
Discretization error [log]
10 −2
Discretization error [log]
10 −2
p = −1.0
10 −3
10 −3
0.4
0.4
0.3
0.2
0.1
0.3
0.2
0.1
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
10 −4
10 0
Quadratic FE
p = 2.1
Linear FE
10 −1
Mesh width [log]
10
Fig. 137 1 10 2 10 3 10 4 10
Fig. 138
5
10 −2
10 −4
Dofs [log]
H 1 (Ω)-semi-norm of discretization error on unit square (−↔p = 1,−↔p = 2)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
# Vertices : 25, # Elements : 32, # Edges : 56
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
# Vertices : 81, # Elements : 128, # Edges : 208
Unstructured triangular meshes ofΩ =]0,1[ 2 (two coarsest specimens)
Recall type of convergence (algebraic convergence vs. exponential convergence) from Def. 1.6.20
and how to detect them in a numerical experiment by inspecting appropriate graphs, see Rem. 1.6.22.
Focus on asymptotics entails studying a
norm of the discretization error as function of a (real, cardinal) discretization parameter.
5.2
p. 474
Observations: • Algebraic rates of convergence in terms ofN andh
• Quadratic Lagrangian FE converge with double the rate of linear Lagrangian FE
5.2
p. 47

Example 5.2.4 (Convergence of linear and quadratic Lagrangian finite elements inL 2 -norm).
✸
Numerical
Methods
for PDEs
✸
Numerica
Methods
for PDEs
Example 5.2.6 (h-convergence of Lagrangian FEM on L-shaped domain).
Setting as above in Ex. 5.2.2,Ω =]0,1[ 2 .
Monitored: asymptotics of the L 2 (Ω)-semi-norm of the Galerkin discretization error (approximate
computation of‖u−u N ‖ L 2 (Ω) by means of local quadrature rule (3.5.38) on a sequence of meshes
created by sucessive regular refinement (→ Ex. 5.1.12) of coarse initial mesh).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Setting: Model problem (5.2.1) onΩ =]−1,1[ 2 \(]0,1[×]−1,0[), exact solution (in polar coordinates)
u(r,ϕ) = r 2/3 sin(2/3ϕ) ➢ f = 0,g = u |∂Ω .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
5.2
p. 477
5.2
p. 47
10 −1 Discretization errors with respect to L 2 norm
10 −1 Discretization errors with respect to L 2 norm
Quadratic FE
Linear FE
Numerical
Methods
for PDEs
Numerica
Methods
for PDEs
10 −2
10 −2
p = 2.0
Discretization error [log]
10 −3
10 −4
10 −5
Discretization error [log]
10 −3
10 −4
10 −5
p = −1.5
p = −1.0
10 −6
10 −7
10 0
Quadratic FE
Linear FE
p = 3.1
10 −1
Mesh width [log]
10 −2
L 2 (Ω)-norm of discretization error on unit square (−↔p = 1,−↔p = 2)
Observations: • Linear Lagrangian FE (p = 1) ➽ ‖u−u N ‖ 0 = O(h 2 M ) = O(N−1 )
• Quadratic Lagrangian FE (p = 2) ➽ ‖u−u N ‖ 0 = O(h 3 M ) = O(N−1.5 )
For the “conversion” of convergence rates with respect to the mesh widthh M andN := dimS 0 p(M),
note that in 2D for Lagrangian finite element spaces with fixed polynomial degree (→ Sect. 3.4) and
meshes created by global (that is, carried out everywhere) regular refinement
10 −6
10 −7
N = O(h −2
M ) .
R. Hiptmair
C. Schwab,
H.
10 1 10 2 10 3 10 4 10 5 Harbrecht
Dofs [log]
V.
Gradinaru
A. Chernov
DRAFT
(5.2.5)
5.2
p. 478
Note:
Exact solutionu
Fig. 139
gradu has a singularity at0, that is, “‖gradu(0)‖ = ∞”.
• Galerkin finite element discretization based on triangulas meshes and
Norm of gradient‖gradu‖
Fig. 140
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
5.2
p. 48

– linear Lagrangian finite elements,V 0,N = S1,0 0 (M) ⊂ H1 0 (Ω) (→ Sect. 3.2),
– quadratic Lagrangian finite elements,V 0,N = S2,0 0 (M) ⊂ H1 0 (Ω) (→ Ex. 3.4.2),
• linear/quadratic interpolation of Dirichlet data to obtain offset function u 0 ∈ Sp,0 0 (M), p = 1,2,
see Sect. 3.5.5, Ex. 3.5.43.
Sequence of meshes created by sucessive regular refinement (→ Ex. 5.1.12) of coarse initial mesh,
see Fig. 141.
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Discretization error [log]
Discretization errors with respect to H 1 semi−norm
10 0 Quadratic FE
Linear FE
p = 0.7
10 −1
10 −2
p = 0.6
10 −3
10 0
10 −1
10 −2
Mesh width [log]
Discretization error [log]
Discretization errors with respect to H 1 semi−norm
Quadratic FE
Linear FE
p = −0.3
10 −1
10 −2
p = −0.3
10 −3
10 1 10 2 10 3 10 4 10 5 10 6
Dofs [log]
H 1 (Ω)-semi-norm of discretization error on “L-shaped” domain (−↔p = 1,−↔p = 2)
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Observations: • For bothp = 1,2: ‖u−u N ‖ 1 = O(N −1/3 )
• No gain from higher polynomial degree
5.2
p. 481
Conjecture: singularity of gradu at x = 0 seems to foil faster algebraic convergence of quadratice
Lagrangian finite element solutions!
5.2
p. 48
1
0.8
2D triangular mesh
1
0.8
2D triangular mesh
Numerical
Methods
for PDEs
Example 5.2.7 (Convergence of Lagrangian FEM forp-refinement).
✸
Numerica
Methods
for PDEs
0.6
0.6
0.4
0.4
0.2
0.2
0
−0.2
0
−0.2
Model BVP as in Ex. 5.2.2 (Ω =]0,1[ 2 ) and Ex. 5.2.6
(L-shaped domainΩ =]−1,1[ 2 \(]0,1[×]−1,0[)).
−0.4
−0.6
−0.8
−1
−0.4
−0.6
−0.8
−1
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Galerkin finite element discretization based on S 0 p(M), p = 1,2,3,5,6,7,8,9,10, built on a
fixed coarse triangular mesh ofΩ.
➤
p-refinement
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
# Vertices : 45, # Elements : 64, # Edges : 108
# Vertices : 153, # Elements : 256, # Edges : 408Fig. 141
Unstructured triangular meshes ofΩ =]−1,1[ 2 \(]0,1[×]−1,0[) (two coarsest specimens)
Monitored: H 1 (Ω)-semi-norm (energy norm) and L 2 (Ω)-norm of discretization error as functions of
polynomial degreepandN := dimS 0 p(M).
Approximate computation of |u−u N | H 1 (Ω)
meshes.
by using local quadrature formula (3.5.38) on FE
5.2
p. 482
(Computation of norms by means of local quadrature rule of order 19!. This renders the error in norm
computations introduced by numerical quadrature negligible.)
Meaningful discretization parameters for asymptotic study of error norms:
5.2
p. 48

polynomial degreepfor Lagrangian finite element space,
N := dimV 0,N as a measure of the “cost” of a discretization, see Sect. 1.6.2.
Numerical
Methods
for PDEs
✸
Numerica
Methods
for PDEs
10 2 Discretization errors for p−FEM
10 2 Discretization errors for p−FEM
5.3 A priori finite element error estimates
10 0
10 0
Discretization error
10 −2
10 −4
10 −6
10 −8
10 −10
10 −12
H 1 semi−norm
L 2 norm
0 50 100 150 200 250 300 350 400
p 2
Fig. 142
Discretization error
10 −2
10 −4
10 −6
10 −8
10 −10
10 −12
H 1 semi−norm
L 2 norm
0 50 100 150 200 250 300 350 400 450
N
Fig. 143
Ω =]0,1[ 2 : behavior of|u−u N | H 1 (Ω) for different polynomial degrees.
Lagrangian FEM:p-convergence for smooth (analytic) solution
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
We are interested in a priori estimates of norms of the discretization error.
A priori estimate: bounds for error norms available before computing approximate solutions.
⇕
A posteriori estimate: bounds for error norms based on an approximate solution already computed.
Results of Sect. 5.1 provide handle on a priori estimate for Galerkin discretization error:
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
5.2
p. 485
✗
✖
Optimality (5.1.11) of Galerkin solution
a priori error estimates
✔
✕
5.3
p. 48
Observation: exponential convergence of FE discretization error, cf. the behavior of the discretization
error of spectral collocation and polynomial spectral Galerkin methods in 1D,
Ex. 1.6.19.
Norm of Discretization error
Discretization errors for p−FEM
H 1 semi−norm
L 2 norm
10 −1
10 −2
10 −3
10 −4
10 0 10 1 10 2
p
Fig. 144
Norm of Discretization error
Discretization errors for p−FEM
H 1 semi−norm
L 2 norm
10 −1
10 −2
10 −3
10 −4
10 0 10 1 10 2 10 3
Lagrangian FEM:p-convergence for solution with singular gradient
N
Fig. 145
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Thm. 5.1.10 ➤ Estimate energy norm of Galerkin discretization erroru−u N
by bounding best approximation error
for exact solutionuin finite element space:
‖u−u N ‖ a ≤ inf ‖u−v N ‖ a , (5.1.11)
v N ∈V 0,N
} {{ } } {{ }
↑ ↑
(norm of) discretization error
How to estimate best approximation error
best approximation error
inf
v N ∈V 0,N
‖u−v N ‖ V ?
➣ Well, given solutionuseek candidate functionw N ∈ V 0,N with
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Observation: Only algebraic convergence of FE discretization error!
The suspect: “singular behavior” ofgradu atx = 0.
5.2
p. 486
Natural choice:
‖u−w N ‖ V ≈ inf
v N ∈V N
‖u−v N ‖ V .
w N by interpolation/averaging of (unknown, but existing)u
5.3
p. 48

5.3.1 Estimates for linear interpolation in 1D
Computational domain (→ Sect. 1.4):
intervalΩ = [a,b]
Given: mesh ofΩ(→ Sect. 1.5.1.2): M := {]x j−1 ,x j [: j = 1,...,M},M ∈ N
a
Goal:
x 1 x 2 x 3 ···
Fig. 146
b
u
Piecewise linear interpolant ofu ∈ C 0 ([a,b])
I 1 u ∈ S1 0 (M) , (5.3.1)
(I 1 u)(x j ) = u(x j ) , j = 0,...,M . (5.3.2)
➣ [18, Sect. 3.5.1]
Bound suitable norm (→ Sect. 1.6.1) of interpolation erroru−I 1 u
in terms of geometric quantities(∗) characterizingM.
(∗): A typical such quantity is the mesh width h M := max|x j −x j−1 |
j
Now we investigate different norms of the interpolation error.
‖u−I 1 u‖ L ∞ ([a,b]), see [18, Sect. 9.1] and [18, Sect. 9.4.1]: from [18, Thm. 9.1.7] for n = 1: for
u ∈ C 2 ([a,b])
max
x j−1 ≤x≤x j
u(t)−(Iu)(x) = 1 4 u′′ (ξ t )(x j −x j−1 ) 2 , for someξ t ∈]x j−1 ,x j [ , (5.3.3)
with local linear interpolant
(5.3.3) interpolation error estimate inL ∞ ([a,b])
(Iu)(x) = x−x j−1
x j −x j−1
u(x j )− x j −x
x j −x j−1
u(x j−1 ) . (5.3.4)
‖u−I 1 u‖ L ∞ ([a,b]) ≤ 4 1 ∥
h2 M
∥u ′′∥ ∥ L ∞ ([a,b]) . (5.3.5)
This is obtained by simply taking the maximum over all local norms of the interpolation error.
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Now all mesh cells contribute to this norm:
‖u−I 1 u‖ 2 M L 2 ([a,b]) = ∑
‖u−I 1 u‖ 2 M L 2 (]x j−1 ,x j [) = ∑ ∫
x j
j=1
➣ Idea: localization
|(u−Iu)(x)| 2 dx , Iu from (5.3.4) .
j=1x j−1
(Estimate error on individual mesh cells and sum local bounds)
By integrating by parts (1.3.22) twice, foru ∈ C 2 ([x j−1 ,x j ]),x ∈ [x j−1 ,x j ],
∫x
(x j −x)(ξ −x j−1 )
u ′′ (ξ)dξ +
x j −x j−1
x j−1
∫
xx j
(x−x j−1 )(x j −ξ)
u ′′ (ξ)dξ
x j −x j−1
(5.3.6)
= x j −x
u(x
x j −x j−1 )+ x−x j−1
u(x
j−1 x j −x j ) −u(x) . (5.3.7)
j−1
} {{ }
=Iu(x)
This is a representation formula for the local interpolation errorIu−uof the form
∫
5.3
xj
(Iu−u)(x) = G(x,ξ)u ′′ (ξ)dξ .
p. 489
x j−1
⎧
(x j −x)(ξ −x j−1 )
⎪⎨
forx
x
with G(x,ξ) = j −x j−1 ≤ ξ < x ,
j−1
,which satisfies
(x−x j−1 )(x j −ξ)
⎪⎩
forx ≤ ξ ≤ x
x j −x j .
j−1
|G(x,ξ)| ≤ |x j −x j−1 | ⇒
∫
x j
x j−1
G(x,ξ) 2 dξ ≤ |x j −x j−1 | 3 .
Kernel functionsGfor 1D linear interpolation for x j−1 = 0 , x j = 1 .
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5.3
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Recall: supremum norm (maximum norm) from Def. 1.6.4
Now, we also want to study other norms of the interpolation error:
‖u−I 1 u‖ L 2 ([a,b]) :
5.3
p. 490
5.3
p. 49

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
G(x,ξ)
0.25
0.2
0.15
0.1
0.05
0
Kernel function G for 1D linear interpolation
ξ
x = 0.1
x = 0.2
x = 0.3
x = 0.4
x = 0.5
x = 0.6
x = 0.7
x = 0.8
x = 0.9
Fig. 147
G(ξ,x)
0.25
0.2
0.15
0.1
0.05
0
1
0.8
0.6
x
0.4
0.2
0
0
0.2
0.4
ξ
0.6
0.8
1
Fig. 148
∣ ∫
x j ∫
x j ∣∣∣∣∣∣ ∫
x j
2
|u(x)−Iu(x)| 2 dx = G(x,ξ)u ′′ (ξ)dξ
dx
(5.3.8)
x j−1 x j−1 x j−1
∣
⎧ ⎫
(2.2.15) ∫
x j ⎪⎨ ∫
x j ∫
x j ⎪⎬
≤ G(x,ξ) 2 dξ · |u ′′ (ξ)| 2 dξ
⎪ ⎩ ⎪⎭ dx , x j−1 x j−1 x j−1
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∫
x j
x j−1
∣ ∣∣∣ d
dx (Iu−u)(x) ∣ ∣∣∣ 2
∣ ∫
x j ∣∣∣∣∣∣ ∫
x j
2
∂G
dx =
∂x (x,ξ)u′′ (ξ)dξ
dx
x j−1 x j−1
∣
∫
x j
⎧⎪ ⎨ ∫
x j ∣ ∣ ∣∣∣ ∂G ∣∣∣ 2 ∫
x j ⎪⎬
≤
∂x (x,ξ) dξ · |u ′′ (ξ)| 2 dξ dx .
x j−1
⎪ ⎩ x j−1 } {{ } x j−1
⎪⎭
≤1
∫
x j
|u−I 1 u| 2 H 1 (]x j−1 ,x j [) ≤ (x j −x j−1 ) 2
⎫
x j−1
|u ′′ (ξ)| 2 dξ . (5.3.11)
As above, apply this estimate on[x j−1 ,x j ], sum over all cells of the meshMand take square root.
(5.3.11) ⇒ |u−I 1 u| H 1 ([a,b]) ≤ h ∥
M
∥u ′′∥ ∥ L 2 ([a,b]) . (5.3.12)
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(5.3.8)
⇒
∫
x j
∫
x j
‖u−iu‖ 2 L 2 (]x j−1 ,x j [) = |u(x)−Iu(x)| 2 dx ≤ |x j −x j−1 | 4
x j−1
x j−1
|u ′′ (ξ)| 2 dξ .
Apply this estimate on[x j−1 ,x j ], sum over all cells of the meshMand take square root.
(5.3.9)
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What we learn from this example:
1. We have to rely on smoothness of the interpolandu to obtain bounds for norms of the interpolation
error.
2. The bounds involve norms of derivatives of the interpoland.
3. For smoothuwe find algebraic convergence (→ Def. 1.6.20) of norms of the interpolation error in
terms of mesh width h M → 0.
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(5.3.9) ⇒ ‖u−I 1 u‖ L 2 ([a,b]) ≤ ∥
h2 M
∥u ′′∥ ∥ L 2 ([a,b]) . (5.3.10)
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5.3.2 Error estimates for linear interpolation in 2D
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|u−I 1 u| H 1 ([a,b]) :
Differentiate representation formula (5.3.7): forx j−1 < x < x j
∫
x j
d
dx (Iu−u)(x) = − ξ −x j−1
u ′′ (ξ)dξ +
x j −x j−1
x j−1
∫
x j
x j−1
x j −ξ
x j −x j−1
u ′′ (ξ)dξ .
5.3
p. 494
Focus on gist of proof and its analogy to 1D; technical details should be skipped
5.3
p. 49

Given: polygonal domainΩ ⊂ R 2
triangular mesh M of Ω (→
Def. 3.3.1)
Ω
Fig. 149
Sect 5.3.1 introduced piecewise linear interpolation on a mesh/grid in 1D. The next definition gives
the natural 2D counterpart on a triangular mesh, which is closely related to the piecewise linear
reconstruction (interpolation) operator from (4.2.6), see Figs. 131, 132.
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Crucial for localization to work: linear interpolation operator I 1 : C 0 (¯Ω) ↦→ S1 0 (M) can be defined
purely locally by
I 1 u |K = u(a 1 )λ 1 +u(a 2 )λ 2 +u(a 3 )λ 3 , (5.3.15)
for each triangle K ∈ M with vertices a 1 , a 2 , a 3 (λ k ˆ= barycentric coordinate functions = local
shape functions for S1 0 (M), see Fig. 72).
Next step, cf. (5.3.7):
representation formula for local interpolation error.
a 3
u ∈ C 2 ( ¯K): by mean value formula∀x ∈ K,
u(a j ) = u(x)+gradu(x)·(a j −x)+
ω 3 ∫ 1
(a j −x) ⊤ D 2 u(x+ξ(a j −x))(a j −x)(1−ξ)dξ ,
K 0
(5.3.16)
ω 1 ⎛
a 1 ∂
ω 2 u
2
D 2 u(x) = ⎝ ∂x 2 (x) ∂ 2 ⎞
u
1 ∂x 1 ∂x (x) 2
∂ 2 u
a 2 ∂x 1 ∂x (x) ∂ 2 ⎠
u
2 ∂x 2 (x) ˆ= Hessian.
2
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The formula (5.3.16) is easily verified by applying integration by parts
5.3
p. 497
f(b)−f(a) = [ ξf ′ (ξ) ] ∫ b
∫ b
b
a − ξf ′′ (ξ)dξ = f ′ (a)(b−a)+ (b−ξ)f ′′ (ξ)dξ .
a
a
5.3
p. 49
Definition 5.3.13 (Linear interpolation in 2D).
The linear interpolation operatorI 1 : C 0 (¯Ω) ↦→ S1 0 (M) is defined by
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to the functionφ(t) = u(ta j +(1−t)x) witha = 0,b = 1.
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I 1 u ∈ S 0 1 (M) , I 1u(p) = u(p) ∀p ∈ V(M) .
Next, use (5.3.16) to replace u(a j ) in the formula (5.3.15) for local linear interpolation. Also use the
identities for the barycentric coordinate functions
3∑
3∑
λ j (x) = 1 , x = a j λ j (x) . (5.3.17)
j=1 j=1
Recalling the definition of the nodal basis B = {b p N :p ∈ V(M)} of S0 1 (M) from (3.2.3), where bp N
is the “tent function” associated with nodep, an equivalent definition is, cf. (3.5.44),
I 1 u = ∑
u(p)b p N , u ∈ C0 (Ω) . (5.3.14)
Task:
p∈V(M)
☞ For “sufficiently smooth”u : Ω ↦→ R (↔u ∈ C ∞ (¯Ω) to begin with) estimate
Idea:
➣
interpolation error norm ‖u−I 1 u‖ H 1 (Ω) .
Localization
I 1 local ➣ first, estimate ‖u−I 1 u‖ 2 H 1 (K)
,K ∈ M,
then, global estimate via summation as in Sect. 5.3.1.
Focus on single triangleK ∈ M
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3∑
3∑
3∑
I 1 u(x) = u(a j )λ j (x) = u(x)· λ j (x) +gradu(x)· (a j −x)λ j (x) +R(x) ,
j=1
j=1
j=1
} {{ } } {{ }
=1
=0
(
3∑ ∫ )
1
with R(x) := (a j −x) ⊤ D 2 u(x+ξ(a j −x))(a j −x)(1−ξ)dξ λ j (x) . (5.3.18)
j=1
0
Again, as in the case of (5.3.7) for 1D linear interpolation we have arrived at an integral representation
formula for the local interpolation error:
(
3∑ ∫ )
1
(u−I 1 u)(x) = (a j −x) T D 2 u(x+ξ(a j −x))(a j −x)(1−ξ)dξ λ j (x) . (5.3.19)
j=1
0
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5.3
p. 50

Together with the triangle inequality, the trivial bound|λ j | ≤ 1 yields
⎛ ⎛
⎞ ⎞
2
3∑
∫ ∫1
‖u−I 1 u‖ L 2 (K) ≤ ⎜ ⎜
⎝ ⎝ (a j −x) T D 2 u(x+ξ(a j −x))(a j ⎟
−x)(1−ξ)dξ⎠
dx⎟
⎠
j=1
K
0
To estimate an expression of the form
∫
K
( ∫ 1
2
(a j −x) T D 2 u(x+ξ(a j −x))(a −x)(1−ξ)dξ) j dx , (5.3.20)
0
we may assume, without loss of generality, thata j = 0.
➣
Denote
✬
✫
Task: estimate terms (where0is a vertex ofK!)
∫
K
( ∫ 1
0
2 ∫
x ⊤ D u((1−ξ)x)x(1−ξ)dξ) 2 dx =
γ ˆ= angle ofK at vertex0,
h ˆ= length of longest edge ofK.
K is contained in the sector
S := {x = ( rcosϕ
rsinϕ) : 0 ≤ r < h,0 ≤ ϕ ≤ γ}
K
( ∫ 1
2
x ⊤ D u(ξx)xξdξ) 2 dx .
0
✩
Lemma 5.3.21. For anyψ ∈ L 2 (S)
⎛ ⎞2
∫ ∫1
⎜
⎝ |y| 2 ⎟
ψ(τy)τ dτ⎠
dy ≤ h4
8 ‖ψ‖2 L 2 (S) .
S 0
0
Using polar coordinates (r,ϕ), ŝ ϕ = ( cosϕ
sinϕ)
, see [29, Bsp. 8.5.3], and Cauchy-Schwarz inequality
(2.2.15):
⎛ ⎞2
⎛ ⎞2
∫
S
⎜
⎝
∫1
|y| 2 ⎟
]ψ(τy)τ dτ⎠
0
∫ γ ∫h
dy =
⎜
⎝
✪
∫1
r 2 ⎟
ψ(τrŝ ϕ )τ dτ⎠
rdrdϕ
0 0 0
∫ γ ⎛ ⎞2
∫h ∫r
∫ γ ∫h ∫r
⎜ ⎟
= ⎝ ψ(σŝ ϕ )σdσ⎠
rdrdϕ ≤
0
0
0
0
0
0
γ
∫r
ψ 2 (σŝ ϕ )σdσ ·
0
S
h
σdσrdrdϕ
1
2
.
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5.3
p. 501
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5.3
p. 502
≤ 2
1 0
∫ γ ∫h
0
∫h
ψ 2 (σŝ ϕ )σdσdϕ·
0
r 3 dr .
Use|z ⊤ Ay| ≤ ‖A‖ F |z||y|,A ∈ R n,n ,z,y ∈ R n , and then apply Lemma 5.3.21 withy := x−a j ,
τ = 1−ξ
‖u−I 1 u‖ 2 L 2 (K) ≤ 3 8 h4 ∥
K ∥∥D 2 ∥ ∥∥ 2
u∥ , F L 2 (K) (5.3.22)
∥
with Frobenius matrix norm ∥D 2 u(x) ∥ 2 2∑
F := ∂ 2 u
2
(x)
(→ [18, Def. 6.5.24]),
∣∂x i,j=1 i ∂x j ∣
size of triangle h K := diamK := max{|p−q|: p,q ∈ K}
Estimate for gradient: from (5.3.16) we infer the local integral representation formula, which can also
be obtained by taking the gradient of (5.3.19).
3∑ 3∑
gradI 1 u(x) = u(x) gradλ j (x) + (a j −x) ⊤ gradλ j (x)·gradu(x)+G(x) ,
j=1
} {{ }
j=1
} {{ }
=0
=I
(
3∑ ∫ )
1
with G(x) := (a j −x) ⊤ D 2 u(x+ξ(a j −x))(a j −x)(1−ξ)dξ gradλ j (x) .
j=1
0
} {{ }
cf. (5.3.20)
Note that
∑
grad 3 λ j (x) = grad1 = 0 and
j=1
3∑
3∑
gradλ j (x)(a j −x) ⊤ = gradλ j (x)(a j ) ⊤ = grad ( ∑ 3 λ j (x)a j) = gradx = I .
j=1
j=1
j=1
(3.5.22) ➤ |gradλ j (x)| ≤ h K
2|K| , x ∈ K . (5.3.23)
‖grad(u−I 1 u)‖ 2 L 2 (K) ≤ h2 (5.3.22)
K
4|K| 2 ‖R‖2 L 2 (K)
≤ 3 h 6 ∥∥ K ∥∥ ∥∥D
84|K| 2 2 ∥ ∥∥ 2
u∥ . F L 2 (K) (5.3.24)
Summary of local interpolation error estimates for linear interpolation according to Def. 5.3.13:
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5.3
p. 50

✬
✩
Lemma 5.3.25 (Local interpolation error estimates for 2D linear interpolation).
For any triangleK andu ∈ C 2 (K) the following holds
‖u−I 1 u‖ 2 L 2 (K) ≤ 3 8 h4 ∥
K ∥∥D 2 ∥ ∥∥ 2
u∥ , F L 2 (K) (5.3.22)
‖grad(u−I 1 u)‖ 2 L 2 (K) ≤ 24
3 h 6 ∥∥ K ∥∥ ∥∥D 2 ∥ ∥∥ 2
|K| 2 u∥ . F L 2 (K) (5.3.24)
✫
✪
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ρ k small
Fig. 150
ρ k large
Fig. 151
ρ k large
Fig. 152
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New aspect compared to Sect. 5.3.1: shape ofK enters error bounds of Lemma 5.3.25.
We aim to extract this shape dependence from the bounds.
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The shape regularity measure ρ M is often used to gauge the quality of meshes produced by mesh
generators.
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Final step: we add up the local estimates from Lemma 5.3.25 over all triangles of the mesh and take
the square root.
5.3
p. 505
✬
✩
5.3
p. 50
Definition 5.3.26 (Shape regularity measures).
For a simplexK ∈ R d we define its shape regularity measure as the ratio
ρ K := h d K : |K| ,
and the shape regularity measure of a simplicial meshM = {K}
ρ M := max
K∈M ρ K .
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Theorem 5.3.27 (Error estimate for piecewise linear interpolation).
For anyu ∈ C 2 (¯Ω)
√
‖u−I 1 u‖ L 2 (Ω) ≤ 38 h 2 ∥
M∥∥D 2 ∥ ∥∥L
u∥ , F 2 (Ω)
√ ∥∥ ‖grad(u−I 1 u)‖ L 2 (Ω) ≤ 3 ∥∥ ∥∥D
24 ρ M h 2 ∥ ∥∥L M u∥ . F 2 (Ω)
✫
✪
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Important:
✗
✖
shape regularity measureρ K is an invariant of a similarity class of triangles.
(= if a triangle is transformed by scaling, rotation, and translation, the shape regularity measure does
not change)
➣
Sloppily speaking,ρ K depends only on the shape, not the size ofK
For triangleK: ρ K large ⇔ K “distorted” ⇔ K has small angles
✔
✕
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5.3
p. 506
Remark 5.3.28 (Energy norm andH 1 (Ω)-norm).
Objection!
Well, Cea’s lemma Thm. 5.1.10 refers to the energy norm, but Thm. 5.3.27 provides
estimates inH 1 (Ω)-norm only!
☞ For uniformly positive definite (→ Def. 2.1.11) and bounded coefficient tensorα : Ω ↦→ R d,d , cf.
(2.1.8),
∃0 < α − < α + : α − ‖z‖ 2 ≤ z T α(x)z ≤ α + ‖z‖ 2 ∀z ∈ R d , x ∈ Ω ,
and the energy norm (→ Def. 2.1.27) induced by
∫
a(u,v) := (α(x)gradu)·gradvdx , u,v ∈ H0 1 (Ω) ,
Ω
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(5.1.6) 5.3
p. 50

we immediately find the equivalence (= two-sided uniform estimate)
√
α − |v| H 1 (Ω) ≤ ‖v‖ a ≤ √ α + |v| H 1 (Ω) . (5.3.29)
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Reassuring: Again, it is only the norms‖u‖ H m (Ω) that matter for us !
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Thus, interpolation error estimates in |·| H 1 (Ω) immediately translate into estimates in terms of the
energy norm.
5.3.3 The Sobolev scales
Bounds in Thm. 5.3.27 involve
∥ ∥
∥∥D 2 ∥∥L
u∥ F 2 (Ω)
measures smoothness ofu
△
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Now, we have come across an additional purpose of Sobolev spaces and their norms:
✗
✖
provide framework for
variational formulation of
Sobolev scale:
elliptic BVP
(→ Sect. 2.2)
Sobolev
spaces
... ⊂ H 3 (Ω) ⊂ H 2 (Ω) ⊂ H 1 (Ω) ⊂ L 2 (Ω)
provide norms‖·‖ H m (Ω) that
measure smoothness of
functions
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➡
Norms of this type are a tool to measure the smoothness of functions (that usually are solutions
of elliptic BVP):
Observation:
bounds in Thm. 5.3.27 = “principal parts” of Sobolev norms, that is, the parts containing
the highest partial derivatives.
5.3
p. 509
5.3
p. 51
Definition 5.3.30 (Higher order Sobolev spaces/norms).
Them-th order Sobolev norm,m ∈ N 0 , foru : Ω ⊂ R d ↦→ R (sufficiently smooth) is defined by
‖u‖ 2 m H m (Ω) := ∑ ∑
∫
|D α u| 2 dx , where D α ∂ |α| u
u :=
k=0α∈N d Ω
∂x α .
1
,|α|=k
1 ···∂xα d
d
Sobolev space H m (Ω) := {v : Ω ↦→ R: ‖v‖ H m (Ω) < ∞} .
Numerical
Methods
for PDEs
Definition 5.3.31 (Higher order Sobolev semi-norms).
Them-th order Sobolev semi-norm,m ∈ N, for sufficiently smoothu : Ω ↦→ R is defined by
|u| 2 H m (Ω) :=
∑
∫
|D α u| 2 dx .
α∈N d Ω
,|α|=m
Numerica
Methods
for PDEs
Recall: multiindex notation (3.3.4), (3.3.5)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Elementary observation: |p| H m (Ω) = 0 ⇔ p ∈ P m−1(R d )
By density arguments we can rewrite the interpolation error estimates of Thm. 5.3.27 in
terms of Sobolev semi-norms:
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✬
✩
Gripe (→ Sect. 2.2): Don’t bother me with these Sobolev spaces !
Corollary 5.3.32 (Error estimate for piecewise linear interpolation in 2D).
Under the assumptions/with notations of Thm. 5.3.27
Response: Well, these concepts are pervasive in the numerical analysis literature and you have to
be familiar with them.
5.3
p. 510
✫
√
‖u−I 1 u‖ L 2 (Ω) ≤ 38 h 2 M |u| H 2 (Ω)
√ ,
∀u ∈ H 2 (Ω) .
|u−I 1 u| H 1 (Ω) ≤ 3
24 ρ M h M |u| H 2 (Ω) ,
✪
5.3
p. 51

Numerical
Methods
for PDEs
We will try to find this out experimentally by computing the best possible constants in the estimates
‖u−I 1 u‖ L 2 (K) ≤ C K,2h 2 k ‖u‖ H 2 (K) , ‖u−I 1u‖ H 1 (K) ≤ C Kh K ‖u‖ H 2 (K) .
Numerica
Methods
for PDEs
Remark 5.3.33 (Continuity of interpolation operators).
Apply△-inequality to estimates of Cor. 5.3.32:
Note: Merely translating, rotating, or scalingK does not affect the constantsC K,2 andC K . Therefore,
we can restrict ourselves to “canonical triangles”. Every general triangle can be mapped to one
of these by translating, rotating, and scaling.
‖I 1 u‖ L 2 (Ω) ≤ ‖u‖ L 2 (Ω) + √
38 h 2 M |u| H 2 (Ω) ≤ 2‖u‖ H 2 (Ω) , (5.3.34)
if lengths are scaled such that h M ≤ 1. Estimate (5.3.34) means that I 1 : H 2 (Ω) ↦→ L 2 (Ω) is a
continuous linear mapping.
The same conclusion could have been drawn from the following fundamental result:
✬
Theorem 5.3.35 (Sobolev embedding theorem).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✩
‖u−I 1 u‖ L
C K,2 := sup
2 (K)
u∈H 2 (K)\{0} ‖u‖ H 2 (K)
on triangleK := convex{ (00 ) ,
( 10 ) ,
( px
p y
) } .
‖u−I 1 u‖ H
, C K := sup
1 (K)
,
u∈H 2 (k)\{0} ‖u‖ H 2 (K)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
m > d 2
✫
⇒ H m (Ω) ⊂ C 0 (Ω) ∧ ∃C = C(Ω) > 0: ‖u‖ ∞ ≤ C‖u‖ H m (Ω) ∀u ∈ H m (Ω) .
✪
On the other handI 1 : H 1 (Ω) ↦→ L 2 (Ω) is not continuous, as we learn from Rem. 2.3.16.
5.3.4 Anisotropic interpolation error estimates
△
5.3
p. 513
Numerical
Methods
for PDEs
Sampling the space of “canonical” triangles
(modulo similarity)
5.3
p. 51
Numerica
Methods
for PDEs
0 ≤ p x ,p y ≤ 1 .
OPTIONAL SECTION
Triangular cells with “bad shape regularity” (ρ K “large”): very small/large angles:
0
p x
p y
K
1
+ Numerical computation ofC K ,C K,2
implementation by A. Inci (spectral polynomial
Galerkin method)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
DRAFT
The estimates of Lemma 5.3.25 might suggest that we face huge local interpolation errors, once ρ K
becomes large.
Issue: are the estimates of Lemma 5.3.25 sharp ?
5.3
p. 514
5.3
p. 51

Numerical
Methods
for PDEs
y
Numerica
Methods
for PDEs
0.18
0.16
0.14
0.12
0.1
C K,2
0.08
0.06
0.04
0.02
0
1
1
0.8
0.8
0.6
0.6
0.4
0.4 p px
y
0.2
0.2
0 0
Fig. 153
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Fig. 156
The interpolant becomes steeper and steeper ash → 0:
h
1 x
{ (00 ) (
triangle K := convex , 10 ) ( 1/2 ) } ,
h
, h > 0,
u(x,y) = x(1−x),0 < x < 1.
✁ linear interpolant ofuonK ash → 0
‖u‖ 2 H 2 (K) = 3031
1440 h , ‖u−I 1u‖ 2 H 1 (K) = 29
2880 h+ 1
12 h+ 1
32 h−1 , ‖u−I 1 u‖ 2 L 2 (K) = 29
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
R. Hiptm
DRAFT
2889 h
5.3
p. 517
5.3
p. 51
14
5
4.5
4
Numerical
Methods
for PDEs
‖u−I 1 u‖ 2 H 1 (K)
‖u‖ 2 H 2 (K)
≥ 269
6062 + 45
3031 h−2 ,
‖u−I 1 u‖ 2 L 2 (K)
‖u‖ 2 H 2 (K)
= 29
6062 .
Numerica
Methods
for PDEs
12
3.5
10
8
3
log(C K
)
2.5
Example 5.3.36 (Good accuracy on “bad” meshes).
C K
Fig. 154 0
6
2
1.5
4
1
2
0.5
0
1
0.8
0.6
p x
0.4
0.2
0
1
0.8
0.6
p y
0.4
0.2
0
0
1
0.8
p x
0.6
0.4
0.2
1
0.8
0.6
0
0.2
0.4
R. Hiptmair
p y
C. Schwab,
Fig. 155 H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Ω =]0,1[ 2 , u(x 1 ,x 2 ) = sin(πx 1 )sin(πx 2 ), BVP −∆u = f, u |∂Ω = 0, finite element Galerkin
discretization on triangular meshes,V N = S 0 1,0 (M).
☞
meshes created by random distortion of tensor product grids
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
5.3
p. 518
5.3
p. 52

1
2D triangular mesh
1
2D triangular mesh
Numerical
Methods
for PDEs
Example 5.3.37 (Gap between interpolation error and best approximation error).
Numerica
Methods
for PDEs
0.9
0.9
0.8
0.7
0.8
0.7
Ex. 5.3.36 raises doubts whether the interpolation error can be trusted to provide good, that is, reasonably
sharp bounds for the best approximation error.
0.6
0.6
0.5
0.4
0.3
0.2
0.5
0.4
0.3
0.2
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
In this example we will see that
inf ‖u−v N ‖ 1 ≪ ∥ ∥u−I p u ∥ v N ∈Sp(M)
0 H 1 (Ω)
is possible !
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
0.1
0.1
DRAFT
DRAFT
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
# Vertices : 41, # Elements : 64, # Edges : 56 Fig. 157
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
# Vertices : 145, # Elements : 256, # Edges : 208Fig. 158
h
δ
h
δ
Elementary cell of “bad mesh”M bad
Elementary cell of “good mesh”M good
5.3
p. 521
5.3
p. 52
H 1 seminnorm error [log]
H 1 seminnorm errors against minimum angles
10 0
10 −1
145 nodes
545 nodes
2113 nodes
8321 nodes
33025 nodes
H 1 seminnorm error [log]
H 1 seminnorm errors against maximum angles
10 0
10 −1
145 nodes
545 nodes
2113 nodes
8321 nodes
33025 nodes
Numerical
Methods
for PDEs
Yet,
‖u−I 1 u‖ H
On “bad” mesh : sup
1 (Ω)
→ ∞ ash/δ → ∞,
u∈H 2 (Ω) ‖u‖ H 2 (Ω)
‖u−I 1 u‖ H
On “good” mesh : sup
1 (Ω)
uniformly bounded inh/δ.
u∈H 2 (Ω) ‖u‖ H 2 (Ω)
inf ‖u−v N ‖
v N ∈S1 0 H (M bad)
1 (Ω) ≤
inf ‖u−v N ‖
v N ∈S1 0 H (M good)
1 (Ω) ∀u ∈ H 2 (Ω) .
Numerica
Methods
for PDEs
10 −2
Fig. 159
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Minimum angle
10 −2
1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
Maximum angle
Fig. 160
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
✸
Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
DRAFT
Monitored: for different mesh resolutions, H 1 (Ω)-seminorm of discretization error as function of
smallest/largest angle in the mesh.
5.3.5 General approximation error estimates
Observation: Accuracy does not suffer much from distorted elements !
✸
5.3
p. 522
In Sect. 5.3.2 we only examined the behavior of norms of the interpolation error for piecewise linear
R.
interpolation intoS1 0 (M), that is, the case of Lagrangian finite elements of degreep = 1.
5.3
p. 52

However, Ex. 5.2.2 sent the clear message that quadratic Lagrangian finite elements achieve faster
convergence of the energy norm of the Galerkin discretization error, see Fig. 137, 138.
⇕
On the other hand quadratic finite elements could not deliver faster convergence in Ex. 5.2.6.
In this section we learn about theoretical results that shed light on these observations and extend the
results of Sect. 5.3.2.
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
whereb l N
are the nodal basis functions.
(3.4.3) ⇒ I p (u)(p l ) = u(p l ) , l = 1,...,N (Interpolation!) .
By virtue of the location of the interpolation nodes, see Ex. 3.4.2, Ex. 3.4.5, and Fig. 103, the nodal
interpolation operators are purely local:
Q∑
∀K ∈ M: I p u |K = u(q K i )bK i , (5.3.40)
i=1
q K i ,i = 1,...,Q = local interpolation nodes in cellK ∈ M, see Ex. 3.4.2, Ex. 3.4.5, and Fig. 103,
b K i ,i = 1,...,Q = local shape functions: bK i (qK j ) = δ ij.
△
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Remark 5.3.38 (L ∞ interpolation error estimate in 1D).
The faster convergence of quadratic Lagrangian FE in Ex. 5.2.2 does not come as a surprise: recall
the esimtate from [18, Eq. 9.4.6]:
∥
∥u−I p u ∥ L ∞ ([a,b]) ≤ hp+1
M ∥
∥u (p+1)∥ ∥ ∥L
(p+1)! ∞ ([a,b])
∀u ∈ C p+1 ([a,b]) ,
whereI p u is theM-piecewise polynomial interpolant ofuof local degreep. It generalizes (5.3.5).
➣
∥
∥u−I p u ∥ ∥ L ∞ ([a,b]) = O(hp+1 M ) !
Remark 5.3.39 (Local interpolation onto higher degree Lagrangian finite element spaces).
M: triangular/tetrahedral/quadrilateral/hybrid mesh of domainΩ(→ Sect. 3.3.1)
Recall (→ Sect. 3.4): nodal basis functions of p-th degree Lagrangian finite element space S 0 p(M)
defined via interpolation nodes, cf. (3.4.3).
△
5.3
p. 525
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Example 5.3.41 (Piecewise quadratic interpolation). → Ex. 3.4.2
triangleK =convex{a 1 ,a 2 ,a 3 },p = 2
⇒ local quadratic interpolation:
3∑
I 2 u |K = − λ i (1−2λ i )u(a i )
i=1
+ ∑
4λ i λ j u( 1 2 (ai +a j )) .
1≤i

✬
✩
Theorem 5.3.42 (Best approximation error estimates for Lagrangian finite elements).
Let Ω ⊂ R d , d = 1,2,3, be a bounded polygonal/polyhedral domain equipped with a mesh M
consisting of simplices or parallelepipeds. Then, for each k ∈ N, there is a constant C > 0
Numerical
Methods
for PDEs
A priori error estimates like (5.3.46) exhibit only the trend of the (norm of) the discretization
error as discretization parametersh M (mesh width),p(polynomial degree) are varied.
Numerica
Methods
for PDEs
depending only onk and the shape regularity measureρ M such that
✫
( hM
inf N ‖
v N ∈Sp(M)‖u−v 0 H 1 (Ω) ≤ C p
) min{p+1,k}−1
‖u‖ H k (Ω) ∀u ∈ H k (Ω) . (5.3.43)
✪
△
This theorem is a typical example of finite element analysis results that you can find in the literature. It
is important to know what kind of information can be gleaned from statements like that of Thm. 5.3.42.
Remark 5.3.44 (“Generic constants”).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 5.3.47. The estimate of Thm. 5.3.42 is sharp: the powers ofh M andpcannot be increased.
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
A statement like (5.3.43) is typical of a priori error estimates in the numerical analysis literature, which
often come in the form
where
‖u−u N ‖ X ≤ C · “discretization parameter”·‖u‖ Y ,
C > 0 is not specified precisely or only claimed to exist (though, in principle, they
could be computed),
C must neither depend on the exact solutionunor the discrete solutionu N ,
the possible dependence ofC on problem parameters or discretization paramters has
Such constantsC > 0 are known as generic constants. Customarily, different generic constants are
even denoted by the same symbol (“C” is most common).
△
5.3
p. 529
Numerical
Methods
for PDEs
What do Thm. 5.3.42, (5.3.46), tell us about the efficiency of a Lagrangian finite element Galerkin
discretization of a 2nd-order elliptic BVP?
Question 5.3.48. What computational effort buys us what error (measured in energy norm)?
Bad luck (→ Rem. 5.3.45):
actual error norm remains elusive! Therefore, rephrase the question so
that it fits the available information about the effect of changing discretization parameters on the error:
5.3
p. 53
Numerica
Methods
for PDEs
Remark 5.3.45 (Nature of a priori estimates). → Sect. 1.6.2
Question 5.3.49. What increase in computational effort buys us a presribed decrease of the (energy
norm of the) error?
Cea’s lemma, Thm. 5.1.10
➣ Thm. 5.3.42 implies a priori estimates of the energy norm of the
finite element Galerkin discretization error (see also Rem. 5.3.28) of
the form
( hM
‖u−u N ‖ a ≤ C
p
) min{p+1,k}−1
‖u‖ H k (Ω) , (5.3.46)
whereuis the exact solution of the discretized 2nd-order elliptic boundary value problem.
Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
The answer to this question offers an a priori gauge of the asymptotic efficiency of a discretization
method.
Convention:
computational effort≈number of unkonwnsN = dimS 0 p(M) (problem size)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
(5.3.46) does not give concrete information about‖u−u N ‖ a , because
Framework:
familyMof simplicial meshes of domainΩ ⊂ R d ,d = 1,2,3, created by
global regular refinement of a single initial mesh
we do not know the value of the “generic constant”C > 0, see Rem. 5.3.44,
R.
asuis unknown, a bound for‖u‖ H k (Ω)
may not be available.
5.3
p. 530
5.3
p. 53

Global regular refinement of a simplicial mesh (→ Ex. 5.1.12)
Numerical
Methods
for PDEs
with constants hidden in≈depending onρ M only.
✸
Numerica
Methods
for PDEs
avoids greater distortion of “child cells” w.r.t. their parents,
spawns meshes with fairly uniform sizeh K of cells.
Now, we merge (5.3.46) and (5.3.50):
∃C > 0: ρ M ≤ C ,
∃C > 0: max{h K /h K ′, K,K ′ ∈ M} ≤ C ,
∀M ∈ M .
Now, for meshes ∈ M, we investigate “N-dependence”, N = dimS 0 p(M), of energy norm of finite
element discretization error:
Counting argument
N = dimS 0 p(M) ≈ p d h −d
M ⇒ h M
p ≈ N−1/d . (5.3.50)
dimensions of local spaces, Lemma 3.3.6
∼ ♯M ∼ ♯V(M),E(M) etc.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
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DRAFT
u ∈ H k (Ω)
Thm. 5.3.42
⇒
withC > 0 depending only ond,p,k, andρ M .
inf ‖u−v N ‖
v N ∈Sp 0 H (M) 1 (Ω) ≤ CN−min{p,k−1} d ‖u‖ H k (Ω) , (5.3.53)
(5.3.53) ➣ algebraic convergence (→ Def. 1.6.20) in problem size
We observe that
(rate
min{p,k −1}
)
d
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Notation:
≈ ˆ= equivalence up to constants only depending onγ (in M γ ),Ω
5.3
p. 533
the rate of convergence is limited by the polynomial degreepof the Lagragian FEM,
5.3
p. 53
Example 5.3.51 (Dimensions of Lagrangian finite element spaces on triangular meshes).
d = 2: for triangular meshesM, by Lemma 3.3.6
dimS 0 p(M) = ♯{nodes(M)}+♯{edges(M)}(p−1)+♯M 1 2 (p−1)(p−2) .
1 basis function per vertex
p−1 basis functions per edge
1
2 (p−1)(p−2) “interior” basis functions
Geometric considerations: the number of triangles sharing a vertex can be bounded in terms ofρ M ,
becauseρ M implies a lower bound for the smallest angles of the triangular cells.
∃C = C(ρ M ): ♯{K j ∈ M: K i ∩K j ≠ ∅} ≤ C (i = 1,2,...,#.M) .
If every vertex belongs only to a small number of triangles, the number♯{nodes(M)} can be bounded
byC ·♯M, whereC > 0 will depend onρ M only. The same applies to the edges.
♯{nodes(M)},♯{edges(M)} ≈ ♯M .
Numerical
Methods
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
the rate of convergence is limited by the smoothness of the exact solutionu, measured by means
of the Sobolev indexk, see Sect. 5.3.3,
the rate of convergence will be worse for d = 3 than for d = 2, the effect being more pronounced
for smallk orp.
Answer to Question 5.3.49:
Assumption:
✬
✫
a priori error estimate (5.3.53) is sharp
∃C = C(u,...) > 0: error norm(N) ≈ CN −min{p,k−1} d ∀M ∈ M .
(
error norm(N 1 )
) min{p,k −1}
N1
−
error norm(N 2 ) ≈ d .
N 2
reduction of (the energy norm of)
the error by a factorρ > 1
requires
increase of the problem size
d
by factorρmin{p,k−1}
✩
✪
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
dimS 0 p(M) ≈ (♯M)p 2 , (5.3.52)
5.3
p. 534
5.3
p. 53

increase in problem size (factor)
error ρ
d = 2, u∈ H 6 (Ω)
p = 1
p = 2
p = 3
10 12
p = 4
p = 5
10 10
10 8
10 6
10 4
10 2
10 0
10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7
reduction factor Fig. 161
10 14
exact solutionu ∈ H 6 (Ω)
10
8
6
4
2
0
0
2
4
6
8
5
10
10
15
20
smoothness index k: u∈ H k (Ω)
polynomial degree p
log(increase in problem size)
error reduction by factorρ = 100
0
Fig. 162
Numerical
Methods
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Note:
arbitrarily fast (super-)algebraic convergence for very smooth solutionsu ∈ C ∞ (Ω)
5.4 Elliptic regularity theory
Crudely speaking, in Sect. 5.3.5 we saw that the asymptotic behavior of the Lagrangian finite element
Galerkin discretization error (for 2nd-order elliptic BVPs) can be predicted provided that
△
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Discussion: Solutionu ∈ H k (Ω) ➣ optimal asymptotic efficiency forp = k −1
• we use families of meshes, whose cells have rather uniform size and whose
shape regularity measure is uniformly bounded,
Remark 5.3.54 (General asymptotic estimates).
5.3
p. 537
• we have an idea about the smoothness of the exact solution u, that is, we knowu ∈ H k (Ω) for a
(maximal)k, see Thm. 5.3.42.
5.4
p. 53
Recall (→ Sect. 1.6.2):
convergence is an asymptotic notion
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Methods
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Knowledge about the mesh can be taken for granted, but
Numerica
Methods
for PDEs
Now we deduce asymptotic estimates for the best approximation errors from Thm. 5.3.42, and
(5.3.53), in particular, for the caseN → ∞:
• h-refinement: p fixed,h M → 0 forM ∈ M:
(5.3.53) ⇒ algebraic convergence w.r.t. N
☞ p ≤ k −1 inf ‖u−v N ‖ 1 = O(N −p/d ) (5.3.55)
v N ∈Sp(M)
0
☞ k ≤ p+1 inf ‖u−v N ‖ 1 = O(N −(k−1)/d ) (5.3.56)
v N ∈Sp(M)
0
Note:
for very smooth solutionu, i.e. k ≫ 1, polynomial degreeplimits speed of convergence
• p-refinement: M ∈ M fixed,p → ∞:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
how can we guess the smoothness of the (unknown !) exact solutionu?
A (partial) answer is given in this section.
Focus:
Scalar 2nd-order elliptic BVP with homogeneous Dirichlet boundary conditions
−div(σ(x)gradu) = f inΩ , u = g on∂Ω .
To begin with, we summarize the available information:
➣ Known: u solves BVP + Informtion about coefficientσ, domain
Ω, source functionf, boundary datag
} {{ }
u will belong to a certain class of functions (e.g. subspaceS ⊂ V )
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
☞ p large inf ‖u−v N ‖ 1 = O(N −(k−1)/d ) (5.3.57)
v N ∈Sp(M)
0
5.3
p. 538
5.4
p. 54

Example 5.4.1 (Elliptic lifting result in 1D).
d = 1,Ω =]0,1[, coefficientσ ≡ 1, homogeneous Dirichlet boundary conditions:
u ′′ = f , u(0) = u(1) = 0 .
Obvious: f ∈ H k (Ω) ⇒ u ∈ H k+2 (Ω) (a lifting theorem)
Can this be generalized to higher dimensionsd > 1?
Partly so:
✸
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
What about non-smooth∂Ω ?
c 4
ω 4
c 3 c 5 These are very common in engineering applications
(“CAD-geometries”).
3 ω
ω 5
✁ polygonal domain with cornersc i
c 2 ω 2
How will the corners affect the smoothness of solutions
of
c 1 ω
ω 6 c 6
1 u ∈ H0 1 (Ω): ∆u = f ∈ C∞ (Ω)?
Example 5.4.3 (Conrner singular functions).
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✬
✩
5.4
p. 541
5.4
p. 54
Theorem 5.4.2 (Smooth elliptic lifting theorem).
If∂Ω isC ∞ -smooth, ie. possesses a local parameterization byC ∞ -functions, andσ ∈ C ∞ (Ω),
then, for anyk ∈ N,
✫
u ∈ H0 1 (Ω) and −div(σgradu)
∈ Hk (Ω)
u ∈ H 1 (Ω) , −div(σgradu) ∈ H k (Ω) and
In addition, for suchuthere isC = C(k,Ω,σ) such that
‖u‖ H k+2 (Ω) ≤ C‖div(σgradu)‖ H k (Ω) .
⇒ u ∈ H k+2 (Ω) .
gradu·n = 0 on∂Ω
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
✪H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
c
ϕ
Ω
ϕ = ω r
ϕ = 0
Fig. 163
corner singular function
u s (r,ϕ) = rω π sin( π ϕ) , (5.4.4)
ω
r ≥ 0 , 0 ≤ ϕ ≤ ω .
(in local polar coordinates)
u s = 0 on∂Ω locally atc!
Straightforward computation: ∆u s = 0 in Ω !
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
5.4
p. 542
To see this recall:
∆ in polar coordinates:
∆u = 1 ∂
r∂r (r∂u ∂r )+ 1 ∂ 2 u
r 2 ∂ϕ 2 . (5.4.5)
=⇒ ∆u s (r,ϕ)= 1 ∂
(r π r∂r
ω rπ ω −1 sin( π )
ω ϕ) + 1 ∂
r 2rπ ω
∂ϕ cos(π ω ϕ)π ω
( π
) 2r π
= ω −2 π
( π
) 2r sin(
ω ω ϕ)− π
ω −2 π sin(
ω ω ϕ) = 0 .
(5.4.4)
5.4
p. 54

What is “singular” about these functions? Plot them forω = 3π 2 , cf. Ex. 5.2.6
Numerical
Methods
for PDEs
✸
Numerica
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Bad news: With the exeption of concocted examples,
corner singular functions like (5.4.4) will be present in the solution of linear scalar
2nd-order elliptic BVP on polygonal domains!
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
The meaning of “being present” is elucidated in the following theorem:
Fig. 164
Fig. 165
u s for ω = 3π 2
‖gradu s ‖ forω = 3π 2
Recall gradient (2.3.19) in polar coordinates
gradu = ∂u
∂r e r + 1 ∂u
r∂ϕ e ϕ . (2.3.19)
(5.4.4)
=⇒ gradu s (r,ϕ) = π ω rπ ω −1( sin( ω πϕ)e r +cos( ω πϕ)) e ϕ .
ω > π (“re-entrant corner”) =⇒ “gradu s (0) = ∞”
How does this “blow-up” of the gradient affect the Sobolev regularity (that is, the smoothness as
expressed through “u s ∈ H k (Ω)”) of the corner singular functionu s ?
We try to compute|u| H 2 (D), with (in polar coordinates, see Fig. 163)
By tedious computations we find
D := {(r,ϕ):0 < r < 1, 0 < ϕ < ω} .
ω > π ⇒
Def. 5.3.30
=⇒
∫
∥
∥D 2 u s (r,ϕ) ∥ 2
D F rd(r,ϕ) = ∞ .
{
ω > π ⇒ u s ∉H 2 }
(D) .
5.4
p. 545
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
5.4
p. 546
✬
Theorem 5.4.6 (Corner singular function decomposition).
Let Ω ⊂ R 2 be a polygon with J corners c i . Denote the polar coordinates in the corner c i by
(r i ,ϕ i ) and the inner angle at the corner c i by ω i . Additionally, let f ∈ H l (Ω) with l ∈ N 0 and
l ≠ λ ik −1, where theλ ik are given by the singular exponents
Thenu ∈ H0 1 (Ω) with−∆u = f inΩcan be decomposed
✫
λ ik = kπ
ω i
fork ∈ N . (5.4.7)
J∑
u = u 0 ∑
+ ψ(r i ) κ ik s ik (r i ,ϕ i ) , κ ik ∈ R , (5.4.8)
i=1 λ ik

★
✧
✬
✫
Ω ⊂ R 2 has re-entrant corners ⇒
ifusolves∆u = f inΩ,u = 0 on∂Ω,
thenu ∉ H 2 (Ω) in general.
Theorem 5.4.10 (Elliptic lifting theorem on convex domains).
If Ω ⊂ R d convex,u ∈ H0 1(Ω),∆u ∈ L2 (Ω) ⇒ u ∈ H 2 (Ω).
✥
✦
✩
✪
Numerical
Methods
for PDEs
Variational crime = replacing (exact) discrete (linear) variational problem
with perturbed variational problem
u N ∈ V 0,N : a(u N ,v N ) = f(v N ) ∀v N ∈ V 0,N , (3.1.4)
ũ N ∈ V 0,N : a N (ũ N ,v N ) = f N (v N ) ∀v N ∈ V 0,N . (5.5.1)
perturbation of Galerkin solutionu N ➥ perturbed solutionũ N ∈ V 0,N
Numerica
Methods
for PDEs
Terminology: if conclusion of Thm. 5.4.10 true → Dirichlet problem 2-regular.
Similar lifting theorems also hold for Neumann BVPs, BVPs with smooth coefficients.
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C. Schwab,
H.
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V.
Gradinaru
A. Chernov
DRAFT
Approximationsa N (·,·) ≈ a(·,·),f N (·) ≈ f(·) due to
• use of numerical quadrature → Sect. 3.5.4,
• approximation of boundary∂Ω → Sect. 3.6.4.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
We are all sinners! Variational crimes are inevitable in practical FEM, recall Rem. 1.5.3!
Remark 5.4.11 (Causes for non-smoothness of solutions of elliptic BVPs).
Causes for poor Sobolev regularity of solutionuof BVPs for−div(σ(x)gradu) = f:
5.4
p. 549
Which “variational petty crimes” can be tolerated?
5.5
p. 55
Numerical
Methods
for PDEs
Guideline 5.5.2. Variational crimes must not affect (type and rate) of asymptotic convergence!
Numerica
Methods
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• Corner of∂Ω, see above
• Discontinuities ofσ
→ singular functions at “material corners”
• Mixed boundary conditions
• Non-smooth source functionf
σ(x) ≡ σ 2
σ(x) ≡ σ 1
c
σ(x) ≡ σ 3
“material corner” atc
△
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
5.5.1 Impact of numerical quadrature
Model problem: on polygonal/polyhedralΩ ⊂ R d :
∫
∫
u ∈ H0 1 (Ω): a(u,v) := σ(x)gradu·gradvdx = f(v) := fvdx . (5.5.3)
Ω
Ω
Assumptions: σ satisfies (2.5.4), σ ∈ C 0 (Ω),f ∈ C 0 (Ω)
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C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• Galerkin finite element discretization,V N := S 0 p(M) on simplicial meshM
5.5 Variational crimes
5.5
p. 550
• Approximate evaluation ofa(u N ,v N ),f(v N ) by a fixed stable local numerical quadrature rule
(→ Sect. 3.5.4)bigskip
➤ perturbed bilinear forma N , right hand sidef N (see (5.5.1))
5.5
p. 55

Focus: h-refinement (key discretization parameter is the mesh widthh M )
Example 5.5.4 (Impact of numerical quadrature on finite element discretization error).
Numerical
Methods
for PDEs
Finite element theory [8, Ch. 4,§4.1] tells us that the Guideline 5.5.2 can be met, if the
local numerical quadrature rule has sufficiently high order. The quantitative results can be condensed
into the following rules of thumb:
Numerica
Methods
for PDEs
Ω =]0,1[ 2 ,σ ≡ 1,f(x,y) = 2π 2 sin(πx)sin(πy),(x,y) T ∈ Ω
‖u−u N ‖ 1 = O(h p M ) at best Quadrature rule of order2p−1 sufficient forf N .
➢ solution u(x,y) = sin(πx)sin(πy),g = 0.
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C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
‖u−u N ‖ 1 = O(h p M ) at best Quadrature rule of order2p−1 sufficient fora N.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
Details of numerical experiment:
DRAFT
DRAFT
• Quadratic Lagrangian FE (V N = S2 0 (M)) on triangular meshesM, obtained by
regular refinement
5.5.2 Approximation of boundary
• “Exact” evaluation of bilinear form by very high order quadrature
•f N from one point quadrature rule (3.5.37) of order 2
5.5
p. 553
We focus on 2nd-order scalar linear variational problems as in the previous section.
5.5
p. 55
Discretization error [log]
10 1 Discretization errors with respect to H 1 semi−norm for quaratic finit elements
10 0
10 −1
10 −2
p = 1.1
Discretization error [log]
10 1 Discretization errors with respect to H 1 semi−norm for quadratic finite elements
10 0
10 −1
10 −2
p = −0.5
Low order quadrature
High order quadrature
Numerical
Methods
for PDEs
Example 5.5.5 (Impact of linear boundary fitting on FE convergence).
Setting:
Ω := B 1 (0) := {x ∈ R 2 : |x| < 1},u(r,ϕ) = cos(rπ/2) (polar coordinates)
➢ f = π 2r sin(r π/2)+ π 2 cos(r π/2)
Numerica
Methods
for PDEs
10 −3
Low order quadrature
p = 2.1
High order quadrature
10 −4
10 0
10 −1
Mesh width [log]
Fig. 166
10 −2
p = −1.0
10 −3
10 −4
10 1 10 2 10 3 10 4 10 5
H 1 (Ω)-norm of discretization error on unit square (−↔rule (3.5.37),−↔rule (3.5.38))
Dofs [log]
Fig. 167
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
• Sequences of unstructured triangular meshesMobtained by regular refinement (of coarse
mesh with 4 triangles) + linear boundary fitting.
• Galerkin FE discretization based onV N := S1,0 0 (M) orV N := S2,0 0 (M).
• Recorded: approximate norm|u−u N | 1,Ωh , evaluated using numerical quadrature rule (3.5.38).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
(FE solution extended beyond the domain covered by M (“mesh interior”) to Ω (“full domain”) by
means of polynomial extrapolation.)
Observation: Use of quadrature rule of order 2 ⇒
Algebraic rate of convergence (w.r.t. N)
drops fromα = 1 to α = 1/2 !
✸
5.5
p. 554
5.5
p. 55

1
2D triangular mesh
1
2D triangular mesh
Numerical
Methods
for PDEs
5.6 Duality techniques
Numerica
Methods
for PDEs
0.8
0.8
0.6
0.6
0.4
0.2
0
0.4
0.2
0
OPTIONAL SECTION
L 2 -estimates are useful occasionally, but not essential.
−0.2
−0.2
−0.4
−0.6
−0.8
−1
−0.4
−0.6
−0.8
−1
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
5.6.1 Linear output functionals
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C. Schwa
H.
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V.
Gradinar
A. Chern
DRAFT
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
# Vertices : 41, # Elements : 64, # Edges : 104Fig. 168
# Vertices : 145, # Elements : 256, # Edges : 400Fig. 169
Linearly boundary fitted unstructured triangular meshes ofΩ = B 1 (0).
Adopt abstract setting of Sect. 5.1:
linear variational problem (1.4.6) in the form
5.5
p. 557
u ∈ V 0 : a(u,v) = l(v) ∀v ∈ V 0 , (3.1.1)
5.6
p. 55
Discretization errors with respect to H 1 semi−norm
10 0 QFE error on full domain
QFE error on mesh interior
LFE error on full domain
LFE error on mesh interior
Discretization errors with respect to H 1 semi−norm
10 0 QFE error on full domain
p = −0.52
QFE error on mesh interior
LFE error on full domain
LFE error on mesh interior
Numerical
Methods
for PDEs
V 0 ˆ= (real) vector space, a space of functions Ω ↦→ R for scalar 2nd-order elliptic variational
problems, usually “energy space”H 1 (Ω)/H0 1 (Ω), see Sect. 2.2
Numerica
Methods
for PDEs
10 −1
p = 1.02
10 −1
a : V 0 ×V 0 ↦→ R ˆ= a bilinear form, see Def. 1.3.13,
Discretization error [log]
10 −2
10 −3
p = 1.57
Discretization error [log]
10 −2
10 −3
p = −0.78
l : V 0 ↦→ R ˆ= a linear form, see Def. 1.3.13,
Assumptions 5.1.1, 5.1.2, 5.1.3 are supposed to hold ➣ existence, uniqueness, and stability of
solutionuby Thm. 5.1.4.
p = −1.01
10 −4
10 −5
10 0
p = 2.04
10 −1
Mesh width [log]
10
Fig. 170 1 10 2 10 3 10 4 10
Fig. 171
5
10 −2
10 −4
10 −5
Dofs [log]
H 1 (Ω)-norm of discretization error on unit ball (−↔p = 1,−↔p = 2)
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
(Examples of 2nd-order linear BVPs discussed in Rem. 5.1.5, Sect. 2.8)
Galerkin discretization usingV 0,N ⊂ V 0 ➢
discrete variational problem
u N ∈ V 0,N : a(u N ,v N ) = f(v N ) ∀v N ∈ V 0,N . (3.1.4)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Theoretical guideline:
IfV 0,N = S 0 p(M), use boundary fitting with polynomials of degreep.
New twist: we are interested mainly/only in the number F(u), where
F : V 0 ↦→ R is an output functional.
Mathematical terminology: functional ˆ= mapping from a function space into R
5.6
p. 558
5.6
p. 56

Example 5.6.1 (Output functionals).
Numerical
Methods
for PDEs
✗
✖
Galerkin solutionu N ∈ V 0,N ➥ approximate output valueF(u N )
✔
✕
Numerica
Methods
for PDEs
Some output functionals for solutions of PDEs commonly encountered in applications:
• mean values, see Ex. 5.6.4 below
• total heat flux through a surface (for heat conduction model→Sect. 2.5), see Ex. 5.6.13 below
• total surface charge of a conducting body (for electrostatics→Sect. 2.1.2)
• total heat production (Ohmic losses) by stationary currents
• total force on a charged conductor (for electrostatics→Sect. 2.1.2)
• lift and drag in computational fluid dynamics (aircraft simulation)
• and many more...
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C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
How accurate isF(u N ), that is, how big is the output error|F(u)−F(u N )|?
Linearity (→ Ass. 5.6.2) and continuity Ass. 5.6.3 conspire to furnish a very simple estimate
|F(u)−F(u N )| ≤ C f ‖u−u N ‖ a .
A priori estimates for‖u−u N ‖ a ➼ estimates for|F(u)−F(u N )|
Hence, Thm. 5.3.42 immediately tells us the asymptotic convergence of linear and continuous output
functionals defined for solutions of 2nd-order scalar elliptic BVPs and Lagrangian finite element
discretization.
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C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✸
Example 5.6.4 (Approximation of mean temperature).
We consider output functionals with special properties, which are rather common in practice:
Assumption 5.6.2 (Linearity of output functional).
The output functionalF is a linear form (→ Def. 1.3.13) onV 0
To put the next assumption into context, please recall Ass. 5.1.2 and Rem. 2.3.11.
5.6
p. 561
Numerical
Methods
for PDEs
Heat conduction model (→ Sect. 2.5), scaled heat conductivity κ ≡ 1, on domain Ω =]0,1[ 2 , fixed
temperatureu = 0 on∂Ω:
−∆u = f inΩ , u = 0 on∂Ω .
Heat source function f(x,y) = 2π 2 sin(πx)sin(πy),(x,y) T ∈ Ω
➢ solution u(x,y) = sin(πx)sin(πy).
5.6
p. 56
Numerica
Methods
for PDEs
Assumption 5.6.3 (Continuity of output functional).
The output functional is continuous w.r.t. the energy norm in the sense that
∃C f > 0: |F(v)| ≤ C f ‖v‖ a ∀v ∈ V 0 .
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C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Details of finite element Galerkin discretization:
mean temperature F(u) = 1 ∫
udx .
|Ω| Ω
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C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• Sequence of triangular meshesMcreated by regular refinement.
Now consider Galerkin discretization of (3.1.1) based on Galerkin trial/test space V 0,N ⊂ V 0 , N :=
dimV 0,N < ∞ ➣ discrete variational problem
u N ∈ V 0,N : a(u N ,v N ) = l(v N ) ∀v N ∈ V 0,N . (3.1.4)
What would you dare to sell as an approximation ofF(u)?
Of course,...
5.6
p. 562
• Galerkin discretization: V 0,N := S1,0 0 (M) (linear Lagrangian finite elements→Sect. 3.2).
• Quadrature rule (3.5.38) of order 6 for assembly of right hand side vector
(more than sufficiently accurate→guidelines from Sect. 5.5.1)
Expected:
algebraic convergence inh M with rate 1 of approximate mean temperature
5.6
p. 56

Mean error [log]
10 0 Errors between the mean and exact solution
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
10 0
Quadratic FE
Linear FE
p = 4.1
10 −1
Mesh width [log]
p = 2.0
10 −2
Mean error [log]
10 0 Error between mean and exact solution
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
p = −1.1
p = −2.0
10 1 10 2 10 3 10 4 10 5
Number of dofs [log]
Error in mean value on unit square (−↔p = 1,−↔p = 2)
Quadratic FE
Linear FE
Observation: Mean value converges twice as fast as expected: algebraic convergenceO(h 2 M )! ✸
Numerical
Methods
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
The mean temperature functional (5.6.6) is obviously linear→Ass. 5.6.2
By the Cauchy-Schwarz inequality (2.2.15) it clearly satisfies Ass. 5.6.3 even with ‖·‖ a =
‖·‖ L 2 (Ω) , let alone for‖·‖ a = |·| H 1 (Ω) onH1 0 (Ω).
What isg F ∈ H0 1 (Ω) in this case? By Thm. 5.6.5 it is the solution of the variational problem
∫
gradg F ·gradvdx = F(v) = 1 ∫
vdx ∀v ∈ H
Ω
|Ω|
0 1 (Ω) .
Ω
The associated 2nd-order BVP reads
−∆g F = 1 inΩ, g
|Ω| F = 0 on∂Ω .
Now recall the elliptic lifting theory Thm. 5.4.10 for convex domains: since Ω =]0,1[ 2 is convex, we
concludeg F ∈ H 2 (Ω).
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
By interpolation estimate of Thm. 5.3.27
(I 1 ˆ= linear interpolation ontoS 0 1 (M))
✬
Theorem 5.6.5 (Duality estimate for linear functional output).
Define the dual solutiong F ∈ V 0 toF as solution of
Then
✫
Proof. For any v N ∈ V 0,N :
g F ∈ V 0 : a(v,g F ) = F(v) ∀v ∈ V 0 .
|F(u)−F(u N )| ≤ ‖u−u N ‖ a
inf
v N ∈V 0,N
‖g F −v N ‖ a . (5.6.6)
F(u)−F(u N ) = a(u−u N ,g F ) (∗) = a(u−u N ,g F −v N ) ≤ ‖u−u N ‖ a ‖g F −v N ‖ a .
(∗)←by Galerkin orthogonality (5.1.7).
✩
✪
✷
5.6
p. 565
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
inf |g F −v N |
v N ∈S1 0 H (M) 1 (Ω) ≤ |g F −I 1 g F | H 1 (Ω) ≤ Ch M|g F | H 2 (Ω) ,
whereC > 0 may depend onΩand the shape regularity measure (→ Def. 5.3.26) ofM.
Plug this into the duality estimate (5.6.6) of Thm. 5.6.5 and note that u ∈ H 2 (Ω) by virtue of
Thm. 5.4.10 andf ∈ L 2 (Ω):
|F(u)−F(u N )| ≤ Ch M· |u−u N | H 1 (Ω) ≤ Ch 2 M ,
} {{ }
≤Ch M ifu∈H 2 (Ω)
where the “generic constant”C > 0 depends only onΩ,u,ρ M .
Again, by the elliptic lifting theory Thm. 5.4.10 we infer that u ∈ H 2 (Ω) holds for this example since
f ∈ L 2 (Ω).
✸
5.6
p. 56
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
If g F can be approximated well in V 0,N , then the output error can converge → 0 (much)
faster than‖u−u N ‖ a .
Example 5.6.7 (Approximation of mean temperature cnt’d). → Ex. 5.6.4
5.6
p. 566
5.6
p. 56

5.6.2 Case study: Boundary flux computation
Numerical
Methods
for PDEs
DomainΩ = B Ro (0)\B Ri (0) := {x ∈ R 2 : R i < |x| < R o } withR o = 1 andR i = 1/2
Numerica
Methods
for PDEs
Model problem (process engineering):
Long pipe carrying turbulent flow of coolant (water)
Ω ⊂ R 2 : cross-section of pipe
κ : (scaled) heat conductivity of pipe material
(assumed homogeneous, κ =
const)
Assumption: Constant temperatures u o , ,u i at
Task:
outer/inner wallΓ o ,Γ i of pipe
Compute heat flow pipe→water
Ω (pipe)
Water
Mathematical model: elliptic boundary value for stationary heat conduction (→ Sect. 2.5)
Γ i
Γ o
Fig. 172
−div(κgradu) = 0 inΩ , u = u x onΓ x , x ∈ {i,o} . (5.6.8)
∫
Heat flux throughΓ i : J(u) := κ gradu·ndS . (5.6.9)
Γ i
Relate to abstract framework: (5.6.8) ∼ = (3.1.1), V 0
∼ = H 1
0 (Ω) (→ Sect. 2.8)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
5.6
p. 569
Dirichlet boundary datau i = 60 ◦ C onΓ i ,u o = 10 ◦ C onΓ o , heat sourcef ≡ 0, heat conductivity
κ ≡ 1.
➢ Exact solution: u(r,ϕ) = C 1 ln(r)+C 2 ,
➢ Exact heat flux: J = 2πκC 1 ,
Details of linear Lagrangian finite element Galerkin discretization:
withC 1 := (u o −u i )/(lnR i −lnR o ),
C 2 := (lnR o u i −lnR i u o )/(lnR i −lnR o ).
• Sequences of unstructured triangular meshesMobtained by regular refinement of coarse mesh
(from grid generator).
• Galerkin FE discretization based onV 0,N := S 0 1,0 (M).
• Approximate evaluation ofa(u N ,v N ),f(v N ) by six point quadrature rule (3.5.38) (“overkill
quadrature”, see Sect. 5.5.1)
• Approximate evaluation ofJ(u N ) by 4 point Gauss-Legendre quadrature rule on boundary edges
ofM.
• Linear boundary approximation (circle replaced by polygon).
• Recorded: errors |J −J(u N )| on sequence of meshes.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
5.6
p. 57
(Actually,u ∈ H 1 (Ω), but by means of offset functions we can switch to the variational spaceH 1 0 (Ω),
see Sects. 2.1.3, 3.5.5.)
Numerical
Methods
for PDEs
1
0.8
2D triangular mesh
1
0.8
2D triangular mesh
Numerica
Methods
for PDEs
0.6
0.6
Numerical method:
finite element computation of heat conduction in pipe
(e.g. linear Lagrangian finite element Galerkin discretization, Sect. 3.2)
0.4
0.2
0.4
0.2
Expectation: Algebraic convergence |J(u)−J(u N )| = O(h 2 M ) for regularh-refinement
This expectation is based on the analogy to Ex. 5.6.4 (Approximation of mean temperature), where
duality estimates yieldedO(h 2 M ) convergence of the mean temperature error in the case of Galerkin
discretization by means of linear Lagrangian finite elements on a sequence of meshes obtained by
regular refinement. Now, it seems, we can follow the same reasoning.
Example 5.6.10 (Computation of heat flux).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
# Vertices : 144, # Elements : 232, # Edges : 376Fig. 173
# Vertices : 520, # Elements : 928, # Edges : 1448 Fig. 174
Unstructured triangular meshes forΩ = B 1 (0)\B 1/2 (0) (two coarsest specimens).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Setting: model problem “heat flux pipeto water”, see (5.6.8) and Fig. 172.
Linear output functional from (5.6.9)
5.6
p. 570
5.6
p. 57

Boundary flux [log scale]
Convergence: standard boundary flux, linear FE
10 2
10 1
p = 1.02
10 0
10 −1
Mesh width [log scale]
10
Fig. 175
−2
Observation:
Algebraic convergence of output error for J from
(5.6.9) only with rate 1 (in mesh widthh M )!
(This is not the fault of the piecewise linear boundary
approximation, which is sufficient when using
piecewise linear Lagrangian finite elements, see
Sect. 5.5.2.)
✸
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
☞ On the other hand, straightforward computation of improper integral using (2.3.20):
|u| 2 H 1 (Ω) = ] ∫
∫1
‖gradu(x)‖ 2 dx = 2π
∫1
|g ′ (r)| 2 rdr = 2πα 2
(1−r) 2α−2 rdr
Ω
0
0
∫1
[ ]
= 2πα 2 s 2α−2 s 2α−1 s=1
(1−s)ds = 2πα
2α−1 − s2α 1
= 2π
2α 2α−1 < ∞ .
0
s=0
Def. 2.2.12
=⇒ u ∈ H 1 (Ω) (u ∈ C 0 (Ω) andu ∈ C ∞ (Ω\{0}) !) .
✸
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Ex. 5.6.11 ➣ Thm. 5.6.5 cannot be applied
Why was our expectation mistaken ?
Suspicion: the output functionalJ fails to meet requirements of duality estimates of Thm. 5.6.5:
5.6
p. 573
(Potentially) poor convergence of flux obtained from straightforward evaluation of
J(u N ) for FE solutionu N ∈ S 0 1,0 (M)!
Apparently there is no remedy, because the boundary flux functional (5.6.9) seems to be enforced on
5.6
p. 57
boundary flux functionalJ from (5.6.9) is not continuous onH 1 (Ω)!
Numerical
Methods
for PDEs
us by the problem: we are not allowed to tinker with it, are we?
Numerica
Methods
for PDEs
Example 5.6.11 (Non-continuity of boundary flux functional).
Idea:
findu ∈ H 1 (Ω), for which “J(u) = ∞”,
cf. investigation of non-continuity of point evaluation functional onH 1 (Ω)→Rem. 2.3.16.
OnΩ = {x ∈ R 2 : ‖x‖ < 1} (unit disk) consider
u(x) = (1−‖x‖) α =: g(‖x‖) ,
and the boundary flux functional (5.6.9) on∂Ω.
1
2 < α < 1 ,
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Trick:
use fixed cut-off functionψ ∈ C 0 (Ω)∩H 1 (Ω),ψ ≡ 1 onΓ i ,ψ |Γo = 0
∫ ∫
∫
κ gradu·ndS = (κ gradu·n) ψdS = div(κgradu) ψ +κgradu·gradψdx
Γ i Γ i Ω} {{ }
∫ =0
use J ∗ (u) := κgradu·gradψdx . (5.6.12)
Ω
Obviously (∗): J ∗ : H 1 (Ω) ↦→ R continuous & J ∗ (u) = J(u) for solution of (5.6.8)
Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
(∗): By the Cauchy-Schwarz inequality (2.2.15), sinceκ = const,
☞ On the one hand, using the expression (2.3.19) for the gradient in polar coordinates,
∫
∂u
J 0 (v) =
∂Ω ∂r (x)dS(x) = 2π α(1−r)α−1 |r=1 “ = ∞′′ .
5.6
p. 574
|J ∗ (u)| ≤ κ‖gradu‖ L 2 (Ω) ‖gradψ‖ L 2 (Ω) ≤ C|u| H 1 (Ω) ,
R.
withC := κ‖gradψ‖ L 2 (Ω), which is a constant independent ofu, asψ is a fixed function.
5.6
p. 57

Objection: You cannot just tamper with the output functional of a problem just because you do not
like it!
Retort: Of course, one can replace the output functionJ with another oneJ ∗ as long as
J(u) = J ∗ (u) for the exact solutionuof the BVP,
Numerical
Methods
for PDEs
Remark 5.6.14 (Finding continuous replacement functionals).
Now you will ask: How can we find good (continuous) replacement functionals, if we are confronted
with an unbounded output functional on the energy space?
Numerica
Methods
for PDEs
because the objective is not to “evaluateJ”, but to obtain an approximation for J(u)!
Example 5.6.13 (Computation of heat flux cnt’d). → Ex. 5.6.13
Further details on flux evaluation:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Unfortunately, there is no recipe, and sometimes it does not seem to be possible to find a suitableJ ∗
at all, for instance in the case of point evaluation, cf. Rem. 2.3.16.
Good news:
another opportunity to show off how smart you are!
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• Galerkin FE discretization based onV 0,N := S 0 1,0 (M) orV 0,N := S 0 2,0 (M).
• Approximate evaluation ofJ ∗ (u N ) by six point quadrature rule (3.5.38) (“overkill quadrature”, see
Sect. 5.5.1)
5.6.3 L 2 -estimates
• Cut-off function with linear decay in radial direction
• Recorded: errors |J −J(u N )| and|J −J ∗ (u N )|.
5.6
p. 577
5.6
p. 57
10 1
Convergence: stable boundary flux, linear FE
p = 1.02
Numerical
Methods
for PDEs
So far we have only studied the energy norm (↔H 1 (Ω)-norm, see Rem. 5.3.28) of the finite element
discretization error for 2nd-order elliptic BVP.
Numerica
Methods
for PDEs
Boundary flux [log scale]
10 2 Mesh width [log scale]
10 0
10 −1
Unstable flux J
Stable flux J ∗
✁ Convergence of |J(u)−J(u N )| and |J(u)−
J ∗ (u N )| for linear Lagrangian finite element
discretization.
The reason was the handy tool of Cea’s lemma Thm. 5.1.10.
What about error estimates in other “relevant norms”, e.g„
10 −2
10 −3
10 −1
Additional observations:
p = 2.02
Fig. 176
10 −2
• Algebraic convergence|J(u)−J ∗ (u N )| = O(h 2 M ) (rate 2 !) for alternative output functionalJ∗
from (5.6.12).
• Dramatically reduced output error!
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
• in the mean square norm orL 2 (Ω)-norm, see Def. 2.2.5,
• in the supremum norm orL ∞ (Ω)-norm, see Def. 1.6.4?
In this section we tackle‖u−u N ‖ L 2 (Ω). We largely reuse the abstract framework of Sect. 5.6.1: linear
variational problem (3.1.1) with exact solutionu ∈ V 0 , Galerkin finite element solutionu N ∈ V 0,N ,
see p. 559, and the special framework of linear 2nd-order elliptic BVPs, see Rem. 5.1.5: concretely,
∫
a(u,v) := κ(x)gradu·gradvdx , u,v ∈ H0 1 (Ω) .
Ω
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✸
5.6
p. 578
Example 5.6.15 (L 2 -convergence of FE solutions). → Ex. 5.2.4
5.6
p. 58

Setting:
Ω =]0,1[ 2 ,D ≡ 1,f(x,y) = 2π 2 sin(πx)sin(πy),(x,y) ⊤ ∈ Ω
➢ u(x,y) = sin(πx)sin(πy).
• Sequence of triangular meshesM, created by regular refinement.
• FE Galerkin discretization based onS 0 1,0 (M) orS0 2 (M).
• Quadrature rule (3.5.38) for assembly of local load vectors (→ Sect. 3.5.4).
• ApproximateL 2 (Ω)-norm by means of quadrature rule (3.5.38).
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 5.6.16 (L 2 interpolation error).
Recall the interpolation error estimate of Thm. 5.3.27
‖u−I 1 u‖ L 2 (Ω) = O(h2 M ) vs. |u−I 1u| H 1 (Ω) = O(h M) ,
on a family of meshes with uniformly bounded shape regularity measure.
☞ Higher rate of algebraic convergence of the interpolation error when measured in the weaker
L 2 (Ω)-norm compared to the strongerH 1 (Ω)-norm.
Therefore a similar observation in the case of the finite element approximation error is not so surprising.
△
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Now we supply a rigorous underpinning and explanation of the behavior of ‖u−u N ‖ L 2 (Ω) that we
have observed and expect.
5.6
p. 581
5.6
p. 58
10 −1 Discretization errors with respect to L 2 norm
10 −2
p = 2.0
10 −1 Discretization errors with respect to L 2 norm
10 −2
Quadratic FE
Linear FE
Numerical
Methods
for PDEs
Idea: Consider special continuous linear output functional
∫
J(v) := v ·(u−u N )dx !
Ω
Numerica
Methods
for PDEs
Discretization error [log]
10 −3
10 −4
10 −5
Discretization error [log]
10 −3
10 −4
10 −5
p = −1.5
p = −1.0
This functional is highly relevant forL 2 -estimates, because
10 −6
10 −7
10 0
10 −6
Quadratic FE
p = 3.1
Linear FE
10 −7
10 −1
10 −2
10 1 10 2 10 3 10 4 10 5
Mesh width [log]
Dofs [log]
L 2 (Ω)-norm of discretization error on unit square (−↔p = 1,−↔p = 2)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
F(u)−F(u N ) = ‖u−u N ‖ 2 L 2 (Ω)
➣ estimates for the output error will provide bounds for‖u−u N ‖ L 2 (Ω) !
Note:
Bothuandu N are fixed functions∈ H 1 (Ω)!
!
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Observations: • Linear Lagrangian FE (p = 1) ➽ ‖u−u N ‖ 0 = O(N −1 )
• Quadratic Lagrangian FE (p = 2) ➽ ‖u−u N ‖ 0 = O(N −1.5 )
➣
Linearity ofJ (→ Ass. 5.6.2) is obvious.
➣ ContinuityJ : H0 1 (Ω) ↦→ R (→ Ass. 5.6.3) is clear, use Cauchy-Schwarz inequality (2.2.15).
✸
5.6
p. 582
Duality estimate of Thm. 5.6.5 can be applied:
5.6
p. 58

Thm. 5.6.5
F(u)−F(u N ) = ‖u−u N ‖ 2 L 2 (Ω) ≤ C|u−u N| H 1 (Ω) inf |g F −v N |
v N ∈V H 1 (Ω) ,
0,N
(5.6.17)
Numerical
Methods
for PDEs
Ass. 5.6.19 ⇒ ‖u−u N ‖ L 2 (Ω) ≤ C h M p ‖u−u N ‖ H 1 (Ω) .
forh-refinement:
gain of one factorO(h M ) (vs. H 1 (Ω)-estimates)
Numerica
Methods
for PDEs
whereC > 0 may depend only onκ, and the dual solutiong F ∈ H0 1 (Ω) satisfies
∫ ∫
∫
Is it important to assume 2-regtularity, Ass. 5.6.19 or merely a technical requirement of the theoretical
a(g F ,v) = F(v) ∀v ∈ V 0 ⇔ κ(x)gradg F ·gradvdx = v(u−u N )dx ∀v ∈ H0 1 (Ω)
R. Hiptmair
C. Schwab,
approach?
H.
Harbrecht
Ω Ω
Ω
V.
Gradinaru Example 5.6.22 (L 2 -estimates on non-convex domain). cf. Ex. 5.2.6
A. Chernov
⇓
DRAFT
−div(κ(x)gradg F ) = u−u N inΩ , g F = 0 on∂Ω . (5.6.18)
Setting: Ω =]−1,1[ 2 \(]0,1[×]−1,0[),D ≡ 1,u(r,ϕ) = r 2/3 sin(2/3ϕ) (polar coordinates)
Assumption 5.6.19 (2-regularity of homogeneous Dirichlet problem).
➢ f = 0, Dirichlet datag = u |∂Ω .
We assume that the homogeneous Dirichlet problem with coefficientκis 2-regular onΩ: There
isC > 0, which depends onΩonly such that
Finite element Galerkin discretization and evaluations as in Ex. 5.6.15.
u ∈ H0 1(Ω)
div(κ(x)gradu) ∈ L 2 ⇒ u ∈ H 2 (Ω) and |u|
(Ω)
H 2 (Ω) ≤ C‖div(κ(x)gradu)‖ L 2 (Ω) . 5.6
p. 585
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
5.6
p. 58
By the elliptic lifting theorem for convex domains Thm. 5.4.10 we know
Numerical
Methods
for PDEs
10 −1 Discretization errors with respect to L 2 norm
Quadratic FE
Linear FE
10 −1 Discretization errors with respect to L 2 norm
p = −0.7
Quadratic FE
Linear FE
Numerica
Methods
for PDEs
Ω convex
Ass. 5.6.19 in conjunction with (5.6.18) yields
whereC > 0 depends only onΩ.
=⇒ Ass. 5.6.19 is satisfied.
|g F | H 2 (Ω) ≤ C‖u−u N‖ L 2 (Ω) , (5.6.20)
Now we can appeal to the general best approximation theorem for Lagrangian finite element spaces
Thm. 5.3.42:
inf |g F −v N |
v N ∈Sp 0 H (M) 1 (Ω) ≤ C h M
p |g (5.6.20)
F| H 2 (Ω) ≤ C h M
p ‖u−u N‖ L 2 (Ω) , (5.6.21)
where the “generic constants” C > 0 depend only on Ω and the shape regularity measure ρ M (→
Def. 5.3.26).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Discretization error [log]
10 −2
10 −3
10 −4
10 −5
10 0
p = 1.2
p = 1.3
10 −1
Mesh width [log]
10 −2
Discretization error [log]
10 −2
10 −3
10 −4
10 −5
10 1 10 2 10 3 10 4 10 5 10 6
Dofs [log]
L 2 (Ω)-norm of discretization error on “L-shaped” domain (−↔p = 1,−↔p = 2)
Observation: For both (p = 1,2) ➽ algebraic convergence‖u−u N ‖ 0 = O(N −2/3 )
p = −0.7
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Combine (5.6.17) and (5.6.21) and cancel one power of‖u−u N ‖ L 2 (Ω) :
Comparison with Ex. 5.2.6: for both linear and quadratic Lagrangian FEM
WithC > 0 depending only onΩ,κ, and the shape regularity measureρ M we conclude
5.6
p. 586
‖u−u N ‖ L 2 (Ω) = O(N−2/3 ) ←→ ‖u−u N ‖ H 1 (Ω) = O(N−1/3 ) ,
5.6
p. 58

that is, we again observe a doubling of the rate of convergence for the weaker norm.
Numerical
Methods
for PDEs
Numerica
Methods
for PDEs
No gain through the use of quadratic FEM, because of limited smoothness of bothuand dual solution
g F . For both the solution and the dual solution the gradient will have a singularity at0.
✸
Intuition:
In the absence of heat sources maximal and minimal temperature attained on surface.
In the presence of a heat source (f ≥ 0) the temperature minimum will be attained on
surface∂Ω.
Iff ≤ 0 (heat sink), then the maximal temperature will be attained on the surface.
5.7 Discrete maximum principle
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
In fact this is a theorem, cf. Sect. 2.7.
✬
Theorem 5.7.2 (Maximum principle for 2nd-order elliptic BVP).
Foru ∈ C 0 (Ω)∩H 1 (Ω) holds the maximum principle
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
✩
DRAFT
KEY SECTION
Discrete maximum principle as gauge for quality of discretization used in Sect. 7.2.2.
✫
−div(κ(x)gradu) ≥ 0 =⇒ min u(x) = min
x∈∂Ω
−div(κ(x)gradu) ≤ 0 =⇒ max
x∈∂Ω
u(x) , x∈Ω
u(x) = max u(x) . x∈Ω
✪
So far we have investigated the accuracy of finite element Galerkin solutions: we studied relevant
norms‖u−u N ‖ of the discretization error.
5.7
p. 589
Numerical
Methods
for PDEs
5.7
p. 59
Numerica
Methods
for PDEs
Now new perspective:
structure preservation by FEM
To what extent does the finite element solutionu N inherit key structural properties of the
solutionuof a 2nd-order scalar elliptic BVP?
This issue will be discussed for a special structural property of the solution of the linear 2nd-order
elliptic BVP (inhomogeneous Dirichlet problem) in variational form (→ Sect. 2.8)
∫
∫
u ∈ ˜g +H0 1 (Ω): a(u,v) := κgradu·gradvdx = fvdx ∀v ∈ H0 1 (Ω) . (5.7.1)
Ω
Ω
where ˜g ˆ= offset function, extension of Dirichlet datag ∈ C 0 (∂Ω), see Sect. 2.3.1, (2.3.5),
κ ˆ= bounded and uniformly positive definite diffusion coefficient, see (2.5.4).
(5.7.1) ←→ BVP (PDE-form)
−div(κ(x)gradu) = f inΩ , u = g on∂Ω .
Recall (→ Sect. 2.5): (5.7.1) models stationary temperature distribution in body, when temperature
on its surface is prescribed byg.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
5.7
p. 590
∆u = 0
Maximum/minimum on∂Ω
Proof. (for the case−div(κ(x)gradu) = 0)
Sect. 2.1.3➣
u solves quadratic minimization problem
∫
u = argmin κ(x)‖gradv(x)‖ 2 dx .
v∈H 1 (Ω)
v=g on∂Ω Ω
Fig. 177
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
5.7
p. 59

Ifuhad a global maximum atx ∗ in the interior ofΩ, that is
Now “chop off” the maximum and define
∃δ > 0: u(x ∗ ) ≥ max
x∈∂Ω u(x)+δ .
w(x) := min{u(x),u(x ∗ )−δ} , x ∈ Ω .
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Methods
for PDEs
also belong toH 1 (Ω). However
∫
∫
κ(x)‖gradw(x)‖ 2 dx < κ(x)‖gradu(x)‖ 2 dx ,
Ω
Ω
which contradicts the definition ofuas the global minmizer of the quadratic energy functional.
✷
Numerica
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Now we consider a finite element Galerkin discretization of (5.7.1) by means of linear Lagrangian
finite elements (→ Sect. 3.4), using offset functions supported near∂Ω as explained in Sect. 3.5.5.
➣
finite element Galerkin solutionu N ∈ S 0 1 (M) ⊂ C0 (Ω)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Issue:
doesu N satisfy a maximum principle, that is, can we conclude
f ≥ 0 =⇒ min u N(x) = min u N(x) ,
x∈∂Ω x∈Ω
f ≤ 0 =⇒ max u N(x) = max u N(x) ?
x∈∂Ω x∈Ω
(5.7.3)
5.7
p. 593
Example 5.7.4 (Maximum principle for finite difference discretization).
5.7
p. 59
Numerical
Methods
for PDEs
Recall from Sect. 4.1: finite difference discretization of
Numerica
Methods
for PDEs
−∆u = 0 inΩ :=]0,1[ 2 , u = g on∂Ω,
on anM ×M tensor product mesh
M := {[(i−1)h,ih]×[(j −1)h,jh], 1 ≤ i,j ≤ M} , M ∈ N .
Unknowns in the finite difference method: µ ij ≈ u((ih,jh) T ),1 ≤ i,j ≤ M −1.
Fig. 178 Fig. 179
∫
∫
κ(x)‖gradw(x)‖ 2 dx ≥ κ(x)‖gradv(x)‖ 2 dx .
Ω
Ω
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Unknowns are solutions of a linear system of equations, see (4.1.2)
1
h 2 (
4µi,j −µ i−1,j −µ i+1,j −µ i,j−1 −µ i,j+1
) = 0 , 1 ≤ i,j ≤ M −1 , (5.7.5)
where values corresponding to points on the boundary are gleaned fromg:
µ 0,j := g(0,hj) , µ M,j := g(1,hj) , µ i,0 := g(hi,0) , µ i,M := g(hi,1) , 1 ≤ i,j < M .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Obviously, w ∈ C 0 (Ω), and as a continuous function which is piecewise in H 1 the function w will
5.7
p. 594
5.7
p. 59

−1
Numerical
Methods
for PDEs
✸
Numerica
Methods
for PDEs
−1
4 −1
Now we try to generalize the considerations of the previous example to the discretization by means of
linear Lagrangian finite elements on a triangular mesh (of a polygonal domainΩ ⊂ R 2 ) see Sect. 3.2.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
à ∈ R M,M ˆ= S1 0 (M)-Galerkin matrix forafrom (5.7.1)
(M := ♯V(M))
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
Fig. 180
−1
Fig. 181
DRAFT
Row of this matrix connects all valuesµ j = u N (x j ) of Galerkin solutionu N ∈ S1 0 (M) according to
(Ã) iiµ i + ∑ j≠i(Ã) ijµ j = (⃗ϕ) i , x i interior node ,
DRAFT
The finite difference solution(µ i,j ) 1≤i,j 0 (positive diagonal) , (5.7.9)
• (Ã) ij ≤ 0 for j ≠ i (non-positive off-diagonal entries) ,(5.7.10)
∑
• (Ã) ij = 0 , ifx i is interior node . (5.7.11)
j
(Recall [18, Def. 2.7.7]:
averaging argument
matrixÃsatisfying (5.7.9)–(5.7.11) is diagonally dominant.)
u N (x i ) = max
y∈V(M) u N(y) can only hold for an interior nodex i ,
ifµ N = const.
Sinceu N ∈ S1 0 (M) attains its extremal values at nodes of the mesh, the maximum principles
holds for it in the casef = 0 provided that (5.7.9)–(5.7.11) are satisfied.
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
The finite difference solution can attain its maximum in the interior only in the case of constant
boundary datag!
Maximum principle satisfied forf = 0!
5.7
p. 598
More general casef ≤ 0:
∫
(⃗ϕ) i = f(x)b i N (x)dx ≤ 0 , sincebi N ≥ 0 .
Ω
Then the averaging argument again rules out the existence of an interior maximum for an nonconstant
solution. The casef ≥ 0 follows similarly.
5.7
p. 60

When will (5.7.9)–(5.7.11) hold forS1 0 (M)-Galerkin matrix?
Numerical
Methods
for PDEs
Remark 5.7.14 (Maximum principle for higher order Lagrangian FEM).
Numerica
Methods
for PDEs
First consider κ ≡ 1, ↔ −∆u = f
(The linear finite element discretization of this BVP was scrutinized in Sect. 3.2)
From formula (3.2.10) for element matrix & assembly, see
Fig. 74:
(Ã) ij = −cotα−cotβ = − sin(α+β)
sinα sinβ .
⇓
(Ã) ij ≤ 0 ⇔ α+β < π .
α
x j
β
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Even when using p-degree Lagrangian finite elements with nodal basis functions associated with
interpolation nodes, see Sect. 3.4.1, the discrete maximum principle will fail to hold on any mesh for
p > 1.
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
x i
Fig. 182
Moreover
∑
x∈V(M)b x N ≡ 1 ⇒ ∑ (Ã) ij = 0 (↔(5.7.11)) .
j
The condition (5.7.9)↔(Ã) ii > 0 is straightforward.
✬
✩
5.7
p. 601
5.7
p. 60
Theorem 5.7.12 (Maximum principle for linear FE solution of Poisson equation).
The linear finite element solution of
−∆u = 0 inΩ ⊂ R 2 , u = g on∂Ω,
Numerical
Methods
for PDEs
6 2nd-Order Linear Evolution Problems
Numerica
Methods
for PDEs
on a triangular mesh M satisfies the maximum principle (5.7.3), if M is a
Delaunay triangulation.
✫
✪
CORE CHAPTER
Remark 5.7.13 (Maximum principle for linear FE for 2nd-order elliptic BVPs).
For S1 0 (M)-Galerkin discretization of (5.7.1) on triangular mesh, the conditions (5.7.9)–(5.7.11) are
fulfilled,
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Takes for granted
• Theory and variational formulation of 2nd-order elliptic BVP, Ch. 2
• Basic concepts and algorithms for finite elements, Sects. 3.1, 3.3, 3.4
• Single step methods for ODEs, [18, Ch. 12].
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
if all angles of triangles ofM≤ π 2 .
The fundamental idea of the methods of lines approach will be used in all other sections dealing with
time-dependent problems.
△
5.7
p. 602
6.0
p. 60

Now we study scalar linear partial differential equations for which one coordinate direction is special
and identified with time and denoted by the independent variable t. The other coordinates are
regarded as spatial coordinates and designated byx = (x 1 ,...,x d ) T .
➣ solution will be a “function of time and space”: u = u(x,t)
Numerical
Methods
for PDEs
Note:
No boundary conditions on Ω×{T} (“final conditions”) are prescribed: time is supposed to
have a “direction” that governs the flow of information in the evolution problem.
evolution problems (on bounded spatial domains) are also known as
initial-boundary value problems (IBVP).
Numerica
Methods
for PDEs
The domain for such PDEs will have tensor product structure (tensor product of spatial domain and a
bounded time interval):
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 6.0.1 (Initial time).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Why do we always pick initial timet = 0?
The modelled physical systems will usually be time-invariant, so that we are free to shift time. Remember
the analoguous situation with autonomous ODE, see [18, Sect. 12.1].
△
Computational domain:
˜Ω := Ω×]0,T[⊂ R d+1 .
T
Ω
6.0
p. 605
Numerical
Methods
for PDEs
6.1 Parabolic initial-boundary value problems
6.1.1 Heat equation
6.1
p. 60
Numerica
Methods
for PDEs
space-time cylinder
Ω ⊂ R d ˆ= spatial domain (satisfying assumptions
of Sect. 2.1.1)
T > 0 ˆ= final time
OnΩ×{0} → initial conditions,
on∂Ω×]0,T[ → (spatial) boundary conditions.
Time
0
Space
Ω
˜Ω
Boundary conditions
Initial conditions
PDE foru(x,t) + initial conditions + boundary conditions
} {{ }
= evolution problem
Fig. 183
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
6.0
p. 606
Sect. 2.5 treated stationary heat conduction: no change of temperature with time (temporal equilibrium)
Now we consider the evolution of a temperature distributionu = u(x,t).
Ω ⊂ R d
: space occupied by solid body (bounded spatial computational domain),
κ = κ(x) : (spatially varying) heat conductivity ([κ] = Km W ),
T > 0
: final time for “observation period”[0,T],
u 0 : Ω ↦→ R : initial temperature distribution inΩ,
g : ∂Ω×[0,T] ↦→ R : surface temperature, varying in space and time: g = g(x,t),
f : Ω×[0,T] ↦→ R : time-dependent heat source/sink ([f] = W m3): f = f(x,t).
Goal:
derive PDE governing transient heat conduction.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
6.1
p. 60

✬
Conservation of energy:
∫ ∫ ∫
d
ρudx+ j·ndS = f dx for all “control volumes”V (6.1.1)
dt V ∂V V
✩
Numerical
Methods
for PDEs
Terminology: (6.1.2) & (6.1.4) & (6.1.5) is a specimen of a
2nd-order parabolic initial-boundary value problem
Numerica
Methods
for PDEs
✫
energy stored inV power flux through∂V heat generation inV
✪
Remark 6.1.6 (Compatible boundary and initial data).
ρ = ρ(x): (spatially varying) heat capacity ([ρ] = JK −1 ), uniformly positive, cf. (2.5.4).
Natural regularity requirements for Dirichlet datag:
As in Sect. 2.5, now apply Gauss’ Theorem Thm. 2.4.5 to the power flux integral in (6.1.1). This
converts the surface integral to a volume integral overdivj and we get
∫ ∫ ∫
d
ρudx+ divjdx = f dx for all “control volumes”V
dt V V V
Now appeal to another version of the fundamental lemma of the calculus of variations, see
Lemma 2.4.10, this time involving piecewise constant test functions.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
g continuous in time and space
Natural compatibility requirement at initial time andu 0 ∈ C 0 (Ω)
g(x,0) = u 0 (x) ∀x ∈ ∂Ω .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Local form of energy balance law
(Heat equation)
∂
∂t (ρu)(x,t)+(div xj)(x,t) = f(x,t) in ˜Ω . (6.1.2)
6.1
p. 609
△
6.1
p. 61
The heat flux is linked to temperature variations by Fourier’s law:
j(x) = −κ(x)gradu(x) , x ∈ Ω . (2.5.3)
From here we let all differential operators like grad and div act on the spatial independent variable
x. As earlier, the independent variablesxandtwill be omitted frequently. Watch out!
Now, plug (2.5.3) into (6.1.2).
∂
∂t (ρu)−div(κ(x)gradu) = f in ˜Ω := Ω×]0,T[ . (6.1.3)
+ Dirichlet boundary conditions (fixed surface temperatur) on∂Ω×]0,T[:
Numerical
Methods
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 6.1.7 (Boundary conditions for 2nd-order parabolic IBVPs).
Physical intuition for transient heat conduction:
On∂Ω]0,T[ we can impose any of the boundary conditions discussed in Sect. 2.6:
• Dirichlet boundary conditionsu(x,t) = g(x,t), see (6.1.4) (fixed surface temperature),
• Neumann boundary conditionsj(x,t)·n = −h(x,t) (fixed heat flux through surface),
• radiation boundary conditionsj(x,t)·n = Ψ(u(x,t)),
and any combination of these as discussed in Ex. 2.6.7, yet, only one of them at any part of
∂Ω×]0,T[, see Rem. 2.6.6.
△
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C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
+ initial conditions fort = 0:
u(x,t) = g(x,t) for (x,t) ∈ ∂Ω×]0,T[ . (6.1.4)
6.1.2 Spatial variational formulation
u(x,0) = u 0 (x) for all x ∈ Ω . (6.1.5)
6.1
p. 610
6.1
p. 61

Now we study the linear 2nd-order parabolic initial-boundary value problem with pure Dirichlet boundary
conditions, introduced in the preceding section:
d
dt (ρu)−div(κ(x)gradu) = f in ˜Ω := Ω×]0,T[ , (6.1.3)
u(x,t) = g(x,t) for (x,t) ∈ ∂Ω×]0,T[ , (6.1.4)
u(x,0) = u 0 (x) for all x ∈ Ω . (6.1.5)
Assume: Homogeneous Dirichlet boundary conditionsg = 0
Numerical
Methods
for PDEs
✬
Spatial variational form of (6.1.3)–(6.1.4): seekt ∈]0,T[↦→ u(t) ∈ H0 1(Ω)
∫ ∫
∫
ρ(x)˙u(t)vdx+ κ(x)gradu(t)·gradvdx = f(x,t)v(x)dx ∀v ∈ H0 1 (Ω) ,
✫
Ω
Ω
Ω
(6.1.8)
u(0) = u 0 ∈ H0 1 (Ω) . (6.1.9)
✩
✪
Numerica
Methods
for PDEs
The general case can be reduced to this by using the offset function trick, see Sect. 3.5.5, and solve
the parabolic initial-boundary value problem for w(x,t) := u(x,t) − ˜g(x,t), where ˜g(·,t) is an
extension of the Dirichlet datag to ˜Ω. Thenw will satisfy homogeneous Dirichlet boundary conditions
and solve an evolution equation with a modified source function ˜f(x,t).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Be aware: u(t) ˆ= function space (H0 1 (Ω)-)valued function on]0,T[.
Also note that grad acts on the spatial independent variables that are suppressed in the notation
u(t).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Now we pursue the formal derivation of the spatial variational formulation of (6.1.3)–(6.1.4).
The steps completely mirror those discussed in Sect. 2.8
✎ Notation: ˙u(t) = ∂u
∂t (t) ˆ= (partial) derivative w.r.t. time.
Shorthand notation (with obvious correspondences):
STEP 1:
test PDE with functionsv ∈ H 1 0 (Ω)
(do not test, where the solution is known, that is, on the boundary∂Ω )
6.1
p. 613
t ∈]0,T[↦→ u(t) ∈ V 0 :
{
m(˙u(t),v)+a(u(t),v) = l(t)(v) ∀v ∈ V0 ,
u(0) = u 0 ∈ V 0 .
(6.1.10)
6.1
p. 61
Note: test function does not depend on time:
v = v(x)!
Numerical
Methods
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Again, herel(t) ˆ= linear form valued function on]0,T[.
Numerica
Methods
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STEP 2:
STEP 3:
STEP 4:
integrate over domainΩ
perform integration by parts in space
(by using Green’s first formula, Thm. 2.4.7)
[optional] incorporate boundary conditions into boundary terms
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Concretely:
∫
m(u,v) :=
ρ(x)˙u(t)vdx , u,v ∈ H 1 0 (Ω) ,
∫Ω
a(u,v) := κ(x)gradu(t)·gradvdx , u,v ∈ H0 1 (Ω) ,
Ω∫
l(t)(v) := f(x,t)v(x)dx , v ∈ H0 1 (Ω) .
Ω
Note that bothmandaare symmetric, positive definite bilinear forms (→ Def. 2.1.25).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
For the concrete PDE (6.1.3) and boundary conditions (6.1.4) refer to Ex. 2.8.1, for more general
boundary conditions to Ex. 2.8.5.
Equivalent formulation, since the bilinear formm does not depend on time:
6.1
p. 614
6.1
p. 61

t ∈]0,T[↦→ u(t) ∈ V 0 :
⎧
⎨ d
dt m(u(t),v)+a(u(t),v) = l(t)(v) ∀v ∈ V 0 ,
⎩
u(0) = u 0 ∈ V 0 .
(6.1.11)
Numerical
Methods
for PDEs
✬
Lemma 6.1.14 (Decay of solutions of parabolic evolutions).
Forρ ≡ 1,κ ≡ 1, andf ≡ 0 the solutionu(t) of (6.1.8) satisfies
‖u(t)‖ L 2 (Ω) ≤ e−γt ‖u 0 ‖ L 2 (Ω) , |u(t)| H 1 (Ω) ≤ e−γt |u 0 | H 1 (Ω) ∀t ∈]0,T[ .
✩
Numerica
Methods
for PDEs
✫
✪
Proof. Multiply the solution of the parabolic IBVP with an exponential weight function:
Now we are concerned with the stability of parabolic evolution problems: We investigate whether
‖u‖ H 1 (Ω) stays bounded for all times in the casef ≡ 0.
For the sake of simplicity: considerρ ≡ 1 andκ ≡ 1
(General case is not more difficult, because both ρ and κ are bounded and uniformly positive, see
(2.5.4). )
By the first Poincaré-Friedrichs inequality Thm. 2.2.16
∃γ > 0: |v| 2 H 1 (Ω) ≥ γ‖v‖2 L 2 (Ω)
In fact, Thm. 2.2.16 revealsγ = diam(Ω) −2 .
∀v ∈ H0 1 (Ω) . (6.1.12)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
6.1
p. 617
w(t) := exp(γt)u(t) ∈ H 1 0
solves the parabolic IBVP
(Ω) ⇒ ẇ := dw
dt (t) = γw(t)+exp(γt)du dt (t) , (6.1.15)
m(ẇ,v)+ã(w,v) = 0 ∀v ∈ V ,
w(0) = u 0 ,
with ã(w,v) = a(w,v) − γm(w,v), γ from (6.1.12). To see this, use that u(t) solves (6.1.11) with
f ≡ 0 (elementary calculation).
Note: (6.1.12) ⇒ ã(v,v) ≥ 0 ∀v ∈ V
Exponential decay of‖·‖ L 2 (Ω)-norm of solution:
d
dt2 1‖w‖2 L 2 (Ω) = dt d 1 2m(w,w) = m(ẇ,w) = −ã(w,w) ≤ 0
(6.1.17)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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(6.1.16)
DRAFT
6.1
p. 61
Remark 6.1.13 (Differentiating bilinear forms with time-dependent arguments).
Numerical
Methods
for PDEs
This confirms thatt ↦→ ‖w‖ L 2 (Ω)(t) is a decreasing function, which involves
(6.1.17) ⇒ ‖w(t)‖ L 2 (Ω) ≤ ‖w(0)‖ L 2 (Ω) ,
Numerica
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Consider (temporally) smooth u : [0,T] ↦→ V 0 , v : [0,T] ↦→ V 0 and a symmetric bilinear form
b : V 0 ×V 0 ↦→ R.
What is
d
dt b(u(t),v(t))?
Formal Taylor expansion:
b(u(t+τ),v(t+τ)) = b(u(t)+ ˙u(t)τ +O(τ 2 ),v(t)+ ˙v(t)τ +O(τ 2 ))
= b(u(t),v(t))+τ(b(˙u(t),v(t))+b(u(t), ˙v(t)))+O(τ 2 ) .
b(u(t+τ),v(t+τ))−b(u(t),v(t))
lim
= b(˙u(t),v(t))+b(u(t), ˙v(t)) .
τ→0 τ
This is a general product rule, see [18, Eq. 4.4.5].
△
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
and the first assertion of the Lemma is evident. Next, we verify the exponential decay of|·| H 1 (Ω) -norm
of solution by a similar trick:
➽
1
2dt d ‖w‖2 ã = ã( dt dw,w) = −m( dt dw, dt dw) ≤ 0 ⇒ ‖w(t)‖ ã ≤ ‖w(0)‖ ã ,
|w(t)| 2 H 1 (Ω) ≤ |w(0)|2 H 1 (Ω) −γ(‖w(0)‖2 L 2 (Ω) −‖w(t)‖2 L 2 (Ω)
) .
} {{ }
≥0 by (6.1.17)
Exponential decrease of energy during parabolic evolution without excitation
MATLAB animation heatevl()
(“Parabolic evolutions dissipate energy”)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
6.1
p. 618
6.1
p. 62

6.1.3 Method of lines
Numerical
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for PDEs
Note:
(6.1.19) is an ordinary differential equation (ODE) fort ↦→ ⃗µ(t) ∈ R N
Numerica
Methods
for PDEs
Idea:
Apply Galerkin discretization (→ Sect. 3.1) to abstract linear parabolic variational problem
(6.1.11).
Conversion (6.1.11) ➙ (6.1.19) through Galerkin discretization in space only is known as
method of lines.
(6.1.19) ˆ= A semi-discrete evolution problem
1st step:
t ∈]0,T[↦→ u(t) ∈ V 0 :
{
m(˙u(t),v)+a(u(t),v) = l(t)(v) ∀v ∈ V0 ,
u(0) = u 0 ∈ V 0 .
replaceV 0 with a finite dimensional subspaceV 0,N ,N := dimV 0,N < ∞
(6.1.11)
Discrete parabolic evolution problem
{
m(˙uN (t),v N )+a(u N (t),v N ) = l(t)(v N ) ∀v N ∈ V 0,N ,
t ∈]0,T[↦→ u(t) ∈ V 0,N :
u N (0) = projection/interpolant ofu 0 inV 0,N .
2nd step: introduce (ordered) basisB N := {b 1 N ,...,bN N } ofV 0,N
⎧ }
⎨
M{ ddt ⃗µ(t) +A⃗µ(t) = ⃗ϕ(t) for0 < t < T ,
(6.1.18) ⇒
⎩⃗µ(0) = ⃗µ 0 .
✄ s.p.d. stiffness matrixA ∈ R N,N ,(A) ij := a(b j N ,bi N ) (independent of time),
✄ s.p.d. mass matrixM ∈ R N,N ,(M) ij := m(b j N ,bi N ) (independent of time),
✄ source (load) vector ⃗ϕ(t) ∈ R N ,(⃗ϕ(t)) i := l(t)(b i N ) (time-dependent),
✄ ⃗µ 0 ˆ= coefficient vector of a projection ofu 0 ontoV 0,N .
(6.1.18)
(6.1.19)
For the concrete linear parabolic evolution problem (6.1.8)–(6.1.9) and spatial finite element discretization
based on a finite element trial/test spaceV 0,N ⊂ H 1 (Ω) we can compute
• the mass matrixMas the Galerkin matrix for the bilinear form
u,v ∈ L 2 (Ω),
• the stiffness matrixAas Galerkin matrix arising from the bilinear form
gradvdx,u,v ∈ H 1 (Ω).
(u,v) ↦→ ∫ Ω ρ(x)uvdx,
(u,v) ↦→ ∫ Ω κ(x)gradu·
The calculations are explained in Sects. 3.5.3 and 3.5.4 and may involve numerical quadrature.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
6.1
p. 621
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
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DRAFT
6.1
p. 622
Discretized in space
Remark 6.1.20 (Spatial discretization options).
←→ but still continuous in time
Beside the Galerkin approach any other method for spatial discretization of 2nd-order elliptic BVPs
can be used in the context of the method of lines: the matrices A, M may also be generated by
finite differences (→ Sect. 4.1), finite volume methods (→ Sect. 4.2), or collocation methods (→
Sect. 1.5.2).
6.1.4 Timestepping
For implementation we need a fully discrete evolution problem. This requires additional discretization
in time:
semi-discrete evolution problem (6.1.19) + timestepping fully discrete evolution problem
Benefit of method of lines: we can apply already known integrators for initial value problems for ODEs
to (6.1.19).
First, refresh central concepts from numerical integration of initial value problems for ODEs, see [18,
Ch. 12], [18, Ch. 13]:
• single step methods of orderp, see [18, Def. 12.2.17] and [18, Thm. 12.3],
• explicit and implicit Runge-Kutta single step methods, see [18, Sect. 12.4], [18, Sect. ??],
encoded by Butcher scheme [18, Eq. 12.4.10], [18, Eq 13.3.8].
• the notion of a stiff problem (→ [18, Notion 13.2.25]),
• the definition of the stability function of a single step method, see [18, Thm. 13.3.9],
• the concept of L-stability [18, Def 13.3.11] and how to verify it for Runge-Kutta methods.
△
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C. Schwa
H.
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V.
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DRAFT
6.1
p. 62
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DRAFT
6.1
p. 62

6.1.4.1 Single step methods
Numerical
Methods
for PDEs
Example 6.1.24 (Crank-Nicolson timestepping).
Numerica
Methods
for PDEs
Recall: single step methods (→ [18, Def. 12.2.17])
are based on a temporal mesh {0 = t 0 < t 1 < ... < t M−1 < t M := T} (with local timestep
sizeτ j = t j −t j−1 ),
compute sequence
(⃗µ (j)) M
j=0 of approximations ⃗µ(j) ≈ µ(t j ) to the solution of (6.1.19) at the
nodes of the temporal mesh according to
⃗µ (j) := Ψ t j−1,t j⃗µ (j−1) := Ψ(t j−1 ,t j ,⃗µ (j−1) ) , j = 1,...,M ,
whereΨis the discrete evolution defining the single step method, see [18, Def. 12.2.17].
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Crank-Nicolson method = implicit midpoint rule: replace
dt d in (6.1.19) with symmetric difference quotient
and average right hand side:
}
M{ ddt ⃗µ(t) +A⃗µ(t) = ⃗ϕ(t)
⇓
M ⃗µ(j) − ⃗µ (j−1)
= − 1 τ 2
(⃗µ A (j) + ⃗µ (j−1)) + 1 2 (⃗ϕ(t j)+ ⃗ϕ(t j−1 )) . (6.1.25)
This yields a method that is 2nd-order consistent.
✸
Generalization of Euler methods:
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DRAFT
Runge-Kutta single step methods → [18, Sect. 12.4], [18, Sect. 13.3]
Example 6.1.21 (Euler timestepping). → [18, Sect. 12.2]
6.1
p. 625
6.1
p. 62
We target the initial value problem
}
M{ ddt ⃗µ(t) +A⃗µ(t) = ⃗ϕ(t) for0 < t < T ,
⃗µ(0) = ⃗µ 0 .
(6.1.19)
Numerical
Methods
for PDEs
Definition 6.1.26 (General Runge-Kutta method). → [18, Def. 13.3.7]
For coefficientsb i ,a ij ∈ R, c i := ∑ s
j=1 a ij , i,j = 1,...,s, s ∈ N, the discrete evolutionΨ s,t
of ans-stage Runge-Kutta single step method (RK-SSM) for the ODEẏ = f(t,y), is defined by
Numerica
Methods
for PDEs
Explicit Euler method [18, Eq. 12.2.4]: replace
dt d in (6.1.19) with forward difference quotient, see
[18, Rem. 12.2.5]:
(6.1.19) M⃗µ (j) = M ⃗ µ (j−1) −τ j
( A⃗µ
(j−1) − ⃗ϕ(tj−1 ) ) , j = 1,...,M −1 . (6.1.22)
Implicit Euler method [18, Eq. 12.2.10]: replace
dt d in (6.1.19) with backward difference quotient
(6.1.19) M⃗µ (j) = M ⃗ µ (j−1) −τ j
( A⃗µ (j) − ⃗ϕ(t j ) ) , j = 1,...,M −1 . (6.1.23)
Note that both (6.1.22) and (6.1.23) require the solution of a linear system of equations in each step
Recall [18, Sect. 12.3]:
(6.1.22): ⃗µ (j) = ⃗µ (j−1) +τ j M −1 (⃗ϕ(t j−1 )−A⃗µ (j−1) ) ,
(
(6.1.23): ⃗µ (j) = (τ j A+M) −1 Mµ (j−1) ⃗
)
+τ j ⃗ϕ(t j ) .
both Euler method are of first order.
✸
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DRAFT
s∑
s∑
k i := f(t+c i τ,y+τ a ij k j ) , i = 1,...,s , Ψ t,t+τ y := y+τ b i k i .
j=1
i=1
Thek i ∈ R d are called increments.
Shorthand notation for s-stage Runge-Kutta methods: Butcher scheme → [18, Eq. 13.3.8]
c A
b T ˆ=
c 1 a 11 a 12 ... ... a 1s
c 2 a 21
... a 2s
. . ... .
c s a s1 . a ss
, c,b ∈ R s , A ∈ R s,s . (6.1.27)
b 1 b 2 ... ... b s
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Concretely for linear parabolic evolution: application ofs-stage Runge-Kutta method to
}
M{ ddt ⃗µ(t) +A⃗µ(t) = ⃗ϕ(t) ⇔ ˙⃗µ = M −1 (⃗ϕ(t)−A⃗µ(t)) . (6.1.19)
6.1
} {{ }
6.1
=f(t,⃗µ)
p. 626
p. 62
DRAFT

Then simply plug this into the formulas of Def. 6.1.26.
Note:
Timestepping scheme for (6.1.19): compute ⃗µ (j+1) from ⃗µ (j) through
s∑
⃗κ i ∈ R N : M⃗κ i + τa im A⃗κ m = ⃗ϕ(t j +c i τ)−A⃗µ (j) , i = 1,...,s , (6.1.28)
m=1
s∑
⃗µ (j+1) = ⃗µ (j) +τ ⃗κ m b m . (6.1.29)
m=1
For an implicit RK-method (6.1.28) is a linear system of equations of sizeNs.
6.1.4.2 Stability
Example 6.1.30 (Convergence of Euler timestepping).
Parabolic evolution problem in one spatial dimension (IBVP):
∂u
∂t = ∂2 u
∂x 2 in[0,1]×]0,1[ , (6.1.31)
u(t,0) = u(t,1) = 0 for0 ≤ t ≤ 1 , u(0,x) = sin(πx) for0 < x < 1 . (6.1.32)
exact solution u(t,x) = exp(−π 2 t)sin(πx) . (6.1.33)
Spatial finite element Galerkin discretization by means of linear finite elements (V 0,N =
S1,0 0 (M)) on equidistant meshMwith meshwidthh := N 1 → Sect. 1.5.1.2.
u N,0 := I 1 u 0 by linear interpolation onM, see Sect. 5.3.1.
Timestepping by explicit and implicit Euler method (6.1.22), (6.1.23) with uniform timestepτ :=
1
M .
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DRAFT
6.1 20
21 %Timestepping
p. 629
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Code 6.1.34: Euler timestepping for (6.1.31)
1 f u n c t i o n [errex,errimp] = sinevl(N,M,u)
2 % Solve fully discrete two-point parabolic evolution problem (6.1.31)
3 % in [0,1]×]0,1[. Use both explicit and implicit Euler method for timestepping
4 % N: number of spatial grid cells
5 % M: number of timesteps
6 % u: handle of type @(t,x) to exact solution
7
8 i f (nargin < 3), u = @(t,x) (exp(-( p i ^2)*t).* s in(pi*x)); end %
Exact solution
9
10 h = 1/N; tau = 1/M; % Spatial and temporal meshwidth
11 x = h:h:1-h; % Spatial grid, interior points
12
13 % Finite element stiffness and mass matrix
14 Amat = g a l l e r y (’tridiag’,N-1,-1,2,-1)/h;
15 Mmat = h/6* g a l l e r y (’tridiag’,N-1,1,4,1);
16 Xmat = Mmat+tau*Amat;
17
18 mu0 = u(0,x)’; % Discrete initial value
19 mui = mu0; mue = mu0;
22 erre = 0; erri = 0;
23 f o r k=1:M
24 mue = mue - tau*(Mmat\(Amat*mue)); % explicit Euler step
25 mui = Xmat\(Mmat*mui); % implicit Euler step
26 utk = u(k*tau,x)’;
27 erre = erre + norm(mue-utk)^2; % Computation of error norm
28 erri = erri + norm(mui-utk)^2;
29 end
30
31 errex = s q r t(erre*h*tau);
32 errimp = s q r t(erri*h*tau);
Numerica
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DRAFT
6.1
p. 63
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Galerkin matrices, see (1.5.60):
⎛
⎞ ⎛
⎞
2 −1 0 0 4 1 0 0
−1 2 −1
1 4 1
A = 1 0 ... ... . ..
h
, M = h 0 . .. . .. ...
⎜
... . .. ... 0
6
.
⎟ ⎜
... ... . .. 0
⎟
⎝ −1 2 −1⎠
⎝ 1 4 1⎠
0 0 −1 2 0 0 1 4
6.1
p. 630
Evaluation of approximate space-timeL 2 -norm of the discretization error:
M∑
err 2 N−1 ∑
:= hτ · |u(t j ,x i )−µ (j)
i | 2 . (6.1.35)
j=1 i=1
Error norm for explicit Euler timestepping:
6.1
p. 63

N\M 50 100 200 400 800 1600 3200
5 Inf 0.009479 0.006523 0.005080 0.004366 0.004011 0.003834
10 Inf Inf Inf Inf 0.001623 0.001272 0.001097
20 Inf Inf Inf Inf Inf Inf 0.000405
40 Inf Inf Inf Inf Inf Inf Inf
80 Inf Inf Inf Inf Inf Inf Inf
160 Inf Inf Inf Inf Inf Inf Inf
320 Inf Inf Inf Inf Inf Inf Inf
Error norm for implicit Euler timestepping:
N\M 50 100 200 400 800 1600 3200
5 0.007025 0.001828 0.000876 0.002257 0.002955 0.003306 0.003482
10 0.009641 0.004500 0.001826 0.000461 0.000228 0.000575 0.000749
20 0.010303 0.005175 0.002509 0.001149 0.000461 0.000116 0.000058
40 0.010469 0.005345 0.002681 0.001321 0.000634 0.000289 0.000116
80 0.010511 0.005387 0.002724 0.001364 0.000677 0.000332 0.000159
160 0.010521 0.005398 0.002734 0.001375 0.000688 0.000343 0.000170
320 0.010524 0.005400 0.002737 0.001378 0.000691 0.000346 0.000172
Numerical
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C. Schwab,
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DRAFT
Code 6.1.37: ode45 applied semi-discrete (6.1.31)
1 f u n c t i o n [Nsteps,err] = peode45(N,tol,u)
2 % Solving fully discrete two-point parabolic evolution problem (6.1.31)
3 % in [0,1]×]0,1[ by means of adaptiv MATLAB standard Runge-Kutta integrator.
4 i f (nargin < 3), u = @(t,x) (exp(-( p i ^2)*t).* s in(pi*x)); end %
Exact solution
5
6 % Finite element stiffness and mass matrix, see Sect. 1.5.1.2
7 h = 1/N; % spatial meshwidth
8 Amat = g a l l e r y (’tridiag’,N-1,-1,2,-1)/h;
9 Mmat = h/6* g a l l e r y (’tridiag’,N-1,1,4,1);
10 x = h:h:1-h; % Spatial grid, interior points
11
12 mu0 = u(0,x)’; % Discrete initial value
13 fun = @(t,muv) -(Mmat\(Amat*muv)); % right hand side of ODE
14
15 opts = odeset(’reltol’,tol,’abstol’,0.01*tol);
16 [t,mu] = ode45(fun,[0,1],mu0,opts);
17
18 Nsteps = l e n g t h(t);
19 [T,X] = meshgrid(t,x); err = norm(mu’-u(T,X),’fro’);
Numerica
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DRAFT
6.1
p. 633
6.1
p. 63
Explicit Euler timestepping: we observe a glaring instability (exponential blow-up) in case of large
timestep combined with fine mesh.
Numerical
Methods
for PDEs
ode45: atol = 0.00001, rtol = 0.001
ode45: atol = 0.00001, rtol = 0.001
Numerica
Methods
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Implicit Euler timestepping: no blow-up at any combination of spatial and temporal mesh width.
✸
No. of ode45 steps h
10 4
10 3
discrete L 2 norm of error
10 −2
ode45
impl Euler, 100 steps
Example 6.1.36 (ode45 for discrete parabolic evolution).
Same IBVP and spatial discretization as in Ex. 6.1.30.
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10 5 spatial mesh width h
10 2
10 −2 10 −1 10 0
Fig. 184
10 −1 spatial mesh width h
10 −2 10 −1 10 0
Fig. 185
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Adaptive Runge-Kutta timestepping by MATLAB standard integrator ode45.
Monitored:
Number of timesteps as a function on spatial meshwidthh,
discreteL 2 -error (6.1.35).
Observations:
ode45: dramatic increase of no. of timesteps forh M → 0 without gain in accuracy.
Implicit Euler achieves better accuracy with only 100 equidistant timesteps!
6.1
p. 634
✸
6.1
p. 63

This reminds us of the stiff initial value problems studied in [18, Thm. 13.2]:
Notion 6.1.38 (Stiff IVP). → [18, Notion 13.2.25]
An initial value problem for an ODE is called stiff, if stability imposes much tighter timestep
constraints on explicit single step methods than the accuracy requirements.
Numerical
Methods
for PDEs
that is, the decoupling of eigencomponents carries over to the explicit Euler method: fori = 1,...,N
η (j)
i = η (j−1)
i
The condition|1−τλ i | < 1 enforces a
−τλ i η (j−1) ⇒ η (j)
i = (1−τλ i ) j η (0)
i . (6.1.40)
|1−τλ i | < 1 ⇔ lim
j→∞ η(j) i = 0 . (6.1.41)
Numerica
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Admittedly, this is a fuzzy notion. Yet, it cannot be fleshed out on the abstract level, but has to be
discussed for concrete evolution problem, which is done next.
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timestep size constraint: τ < 2 λ i
(6.1.42)
in order to achieve the qualitatively correct behavior lim
j→∞ η(j) i = 0 and to avoid blow-up lim
j→∞ |η(j) i | =
∞: the timestep size constraint (6.1.42) is necessary only for the sake of stability (not in order to
guarantee a prescribed accuracy).
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DRAFT
Let us try to understand, why semi-discrete parabolic evolutions (6.1.19) arising from the method of
lines lead to stiff initial value problems.
This accounts to the observed blow-ups in Ex. 6.1.30. On the other hand, adaptive stepsize control
[18, Sect. 12.5] manages to ensure the timestep constraint, but the expense of prohibitively small
timesteps that render the method grossly inefficient, if some of theλ i are large.
Technique: Diagonalization, cf. [18, Eq. 13.2.10]
(Recall the concept of a “square root”M 1/2 of an s.p.d. matrixM, see [18, Sect. 5.3])
A,M symmetric positive definite ⇒ M −1/2 AM −1/2 symmetric positive definite .
[18, Cor. 6.1.9]
⇒ ∃ orthogonalT ∈ R N,N : T ⊤ M −1/2 AM −1/2 T = D := diag(λ 1 ....,λ N ) ,
where the λ i > 0 are generalized eigenvalues for A ⃗ ξ = λM ⃗ ξ ➤
constant introduced in (6.1.12)).
λ i ≥ γ for all i (γ is the
6.1
p. 637
Numerical
Methods
for PDEs
The next numerical demonstrations and Lemma show that λ max := max i λ i will inevitably become
huge for finite element discretization on fine meshes.
Example 6.1.43 (Behavior of generalized eigenvalues ofA⃗µ = λM⃗µ).
Bilinear forms associated with parabolic IBVP and homogeneous Dirichlet boundary conditions
∫
∫
a(u,v) = gradu·gradvdx , m(u,v) = u(x)v(x)dx , u,v ∈ H0 1 (Ω) .
Ω
Ω
6.1
p. 63
Numerica
Methods
for PDEs
➤
Transformation (“diagonalization”) of (6.1.19) based on substitution⃗η := T ⊤ M 1/2 ⃗µ:
(6.1.19)
⃗η:=T ⊤ M 1/2 ⃗µ
=⇒ d
dt ⃗η(t)+D⃗η = T ⊤ M −1/2 ⃗ϕ(t) . (6.1.39)
Since D is diagonal, (6.1.39) amounts to N decoupled scalar ODEs (for eigencomponentsη i
of ⃗µ).
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Linear finite element Galerkin discretization, see Sect. 1.5.1.2 for 1D, and Sect. 3.2 for 2D.
Numerical experiments in 1D & 2D:
•Ω =]0,1[, equidistant meshes→Ex. 6.1.30
• “disk domain”Ω = {x ∈ R 2 : ‖x‖ < 1}, sequence of regularly refined meshes.
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Note: for ⃗ϕ ≡ 0,λ > 0 : η i (t) = exp(−λ i t)η i (0) → 0 fort → ∞
As in [18, Thm. 13.2.12] this transformation can be applied to the explicit Euler timestepping (6.1.22)
(for ⃗ϕ ≡ 0, uniform timestepτ > 0)
Monitored:
largest and smallest generalized eigenvalue
⃗µ (j) = ⃗µ (j−1) −τM −1 A⃗µ (j−1) ⃗η:=T ⊤ M 1/2 ⃗µ
⃗η (j) = ⃗η (j−1) −τD⃗η (j−1) ,
6.1
p. 638
6.1
p. 64

Code 6.1.44: Computation of extremal generalized eigenvalues
1 % LehrFEM MATLAB script for computing Dirichlet eigenvalues of Laplacian
2 % on a unit disc domain.
3
4 GD_HANDLE = @(x,varargin)zeros( s i z e (x,1),1); % Zero Dirichlet data
5 H0 =[ .25 .2 .1 .05 .02 .01 0.005]’; % target mesh widths
6 NRef = l e n g t h(H0); % Number of refinement steps
7
8 % Variables for mesh widths and eigenvalues
9 M_W = zeros(NRef,1); lmax = M_W; lmin = M_W;
10
11 % Main refinement loop
12 f o r iter = 1:NRef
13
14 % Set parameters for mesh
15 C = [0 0]; % Center of circle
16 R = 1; % Radius of circle
17 BBOX = [-1 -1; 1 1]; % Bounding box
18 DHANDLE = @dist_circ; % Signed distance function
19 HHANDLE = @h_uniform; % Element size function
20 FIXEDPOS = []; % Fixed boundary vertices of the mesh
21 DISP = 0; % Display flag
22
23 % Mesh generation
24 Mesh =
init_Mesh(BBOX,H0(iter),DHANDLE,HHANDLE,FIXEDPOS,DISP,C,R);
25 Mesh = add_Edges(Mesh); % Provide edge information
26 Loc = get_BdEdges(Mesh); % Obtain indices of edges on ∂Ω
27 Mesh.BdFlags = zeros( s i z e (Mesh.Edges,1),1);
28 Mesh.BdFlags(Loc) = -1; % Flag boundary edges
29 Mesh.ElemFlag = zeros( s i z e (Mesh.Elements,1),1);
30 M_W(iter) = get_MeshWidth(Mesh); % Get mesh width
31
32 f p r i n t f (’Mesh on level %i: %i elements, h =
%f\n’,iter, s i z e (Mesh,1),M_W(iter));
33 % Assemble stiffness matrix and mass matrix
34 A = assemMat_LFE(Mesh,@STIMA_Lapl_LFE,P7O6());
35 M = assemMat_LFE(Mesh,@MASS_LFE,P7O6());
36 % Incorporate Dirichlet boundary data (nothing to do here)
37 [U,FreeNodes] = assemDir_LFE(Mesh,-1,GD_HANDLE);
38 A = A(FreeNodes,FreeNodes);
39 M = M(FreeNodes,FreeNodes);
40
41 % Use MATLAB’s built-in eigs-function to compute the
42 % extremal eigenvalues, see [18, Sect. 6.4].
43 NEigen = 6;
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DRAFT
6.1
p. 641
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DRAFT
6.1
p. 642
44 d = eigs(A,M,NEigen,’sm’); lmin(iter) = min(d);
45 d = eigs(A,M,NEigen,’lm’); lmax(iter) = max(d);
46 end
47
48 f i g u r e ; p l o t (M_W,lmin,’b-+’,M_W,lmax,’r-*’); g r i d on;
49 s e t(gca,’XScale’,’log’,’YScale’,’log’,’XDir’,’reverse’);
50 t i t l e (’\bf Eigenvalues of Laplacian on unit disc’);
51 x l a b e l (’{\bf mesh width h}’,’fontsize’,14);
52 y l a b e l (’{\bf generalized eigenvalues}’,’fontsize’,14);
53 legend(’\lambda_{min}’,’\lambda_{max}’,’Location’,’NorthWest’)
54 p = p o l y f i t (log(M_W),log(lmax),1);
55 add_Slope(gca,’east’,p(1));
56
57 p r i n t -depsc2 ’../../../Slides/NPDEPics/geneigdisklfe.eps’;
generalized eigenvalues
Linear FE in 1D: lambda i
10 7 λ min
λ max
10 6 O(h −2 )
10 5
10 4
10 3
10 2
10 1
10 0
10 0
10 −1
10 −2
spatial mesh width h
Observation:
Ω =]0,1[
10 −3
Fig. 186
λ min := min i λ i does hardly depend on the mesh width.
λ max := max i λ i displays aO(h −2
M ) growth ash M → 0
generalized eigenvalues
Eigenvalues of Laplacian on unit disc
10 6 λ min
10 5
10 4
10 3
10 2
10 1
10 0
10 0
λ max
p = −2.00
10 −1
10 −2
10 −3
Fig. 187
mesh width h
Ω = {x ∈ R 2 : ‖x‖ < 1}
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DRAFT
6.1
p. 64
Numerica
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for PDEs
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C. Schwa
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DRAFT
6.1
p. 64

Remark 6.1.45 (Spectrum of elliptic operators).
The observation made in Ex. 6.1.43 is not surprising!
To understand why, let us translate the generalized eigenproblem “back to the ODE/PDE level”:
A⃗µ = λM⃗µ (6.1.46)
⇕
u N ∈ V 0,N : a(u N ,v N ) = λm(u N ,v N ) ∀v N ∈ V 0,N .
← “undo Galerkin discretization”
∫
∫
u ∈ H0 1 (Ω): gradu·gradvdx = λ u·vdx ∀v ∈ H0 1 (Ω) .
Ω
Ω
⇓
which is a so-called elliptic eigenvalue problem.
It is easily solved in 1D onΩ =]0,1[:
−∆u = λu inΩ , u = 0 on∂Ω , (6.1.47)
d 2 u
(6.1.47) ˆ=
dx2(x) = λu(x) , 0 < x < 1 , u(0) = u(1) = 0 .
⇒ u k (x) = sin(kπx) ↔ λ k = (πk) 2 , k ∈ N .
Note that we find an infinite number of eigenfunctions and eigenvalues, parameterized byk ∈ N. The
eigenvalues tend to∞fork → ∞:
λ k = O(k 2 ) for k → ∞ .
Of course, (6.1.46) can have a finite number of eigenvectors only. Crudely speaking, they correspond
to those eigenfunctions u k (x) = sin(kπx) that can be resolved by the mesh (if u k “oscillates too
much”, then it cannot be represented on a grid). These are the first N so that we find in 1D for an
equidistant mesh
λ max = O(N 2 ) = O(h −2
M ) .
This is heuristics, but the following Lemma will a precise statement.
✸
△
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
6.1
p. 645
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
6.1
p. 646
✬
Lemma 6.1.48 (Behavior of of generalized eigenvalues).
Let M be a simplicial mesh and A, M denote the Galerkin matrices for the bilinear forms
a(u,v) = ∫ Ω gradu · gradvdx and m(u,v) = ∫ Ω u(x)v(x)dx, respectively, and V 0,N :=
Sp,0 0 (M). Then the smallest and largest generalized eigenvalues of A⃗µ = λM⃗µ, denoted by
λ min andλ max , satisfy
✫
1
diam(Ω) 2 ≤ λ min ≤ C , λ max ≥ Ch −2
M ,
where the “generic constants” (→ Rem. 5.3.44) depend only on the polybomial degree p and
the shape regularity measureρ M .
Proof.
(partial) We rely on the Courant-Fischer min-max theorem [18, Thm. 6.3.35] that, among
other consequencees, expresses the boundaries of the spectrum of a symmetric matrix through the
extrema of its Rayleigh quotient
T = T T ∈ R N,N ⃗T ξ T ⃗ ξ
⃗T ξ T ⃗ ξ
⇒ λ min (T) = min
⃗ ξ∈R N \{0} ⃗ ξ T , λ max (T) = max
⃗ ξ ⃗ ξ∈R N \{0} ⃗ ξ T .
⃗ ξ
Apply this to the generalized eigenvalue problem
A⃗µ = λM⃗µ ⃗ ζ:=M 1/2 ⃗µ
⇔
}
M −1/2 {{
AM −1/2 ⃗
} ζ = λ ⃗ ζ .
=:T
⃗µ T A⃗µ
λ min = min
⃗µ≠0 ⃗µ T M⃗µ , λ ⃗µ T A⃗µ
max = max
⃗µ≠0 ⃗µ T M⃗µ . (6.1.49)
As a consequence we only have to find bounds for the extrema of a generalized Rayleigh quotient,
cf. [18, Eq. 6.3.33]. This generalized Rayleigh quotient can be expressed as
⃗µ T A⃗µ
⃗µ T M⃗µ = a(u N,u N )
m(u N ,u N ) , ⃗µˆ= coefficient vector for u N . (6.1.50)
Now we discuss a lower bound for λ max , which can be obtained by inserting a suitable candidate
function into (6.1.50).
✩
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R. Hiptm
C. Schwa
H.
Harbrech
✪V.
Gradinar
A. Chern
R. Hiptm
DRAFT
Discussion for special setting: V 0,N = S1 0 (M) on triangular meshM
Candidate function: “tent function” u N = b i N (→ Sect. 3.2.3) for some node xi ∈ V(M) of the
mesh!
By elementary computations as in Sect. 3.2.5 we find
(6.1.51)
a(b i N ,bi N ) ≈ C , m(bi N ,bi N ) ≤ C max h 2
K∈U(x i K ,
)
DRAFT
6.1
p. 64
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C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
6.1
p. 64

where the generic constantsC > 0 depend on the shape regularity measureρ M only.
(6.1.49) & (6.1.51) ⇒ λ max ≥ Ch −2
M .
✷
Numerical
Methods
for PDEs
There we saw that everything boils down to inspecting the modulus of a rational stability function on
C, see [18, Thm. 13.3.9]. This gave rise to the concept of L-stability, see [18, Def. 13.3.11]. Here, we
will not delve into a study of stability functions.
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Necessary condition for suitability of a single step method for semi-discrete parabolic evolution
problem (6.1.19)(“method of lines”):
Lemma 6.1.48 timestep constraint (6.1.42) unacceptable to semi-discrete parabolic evolutions!
C. Schwab,
The discrete evolution Ψ τ λ : R ↦→ R of the single step method applied to the scalar ODE
R. Hiptmair
ẏ = −λy satisfies
H.
Harbrecht
V.
Gradinaru λ > 0 ⇒ lim
A. Chernov
j→∞ (Ψτ λ )j y 0 = 0 ∀y 0 ∈ R, ∀τ > 0 . (6.1.55)
From [18, Sect. 13.3] we already know that some implicit single step methods are not affected by DRAFT
stability induced timestep constraints.
Definition 6.1.56 (L(π)-stability).
A single step method satisfying (6.1.55) is called L(π)-stable.
Recall [18, Ex. 13.3.1]: apply diagonalization technique, see (6.1.39), to implicit Euler timestepping
with uniform timestepτ > 0
Example 6.1.57 (L(π)-stable Runge-Kutta single step methods).
⃗µ (j) = ⃗µ (j−1) −τM −1 A⃗µ (j) ⃗η:=T ⊤ M 1/2 ⃗µ
⃗η
(j) = ⃗η
(j−1) −τD⃗η
(j) ,
that is, the decoupling of eigencomponents carries over to the implicit Euler method: fori = 1,...,N
η (j)
i = η (j−1)
i
[ ∣ ∣ ∣∣
1
1+τλ i
∣ ∣∣∣
< 1 and
−τλ i η (j) ⇒ η (j) ( )
i = 1 jη (0)
1+τλ i i . (6.1.52)
λ i > 0 ⇒
]
lim
j→∞ η(j) i = 0 ∀τ > 0 . (6.1.53)
This diagonalization trick can be applied to general Runge-Kutta single step methods (RKSSM, →
Def. 6.1.26). Loosely speaking, the following diagram commutes
Mdt d⃗µ+Aµ = 0 transformation ⃗η = TT M 1 /2
⃗µ
−−−−−−−−−−−−−−−−−→
⏐
RK-SSM↓
dt dη i = −λ i η i , i = 1,...,N
⏐
↓RK-SSM
⃗µ (j) = Ψ τ ⃗µ (j−1) transformation ⃗η = T T M 1 /2
⃗µ
−−−−−−−−−−−−−−−−−→ ⃗η (j)
i = ˜Ψ τ ⃗η (j−1)
i , i = 1,...,N .
The bottom line is
(6.1.54)
that we have to study the behavior of the RK-SSM only for linear scalar ODEsẏ = −λy,λ > 0.
6.1
p. 649
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R. Hiptmair
C. Schwab,
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V.
Gradinaru
A. Chernov
DRAFT
Simplest example: implicit Euler timestepping (6.1.23).
Some commonly used higher order methods, specified through their Butcher schemes, see (6.1.27):
1 5
3 12 − 12
1
1 3 1
4 4
(6.1.58)
3 1
4 4
RADAU-3 scheme (order 3)
More examples → [18, Ex. 13.3.20]
6.1.5 Convergence
λ λ 0
1 1−λ λ
1−λ λ
, λ := 1− 1 2√
2 , (6.1.59)
SDIRK-2 scheme (order 2)
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
6.1
p. 65
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Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
This is the gist of the model problem analysis discussed in [18, Sect. 13.3].
6.1
p. 650
6.1
p. 65

Why should one prefer complicated implicit L(π)-stable Runge-Kutta single step methods (→
Ex. 6.1.57) to the simple implicit Euler method?
Silly question! Because these methods deliver “better accuracy”!
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Since the implicit Euler method is first order consistent we expect
temporal timestepping error
O(τ)
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However, we need some clearer idea of what is meant by this. To this end, we now study the dependence
of (a norm of) the discretization error for a parabolic IBVP on the parameters of the spatial and
temporal discretization.
(6.1.61) ➣ conjecture: total error is sum of spatial and temporal discretization error.
R. Hiptmair
C. Schwab,
H.
Example 6.1.60 (Convergence of fully discrete timestepping in one spatial dimension).
Harbrecht From Fig. 188 we draw the compelling conclusion:
V.
Gradinaru
A. Chernov
d
dt u−u′′ DRAFT
= f(t,x) on]0,1[×]0,1[
exact solutionu(x,t) = (1+t 2 )e −π2t • for big mesh widthh M (spatial error dominates) further reduction of timestep sizeτ is useless,
sin(πx), source term accordingly
Linear finite element Galerkin discretization equidistant mesh, see Sect. 1.5.1.2,V 0,N = S1,0 0 (M),
yield a reduction of the total error.
piecewise linear spatial approximation of source termf(x,t)
implicit Euler timestepping (→ Ex. 6.1.21) with uniform timestepτ > 0
)1 2.
Monitored:
error norm
(
τ
M∑
|u−u N (τj)| 2 H 1 (Ω)
j=1
The norms |u−u N (τj)| H 1 (Ω) were approximated by high order local quadrature rules, whose impact
can be neglected.
l 2 −mean of H 1 −seminorm of error
✁ h M - andτ-dependence of error norm
6.1
p. 653
Numerical
Methods
for PDEs
• if timestep τ is large (temporal error dominates), refinement of the finite element space does not
Example 6.1.62 (Higher order timestepping for 1D heat equation).
same IBVP as in Ex. 6.1.60
spatial discretization on equidistant grid, very small meshwidthh = 0.5·10 −4 ,V N = S 0 1,0 (M)
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
6.1
p. 65
Numerica
Methods
for PDEs
error norm
10 −2
10 −3
10 −4
10 −1
10 −2
timestep ∆ t
10 −3
10 −4
Fig. 188
10 −4 10 −3 10 −2 10 −1
Obervation:
τ small: error norm≈ h M
h M small: error norm≈ τ
The error seems to behave like
error norm ≈ C 1 h M +C 2 τ . (6.1.61)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Various timestepping methods
(➣ different orders of consistency)
• implicit Euler timestepping (6.1.23), first order
• Crank-Nicolson-method (6.1.25), order 2
• SDIRK-2 timestepping (→ Ex. 6.1.57), order 2
• Gauss-Radau-Runge-Kutta collocation methods
withsstages, order2s−1
Maximal L 2 −error
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
Discretization error for heat equation
Implicit Euler
Crank−Nicolson
SDIRK2
Radau, s=2
Radau, s=3
Radau, s=4
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
10 −1 meshwidth h
Recall from Sect. 5.3.5, Thm. 5.1.10, Thm. 5.3.42:
Note: all methods L(π)-stable (→ Def. 6.1.56),
except for Crank-Nicolson-method.
10 0 No. of timesteps
10 −9
10 −10
10 1 10 2
Fig. 189
energy norm of spatial finite element discretization error O(h M ) forh M → 0
6.1
p. 654
6.1
p. 65

Monitored:
max
j
∥
∥u(t j )−u (j)
N ∥ L 2 (evaluated by high order quadrature)
(]0,1[)
✸
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Methods
for PDEs
A message contained in (6.1.64):
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total discretization error = spatial error + temporal error
Rem. 5.3.45 still applies:
(6.1.64) does not give information about actual error, but only about the
trend of the error, when discretization parametersh M andτ are varied.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Nevertheless, as in the case of the a priori error estimates of Sect. 5.3.5, we can draw conclusions
about optimal refinement strategies in order to achieve prescribed error reduction.
As in Sect. 5.3.5 we make the assumption that the estimates (6.1.64) are sharp for all contributions
to the total error and that the constants are the same (!)
contribution of spatial error ≈ Ch p M , h M ˆ= mesh width (→ Def. 5.2.3) ,
contribution of temporal error ≈ Cτ q , τ ˆ= timestep size .
(6.1.65)
This suggests the following change ofh M ,τ in order to achieve error reduction by a factor ofρ > 1:
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✬
✩
6.1
p. 657
reduce mesh width by factor ρ 1/p
reduce timestep by factor ρ 1/q (6.1.65)
=⇒ error reduction byρ > 1 . (6.1.66)
6.1
p. 65
“Meta-theorem” 6.1.63 (Convergence of solutions of fully discrete parabolic evolution problems).
Assume that
the solution of the parabolic IBVP (6.1.3)–(6.1.5) is “sufficiently smooth”,
its spatial Galerkin finite element discretization relies on degree p Lagrangian finite elements
(→ Sect. 3.4) on uniformly shape-regular families of meshes,
timestepping is based on an L(π)-stable single step method of orderq with uniform timestep
τ > 0.
Then we can expect an asymptotic behavior of the total discretization error according to
(
τ
M∑
j=1
whereC > 0 must not depend onh M ,τ.
|u−u N (τj)| 2 H 1 (Ω))1 2
≤C(h p M +τq ) , (6.1.64)
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Guideline:
spatial and temporal resolution have to be adjusted in tandem
Remark 6.1.67 (Potential inefficiency of conditionally stable single step methods).
Terminology:
A timestepping scheme is labelled conditionally stable, if blow-up can be avoided by
using sufficient small timesteps (timestep constraint).
Now we can answer the question, why a stability induced timestep constraint like
τ ≤ O(h −2
M ) (6.1.68)
can render a single step method grossly inefficient for integrating semi-discrete parabolic IBVPs.
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✫
✪
(6.1.66) ➣ in order to reduce the error by a fixed factor ρ one has to reduce both timestep and
meshwidth by some other fixed factors (asymptotically). More concretely, for the timestepτ:
This has been dubbed a “meta-theorem”, because quite a few technical assumptions on the exact
solution and the methods have been omitted in its statement. Therefore it is not a mathematically
rigorous statement of facts. More details in [19].
6.1
p. 658
(6.1.66) ➣ accuracy requires reduction ofτ by a factorρ 1/q
(6.1.68) ➣ stability entails reduction ofτ by a factor(ρ 1/p ) 2 = ρ 2/p .
6.1
p. 66

1
q < 2 p
⇒ stability enforces smaller timestep than required by accuracy
⇒ timestepping is inefficient!
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6.2.1 Vibrating membrane
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Faced with conditional stability (6.1.68), then for the sake of efficiency
use high-order spatial discretization combined with low order timestepping.
However, this may not be easy to achieve
because high-order timestepping is much simpler than high-order spatial discretization,
because limited spatial smoothness of exact solution (→ results of Sect. 5.4 apply!) may impose
a limit onq in (6.1.64).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Recall:
Tense string model (→ Sect. 1.4), shape of string described by continuous displacement function
u : [a,b] ↦→ R,u ∈ H 1 ([a,b]).
Taut membrane model (→ Sect. 2.1.1), shape of membrane given by displacement function u :
Ω ↦→ R,u ∈ H 1 (Ω), over base domainΩ ⊂ R 2 .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Concretely: 5th-order ode45 timestepping (q = 5)
1
q = 2 p
➣
use degree-10 Lagrangian FEM!
6.2 Wave equations
△
6.2
p. 661
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for PDEs
u
6.2
p. 66
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for PDEs
5
OPTIONAL SECTION
Lemma 6.1.14 teaches that in the absence of time-dependent sources the rate of change of temperature
will decay exponentially in the case of heat conduction.
Now we will encounter a class of evolution problems where temporal and spatial fluctuations will not
be demped and will persist for good:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Forcef(x)
(a,u l )
a
Tense string↔u : [a,b] ↦→ R
u(x)
(b,u r )
b
Fig. 190
x
u(x 1
,x 2
)
4
3
2
1
0
−1
−2
−3
0
u(x,t)
1
0.8
0.6
0.4
0.2
0.4
0.2
0.6
0.8
0
1
x 2
x 1
Fig. 191
Taut membrane↔u : Ω ↦→ R
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
This will be the class of linear conservative wave propagation problems
As before these initial-boundary value problems (IBVP) will be posed on a space time cylinder ˜Ω :=
Ω×]0,T[⊂ R d+1 (→ Fig. 183), whereΩ ⊂ R d , d = 2,3, is a bounded spatial domain as introduced
in the context of elliptic boundary value problems, see Sect. 2.1.1.
In Sect. 2.1.3 we introduced the general variational formulation: with Dirichlet data (elevation of frame)
given byg ∈ C 0 (∂Ω),
V := {v ∈ H 1 (Ω): v |∂Ω = g}
we seek
The unknown will be a functionu = (x,t) : ˜Ω ↦→ R.
6.2
p. 662
6.2
p. 66

where
u ∈ V:
∫
∫
σ(x)gradu·gradvdx =
Ω
Ω
f : Ω ↦→ R ˆ= density of vertical force,
f(x)v(x)dx , ∀v ∈ H0 1 (Ω) , (6.2.1)
σ : Ω ↦→ R ˆ= uniformly positive stiffness coefficient (characteristic of material of the
membrane).
Now we switch to a dynamic setting: we allow variation of displacement with time, u = u(x,t), the
membrane is allowed to vibrate.
Recall (secondary school):
Newton’s second law of motion (law of inertia)
Apply this in a local version (stated for densities) to membrane
F = ma (6.2.2)
force = mass · acceleration (6.2.3)
force density
f(x,t) = ρ(x)· ∂2 u
∂t 2(x,t) ,
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C. Schwab,
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Gradinaru
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DRAFT
(6.2.4) 6.2
p. 665
where V(t) := {v :]0,T[↦→ H 1 (Ω): v(x,t) = g(x,t) for x ∈ ∂Ω, 0 < t < T}
(with continuous time-dependent Dirichlet datag : ∂Ω×]0,T[↦→ R.)
Undo integration by parts by reverse application of Green’s first formula Thm. 2.4.7:
∫ {
∂ 2 }
u
(6.2.5) ⇒
Ω ∂t 2(x,t)−div x(σ(x)(grad x u)(x,t)) v(x)dx = 0 ∀v ∈ H0 1 (Ω) .
(6.2.7)
Here it is indicated that the differential operatorsgrad anddiv act on the spatial independent variable
x only. This will tacitly be assumed below.
Now appeal to the fundamental lemma of calculus of variations in higher dimensions Lemma 2.4.10.
(6.2.7)
Lemma 2.4.10
=⇒
∂ 2 u
∂t 2 −div(σ(x)gradu) = 0 in ˜Ω . (6.2.8)
(6.2.8) is called a (homogeneous) wave equation. A general wave equation is obtained, when an
addition exciting vertical force densityf = f(x,t) comes into play:
∂ 2 u
∂t 2 −div(σ(x)gradu) = f(x,t) in ˜Ω . (6.2.9)
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C. Schwa
H.
Harbrech
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Gradinar
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DRAFT
6.2
p. 66
where ρ : Ω ↦→ R + ˆ= uniformly positive mass density of membrane,[ρ] = kgm −2 ,
ü := ∂2 u
∂t 2 ˆ= vertical aceleration (second temporal derivative of position).
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The wave equations (6.2.8), (6.2.9) have to be supplemented by
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Now, we assume that the force f in (2.3.2) is due to inertia forces only and express these using
(6.2.4):
(2.3.2)
(6.2.4)
∫
Ω
∫
σ(x)gradu(x,t)·gradv(x)dx = −
Why the “−”-sign? Because, here the inertia force enters as a reaction force.
Ω
ρ(x)· ∂2 u
∂t 2(x,t)dx ∀v ∈ H1 0 (Ω) .
Linear wave equation in variational form (Dirichlet boundary conditions):
∫
u ∈ V(t): ρ(x)· ∂2 ∫
u
∂t 2(x,t)v(x)dx+ σ(x)gradu(x,t)·gradv(x)dx = 0 ∀v ∈ H0 1 (Ω)
Ω
Ω
stiffness
mass density
(6.2.5)
↕
u ∈ V(t): m(ü,v)+a(u,v) = 0 ∀v ∈ V 0 . (6.2.6)
R. Hiptmair
C. Schwab,
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DRAFT
6.2
p. 666
• spatial boundary conditions: v(x,t) = g(x,t) for x ∈ ∂Ω, 0 < t < T ,
• two initial conditions
u(x,0) = u 0 (x) ,
∂u
∂t (x,0) = v 0 for x ∈ Ω ,
with initial data u 0 ,v 0 ∈ H 1 (Ω), satisfying the compatibility conditions u 0 (x) = g(x,0) for x ∈
∂Ω.
(6.2.8) & boundary conditions & initial conditions = hyperbolic evolution problem
Hey, why do we need two initial conditions in contrast to the heat equation?
Remember that
• (6.2.8) is a second-order equation also in time (whereas the heat equation is merely first-order),
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
6.2
p. 66

• for second order ODEsÿ = f(y) we need two initial conditions
y(0) = y 0 and ẏ(0) = v 0 , (6.2.10)
in order to get a well-posed initial value problem, see [18, Rem. 12.1.15].
The physical meaning of the initial conditions (6.2.10) in the case of the membrane model is
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Additional unknown: velocity v(x,t) = ∂u
∂t (x,t)
∂ 2 {
u
˙u = v ,
∂t 2 −div(σ(x)gradu) = 0 ˙v = div(σ(x)gradu)
with initial conditions
in ˜Ω (6.2.13)
u(x,0) = u 0 (x) , v(x,0) = v 0 (x) for x ∈ Ω . (6.2.14)
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•u 0 ˆ= initial displacement of membrane,u 0 ∈ H 1 (Ω) “continuous”,
•v 0 ˆ= initial vertical velocity of membrane.
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6.2.2 Wave propagation
Remark 6.2.11 (Boundary conditions for wave equation).
Constant coefficient wave equation ford = 1,Ω = R (“Cauchy problem”)
Rem. 6.1.7 also applies to the wave equation (6.2.8):
6.2
p. 669
c > 0:
∂ 2 u
∂t 2 −c2∂2 u
∂x 2 = 0 , u(x,0) = u 0(x) ,
∂u
∂t (x,0) = v 0(x) , x ∈ R . (6.2.15)
6.2
p. 67
On∂Ω×]0,T[ we can impose any of the boundary conditions discussed in Sect. 2.6:
• Dirichlet boundary conditions u(x,t) = g(x,t) (membrane attached to frame),
• Neumann boundary conditions j(x,t)·n = 0 (free boundary, Rem. 2.4.16)
• radiation boundary conditionsj(x,t)·n = Ψ(u(x,t)),
and any combination of these as discussed in Ex. 2.6.7, yet, only one of them at any part of
∂Ω×]0,T[, see Rem. 2.6.6.
Remark 6.2.12 (Wave equation as first order system in time).
△
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Change of variables: ξ = x+ct, τ = x−ct: ũ(ξ,τ) := u( ξ+τ
2 , ξ−τ
2c ). Applying the chain
rule we immediately see
u satisfies (6.2.15)
for anyF,G ∈ C 2 (R) !
∂ 2 ũ
= 0 ⇒ ũ(ξ,τ) = F(ξ)+G(τ) ,
∂ξ∂τ
← matching initial data
∫ x+ct
u(x,t) = 2 1(u 0(x+ct)+u 0 (x−ct))+ 1 2 v 0 (s)ds . (6.2.16)
x−ct
(6.2.16) = d’Alembert solution of Cauchy problem (6.2.15).
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Usual procedure [18, Rem. 12.1.15]: higher-order ODE can be converted into first-order ODEs by
introducing derivatives as additional solution components. This approach also works for the secondorder
(in time) wave equation (6.2.8):
6.2
p. 670
6.2
p. 67

u(x,t j )
v 0 = 0 ➤ initial data u 0 travel with speed c in
opposite directions
t 6
t 5
t 4
t 3
finite speed of propagation is typical feature
of solutions of wave equations
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Methods
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Domain of influence: intial data in I 0 will be relevant for the solution only in the yellow triangle in
Fig. 193.
✸
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Methods
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t 2
t 1
0
finite speed of propagation
x
Note: (6.2.16) meaningful even for discontinuous
u 0 ,v 0 !
➡ “generalized solutions” !
“point value”u(¯x,¯t),(¯x,¯t) ∈ ˜Ω, may not depend on initial values
outside proper subdomain ofΩ!
Example 6.2.17 (Domain of dependence/influence for 1D wave equation, constant coefficient case).
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✬
Theorem 6.2.18 (Domain of dependence for isotropic wave equation). → [12, 2.5, Thm. 6]
Letu : ˜Ω ↦→ R be a (classical) solution of ∂2 u
∂t2 −c∆u = 0. Then
(|x−x 0 | ≥ R ⇒ u(x,0) = 0 , )
∂u ⇒ u(x,t) = 0 , if |x−x
∂t (x,0) = 0
0 | ≥ R+ct .
✫
The solution formula (6.2.16) clearly indicates that in 1D and in the absence of boundary conditions
the solution of the wave equation will persist undamped for all times.
✩
✪
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Considerd = 1, initial-boundary value problem (6.2.15) for wave equation:
c > 0:
∂ 2 u
∂t 2 −c2∂2 u
∂x 2 = 0 , u(x,0) = u 0(x) ,
Intuitive: from D’Alembert formula (6.2.16)
∂u
∂t (x,0) = v 0(x) , x ∈ R . (6.2.15)
6.2
p. 673
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Methods
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This absence of damping corresponds to a conservation of total energy, which is a distinguishing
feature of conservative wave propagation phenomena.
Now, we examine this for the model problem
6.2
p. 67
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Methods
for PDEs
t
(¯x,¯t)
c
1
D − (¯x,¯t)
domain of dependence of(¯x,¯t) ∈ ˜Ω
Fig. 192
x
t
D + (I 0 )
c
I 0
domain of influence ofI 0 ⊂ R
1
Fig. 193
x
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u ∈ H 1 0 (Ω): ∫
Ω
ρ(x)· ∂2 u
∂t 2 vdx+ ∫
Ω
σ(x)gradu·gradvdx = 0 ∀v ∈ H0 1 (Ω) (6.2.19)
↕
u ∈ V 0 : m(ü,v)+a(u,v) = 0 ∀v ∈ V 0 . (6.2.20)
Here we do not include the case of non-homogeneous spatial Dirichlet boundary conditions through
an affine trial space. This can always be taken into account by offset functions, see the remark after
(6.1.5).
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Domain of dependence: the value of the solution in (¯x,¯t) (•) will depend only on data in the yellow
triangle in Fig. 192.
6.2
p. 674
6.2
p. 67

✬
✩
Theorem 6.2.21 (Energy conservation in wave propagation).
Ifu : ˜Ω ↦→ R solves (6.2.20), then
Numerical
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2nd step: introducce (ordered) basisB N := {b 1 N ,...,bN N } ofV 0,N
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✫
t ↦→ 2 1m(∂u
∂t ,∂u ∂t )+ 2 1 a(u,u) ≡ const .
kinetic energy elastic (potential) energy, see (2.1.3)
✪
(6.2.23) ⇒
⎧ { }
⎨M
d 2
dt 2⃗µ(t) +A⃗µ(t) = 0 for0 < t < T ,
⎩ d⃗µ
⃗µ(0) = ⃗µ 0 , dt (0) = ⃗ν 0 .
(6.2.24)
Proof. A “formal proof” boils down to a straightforward application of the product rule (→
Rem. 6.1.13) together with the symmetry of the bilinear formsmanda.
Introduce the total energy
E(t) := 1 2 m(∂u ∂t ,∂u ∂t )+ 1 2 a(u,u) .
dE
(t) = m(ü, ˙u)+a(˙u,u) = 0 for solutionuof (6.2.20) ,
dt
because this is what we conclude from (6.2.20) for the special test function v(x) = ˙u(x,t) for any
t ∈]0,T[.
✷
6.2.3 Method of lines
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6.2
p. 677
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✄ s.p.d. stiffness matrixA ∈ R N,N ,(A) ij := a(b j N ,bi N ) (independent of time),
✄ s.p.d. mass matrixM ∈ R N,N ,(M) ij := m(b j N ,bi N ) (independent of time),
✄ source (load) vector ⃗ϕ(t) ∈ R N ,(⃗ϕ(t)) i := l(t)(b i N ) (time-dependent),
✄ ⃗µ 0 ˆ= coefficient vector of a projection ofu 0 ontoV 0,N .
✄ ⃗ν 0 ˆ= coefficient vector of a projection ofv 0 ontoV 0,N .
Note:
(6.2.24) is a 2nd-order ordinary differential equation (ODE) fort ↦→ ⃗µ(t) ∈ R N
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6.2
p. 67
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The method of lines approach to the wave equation (6.2.19), (6.2.20) is exactly the same as for the
heat equation, see Sect. 6.1.3.
Remark 6.2.25 (First-order semidiscrete hyperbolic evolution problem).
Idea:
Apply Galerkin discretization (→ Sect. 3.1) to abstract linear parabolic variational problem
(6.1.11).
⎧
⎪⎨ m( d2 u
t ∈]0,T[↦→ u(t) ∈ V 0 : dt 2(t),v)+a(u(t),v) = 0 ∀v ∈ V 0 ,
du
⎪⎩ u(0) = u 0 ∈ V 0 ,
dt (0) = v 0 ∈ V 0 .
1st step: replaceV 0 with a finite dimensional subspaceV 0,N ,N := dimV 0,N < ∞
(6.2.22)
Discrete hyperbolic evolution problem
⎧
m( d2 u N
⎪⎨ dt 2 (t),v N)+a(u N (t),v N ) = 0 ∀v N ∈ V 0,N ,
t ∈]0,T[↦→ u(t) ∈ V 0,N : u N (0) = projection/interpolant ofu 0 inV 0,N , (6.2.23)
du ⎪⎩ N
dt (0) = projection/interpolant ofv 0 inV 0,N .
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DRAFT
6.2
p. 678
Completely analoguous to Rem. 6.2.12:
with intial conditions
M
{ d 2
dt 2⃗µ(t) }
+A⃗µ(t) = 0
← auxiliary unknownν = ˙µ
⎧
d ⎪⎨ ⃗µ(t) = ⃗ν(t) ,
dt
⎪⎩ M d , 0 < t < T . (6.2.26)
dt ⃗ν(t) = −A⃗µ(t) ,
⃗µ(0) = ⃗µ 0 , ⃗ν(0) = ⃗ν 0 . (6.2.27)
△
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DRAFT
6.2
p. 68

6.2.4 Timestepping
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A =N ×N Poisson matrix, see (4.1.3), scaled withh := n −1 ,
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mass matrixM = hI, thanks to quadrature formula, see Rem. 6.2.34.
The method of lines approach gives us the semi-discrete hyperbolic evolution problem =
ODE:
{ }
M d 2
dt 2⃗µ(t) +A⃗µ(t) = 0 , ⃗µ(0) = ⃗µ 0 ,
Key features of (6.2.28) ➡ to be respected “approximately” by timestepping:
reversibility: (6.2.28) invariant under time-reversal t ← −t
2nd-order
d⃗µ
dt (0) = ⃗η 0 . (6.2.28)
energy conservation, cf. Thm. 6.2.21: E N (t) := 1 d⃗µ
2dt ·Md⃗µ dt + 1 2⃗µ·A⃗µ = const
Example 6.2.29 (Euler timestepping for 1st-order form of semi-discrete wave equation).
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Timestepping:
(6.2.26), timestepτ > 0:
implicit and explicit Euler method (→ Ex. 6.1.21, [18, Sect. 12.2]) for 1st-order ODE
⃗µ (j) − ⃗µ (j−1) = τ⃗ν (j−1) ,
M(⃗ν (j) −⃗ν (j−1) ) = −τA⃗µ (j−1) .
explicit Euler
Monitored: behavior of (discrete) kinetic, potential, and total energy
⃗µ (j) − ⃗µ (j−1) = τ⃗ν (j) ,
M(⃗ν (j) −⃗ν (j−1) ) = −τA⃗µ (j) .
implicit Euler
E (j)
kin = (⃗ν(j) ) T M⃗ν (j) , E (j)
pot = (⃗µ(j) ) T A⃗µ (j) , j = 0,1,... .
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Explicit Euler timestepping:
6.2
p. 681
6.2
p. 68
u(x 1
,x 2
)
5
4
3
2
1
0
−1
−2
−3
0
0.2
0.4
0.6
x 1
0.8
1
u(x,t)
1
0.8
0.6
0.4
0.2
0
x 2
Fig. 194
Model problem:
membrane
wave propagation on a square
∂ 2 u
∂t 2 −∆u = 0 on ]0,1[2 ×]0,1[ ,
u(x,t) = 0 on ∂Ω×]0,T[ ,
∂u
u(x,0) = u 0 (x) ,
∂t (x,0) = 0 .
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u
0.06
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
1
0.8
0.6
0.4
0.2
0
t = 1.000000
0
0.2
0.4
0.6
0.8
1
energy
4.5
4
3.5
3
2.5
2
1.5
1
0.5
kinetic energy
potential energy
total energy
Spatial resolution n = 30, 3000 timesteps
Fig. 195 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time t
Fig. 196
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Implcit Euler timestepping:
DRAFT
Initial data u 0 (x) = max{0, 1 5 −‖x‖},v 0(x) = 0,
M ˆ= “structured triangular tensor product mesh”, see Fig. 123,nsquares in each direction,
linear finite element spaceV N,0 = S1,0 0 (M),N := dimS0 1,0 (M) = (n−1)2 ,
All local computations (→ Sect. 3.5.4) rely on 3-point vertex based local quadrature formula “2D
trapezoidal rule” (3.2.17). More explanations will be given in Rem. 6.2.34 below.
6.2
p. 682
6.2
p. 68

0
t = 1.000000
4
3.5
Spatial resolution n = 30, 3000 timesteps
kinetic energy
potential energy
total energy
Numerical
Methods
for PDEs
{ }
M d 2
dt 2⃗µ(t) +A⃗µ(t) = 0 (6.2.28)
Numerica
Methods
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0.04
3
u
0.02
0
−0.02
−0.04
−0.06
energy
2.5
2
1.5
M ⃗µ(j+1) −2⃗µ (j) + ⃗µ (j−1)
τ 2 = −A⃗µ (j) , j = 0,1,... . (6.2.30)
This is a two-step method, the Störmer scheme/explicit trapezoidal rule
−0.08
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
1
0.5
Fig. 197 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time t
Fig. 198
Also runs animations for numerical wave propagation based on expliiti/implicit Euler timestepping
weeul(30,3000).
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DRAFT
By Taylor expansion:
Störmer scheme is a 2nd-order method
However, from where do we get ⃗µ (−1) ? Two-step methods need to be kick-started by a special initial
step: This is constructed by approximating the second initial condition by a symmetric difference
quotient:
d
dt ⃗µ(0) = ⃗ν 0
⃗µ (1) − ⃗µ (−1)
= ⃗ν
2τ 0 . (6.2.31)
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Parallel animation in windows next to each other.
6.2
p. 685
Example 6.2.32 (Leapfrog timestepping).
6.2
p. 68
Observation:
neither method conserves energy,
Numerical
Methods
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For the semi-discrete wave equation we again consider the explicit trapezoidal rule (Störmer scheme):
Numerica
Methods
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☞
☞
explicit Euler timestepping ➣ steady increase of total energy
implicit Euler timestepping ➣ steady decrease of total energy
Ex. 6.2.29 ➣
Euler methods violate energy conservation!
(The same is true of all explicit Runge-Kutta methods, which lead to an increase of
the total energy over time, and L(π)-stable implicit Runge-Kutta method, which make
the total energy decay.)
Let us try another simple idea for the 2nd-order ODE (6.2.24):
Replace
R.
d 2
dt2⃗µ with symmetric difference quotient (1.5.96)
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6.2
p. 686
M ⃗µ(j+1) −2⃗µ (j) + ⃗µ (j−1)
τ 2 = −A⃗µ (j) , j = 1,... . (6.2.30)
Inspired by Rem. 6.2.25 we introduce the auxiliary variable
⃗ν (j+1/2) := ⃗µ(j+1) − ⃗µ (j)
,
τ
which can be read as an approximation of the velocityv := ˙u.
This leads to a timestepping scheme, which is algebraically equivalent to the explicit trapezoidal rule:
leapfrog timestepping (with uniform timestepτ > 0):
M ⃗ν(j+1 2 ) −⃗ν (j−1 2 )
τ
⃗µ (j+1) − ⃗µ (j)
τ
= −A⃗µ (j) ,
= ⃗ν (j+1 2 ) ,
+ initial step ⃗ν (−1 2 ) +⃗ν (1 2 ) = 2⃗ν 0 .
j = 0,1,... , (6.2.33)
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6.2
p. 68

t
⃗µ (j−1) ⃗ν (j−1 2 ) ⃗µ (j) ⃗ν (j+1 2 ) ⃗µ (j+1)
Remark 6.2.34 (Mass lumping).
work per step:
1× evaluationA×vector,
1× solution of linear system forM
✸
Required in each step of leapfrog timestepping: solution of linear system of equations with (large
sparse) system matrixM ∈ R N,N ➣ expensive!
Trick for (bi-)linear finite element Galerkin discretization:
V 0,N ⊂ S 0 1 (M):
use vertex based local quadrature rule
(e.g. “2D trapezoidal rule” (3.2.17) on triangular mesh)
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R. Hiptmair
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DRAFT
6.2
p. 689
Code 6.2.36: Computing behavior of energies for Störmer timestepping
1 f u n c t i o n lfen(n,m)
2 % leapfrom timestepping for 2D wave equation, computation of energies
3 % n: spatial resolution (no. of cells in one direction)
4 % m: number of timesteps
5
6 % Assemble stiffness matrix, see Sect. 4.1, (4.1.3)
7 N = (n-1)^2; h = 1/n; A = g a l l e r y (’poisson’,n-1)/(h*h);
8
9 % initial displacement u 0 (x) = max{0, 1 5 −‖x‖}
10 [X,Y] = meshgrid(0:h:1,0:h:1);
11 U0 = 0.2- s q r t((X-0.5).^2+(Y-0.5).^2);
12 U0( f i n d (U0 < 0)) = 0.0;
13 u0 = reshape(U0(2:end-1,2:end-1),N,1);
14 v0 = zeros(N,1); % initial velocity
15
16 % loop for Störmer timestepping, see (6.2.30)
17 tau = 1/m; % uniform timestep size
18 u = u0+tau*v0-0.5*tau^2*A*u0; % special initial step
19 u_old = u0;
20 [pen,ken] = geten(A,tau,u0,u); % compute potential and kinetic energy
21 E = [0.5*tau,pen,ken,pen+ken];
22 f o r k=1:m-1
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6.2
p. 69
∫
f(x)dx ≈ |K| ∑
f(p) , V(K) := set of vertices ofK .
K ♯V(K)
p∈V(K)
(For a comprehensive discussion of local quadrature rules see Sect. 3.5.4)
Mass matrixMwill become a diagonal matrix (due to defining equation (3.2.3) for nodal basis
functions, which are associated with nodes of the mesh).
This so-called mass lumping trick was was used in the finite element discretization of Ex. 6.2.29.
Example 6.2.35 (Energy conservation for leapfrog).
△
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23 u_new = -(tau^2)*(A*u) + 2*u - u_old;
24 [pen,ken] = geten(A,tau,u,u_new);
25 E = [E;(k+0.5)*tau,pen,ken,pen+ken];
26 u_old = u; u = u_new;
27 end
28
29 f i g u r e (’name’,’Leapfrog energies’);
30 p l o t (E(:,1),E(:,3),’r-’,E(:,1),E(:,2),’b-’,E(:,1),E(:,4),’m-’);
31 x l a b e l (’{\bf time t}’,’fontsize’,14);
32 y l a b e l (’{\bf energies}’,’fontsize’,14);
33 legend(’kinetic energy’,’potential energy’,’total
energy’,’location’,’south’);
34 t i t l e ( s p r i n t f (’Spatial resolution n = %i, %i timesteps’,n,m));
35
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36 p r i n t (’-depsc’, s p r i n t f (’../../../../rw/Slides/NPDEPics/leapfrogen%d.eps
DRAFT
Model problem and discretization as in Ex. 6.2.29.
Leapfrog timestepping with constant timestep sizeτ = 0.01
6.2
p. 690
Code 6.2.37: Computing potential and kinetic energiy for Störmer timestepping
1 f u n c t i o n [pen,ken] = geten(A,ts,u_old,u_new)
2 % Compute the current approximate potential and kinetic energies for u_old
6.2
p. 69

3 % and u_new from Sörmer timestepping
4 %E (j)
kin = τ−2 (⃗µ (j) − ⃗µ (j−1) ) T M(⃗µ (j) − ⃗µ (j−1) ) , E (j)
pot = 1 4 (⃗µ(j) + ⃗µ (j−1) ) T A(⃗µ (j) + ⃗µ (j−1) ) , j = 0,1,... .
5 meanv = 0.5*(u_old+u_new); pen = dot(meanv,A*meanv); % potential
energy
6 dtemp = (u_new-u_old)/ts; ken = dot(dtemp,dtemp); % kinetic energy
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➣
(as in Sect. 6.1.4.2) Stability analysis of leapfrog timestepping based on diagonalization:
∃ orthogonalT ∈ R N,N : T ⊤ M −1/2 AM −1/2 T = D := diag(λ 1 ....,λ N ) .
where the λ i > 0 are generalized eigenvalues for A ⃗ ξ = λM ⃗ ξ ➤
constant introduced in (6.1.12)).
λ i ≥ γ for all i (γ is the
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60
Spatial resolution n = 30, 100 timesteps
Next, apply transformation⃗η := T T M 1/2 ⃗µ to the 2-step formulation (6.2.30)
energies
50
40
30
20
10
kinetic energy
potential energy
total energy
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time t
Fig. 199
Leapfrog is (nearly) energy conserving
(no energy drift, only small oscillations)
This behavior is explained by the deep mathemtical
theorie of symplectic integrators, see [17].
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
(6.2.30)
⃗η:=T ⊤ M 1/2 ⃗µ
=⇒ ⃗η (j+1) −2⃗η (j) +⃗η (j−1) = −τ 2 D⃗η (j) .
Again, we have achieved a complete decoupling of the timestepping for the eigencomponents.
η (j+1)
i −2η (j)
i +η (j−1)
i = −τ 2 λ i η (j)
i , i = 1,...,N , j = 1,2,... . (6.2.39)
In fact, (6.2.39) is what we end up with then applying Störmers scheme to the scalar linear 2nd-order
ODE ¨η i = −λ i η i . In a sense, the commuting diagram (6.1.54) remains true for 2-step methods and
second-order ODEs.
(6.2.39) is a linear two-step recurrence formula for the sequences(η (j)
i ) j
.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
6.2.5 CFL-condition
Example 6.2.38 (Blow-up for leapfrog timestepping).
6.2
p. 693
Numerical
Methods
for PDEs
Try:
Plug this into (6.2.39)
η (j)
i = ξ j for some
ξ ∈ C\{0}
ξ 2 −2ξ +1 = −τ 2 λ i ξ ⇔ ξ 2 −(2−τ 2 λ i )ξ +1 = 0 .
( √ )
⇒ two solutions ξ ± = 2
1 2−τ 2 λ i ± (2−τ 2 λ i ) 2 −4 .
6.2
p. 69
Numerica
Methods
for PDEs
energies
Spatial resolution n = 30, 40 timesteps
6 x 1019 kinetic energy
potential energy
total energy
5
4
3
2
1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time t
Fig. 200
✁ Ex. 6.2.35 repeated withτ = 0.04
Observation:
Leapfrog suffers a blow-up: exponential increase
of energies!
A similar behavior is observed with the explicit Euler
scheme for the semi-discrete heat equation,
in case the timestep constraint is violated, see
Sect. 6.1.4.2.
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
We can get a blow-up of some solutions of (6.2.39), if |ξ + | > 1 of |ξ − | > 1. From secondary school
we know Vieta’s formula
{
}
ξ ± ∈ R and ξ + ≠ ξ − ⇒ |ξ + | > 1 or |ξ − | > 1 ,
ξ +·ξ − = 1 ⇒ {
ξ − = ξ+ ∗ }
⇒ |ξ − | = |ξ + | = 1 ,
where ξ+ ∗ designates complex conjugation. So the recurrence (6.2.39) has only bounded solution, if
and only if
←→
discriminant D := (2−τ 2 λ i ) 2 −4 ≤ 0 ⇔ τ < 2 √ λi
. (6.2.40)
stability induced timestep constraint for leapfrog timestepping
Special setting: spatial finite element Galerkin discretization based on fixed degree Lagrangian finite
element spaces (→ Sect. 3.4), meshes created by uniform regular refinement.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
6.2
p. 694
Under these conditions a generalization of Lemma 6.1.48 shows
6.2
p. 69

Stability of leapfrog timestepping entails τ ≤ O(h M ) forh M → 0
This is known as
Courant-Friedrichs-Lewy (CFL) condition
Remark 6.2.41 (Geometric interpretation of CFL condition in 1D).
Numerical
Methods
for PDEs
t
(¯x,¯t)
h
τ
t
(¯x,¯t)
h
τ
Numerica
Methods
for PDEs
Setting:
t
1D wave equation, (spatial) boundary conditions ignored
c > 0:
∂ 2 u
∂t 2 −c2∂2 u
∂x 2 = 0 , u(x,0) = u 0(x) ,
(“Cauchy problem”),
∂u
∂t (x,0) = v 0(x) , x ∈ R . (6.2.15)
Linear finite element Galerkin discretization on equidistant spatial meshM := {[x j−1 ,x j ]:j ∈
Z},x j := hj (meshwidthh), see Sect. 1.5.1.2.
Mass lumping for computation of mass matrix, which will becomeh·I, see Rem 6.2.34.
Timestepping by Sörmer scheme (6.2.30) with constant timestepτ > 0.
t k+1
t k
t k−1
x j−1 x j x j+1
Fig. 201x
✁ flow of information in one step of Störmer
scheme
Since the method is a two-step method, information
from time-slicest k andt k−1 is needed.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
6.2
p. 697
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
▲
✗
✖
x
Fig. 202
cτ < h: numerical domain of dependence
t
(marked —) contained in d.o.d.
➪ CFL-condition met
(¯x,¯t)
h
x
τ
u 0
Fig. 204
x
Fig. 203
cτ > h: numerical domain of dependence
(marked —) not contained in d.o.d.
➪ CFL-condition violated
(• ˆ= coarse grid, ■ ˆ= fine grid,
✁ 1D consideration:
ˆ= d.o.d)
sequence of equidistant space-time grids of ˜Ω with
τ = γh (τ/h = meshwidth in time/space)
Ifγ > CFL-constraint (hereγ > c −1 ), then
analytical domain
of dependence
numerical domain
⊄
of dependence
initial data u 0 outside numerical domain of dependence cannot influence approximation at grid
point(¯x,¯t) on any mesh no convergence !
CFL-condition ⇔ analytical domain of dependence ⊂ numerical domain of dependence
Will the CFL-condition thwart the efficient use of leapfrog, see Rem. 6.1.67 ?
✔
✕
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
6.2
p. 69
Numerica
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for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Below: yellow region ˆ= domain of dependence (d.o.d.) of(¯x,¯t)
To this end we need an idea about the convergence of the solutions of the fully discrete method:
6.2
p. 698
6.2
p. 70

✬
“Meta-theorem” 6.2.42 (Convergene of fully discrete solutions of the wave equation).
Assume that
the solution of the IBVP for the wave equation (6.2.19) is “sufficiently smooth”,
its spatial Galerkin finite element discretization relies on degree p Lagrangian finite elements
(→ Sect. 3.4) on uniformly shape-regular families of meshes,
timestepping is based on the leapfrog method (6.2.33) with uniform timestepτ > 0.
Then we can expect an asymptotic behavior of the total discretization error according to
( M∑
τ ‖u−u N (τj)‖ 2 2
H
≤C(h p (Ω))1 1 M +τ2 ) ,
j=1
(6.2.43)
( M∑
τ ‖u−u N (τj)‖ 2 )1
2
L 2 (Ω)
≤ C(h p+1
M +τ2 ) , (6.2.44)
j=1
whereC > 0 must not depend onh M ,τ. L.F. is 2nd-order !
✩
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Guideline:
spatial and temporal resolution have to be adjusted in tandem
Parallel zu Rem. 6.1.67 we may wonder whether the timestep constraintτ < O(h M ) (asymptotically)
enforces small timesteps not required for accuracy:
When interested in error in energy norm (↔H 1 (Ω)-norm):
Only for p = 1 (linear Lagrangian finite elements) the requirementτ < O(h M ) stipulates the
use of a smaller timestep than accuracy balancing according to (6.2.46).
When interested inL 2 (Ω)-norm:
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
✫
✪
No undue timestep constraint enforced by CFL-conditon for any (h-version) of Lagrangian finite
element Galerkin discretization.
“expect”: unless lack of regularity of the solutionuinterferes, cf. 5.4, Rem. 5.4.11.
6.2
p. 701
★
✥
6.2
p. 70
As in the case of Metatheorem 6.1.63 (☞ nothing new!) we find:
Numerical
Methods
for PDEs
The leapfrog timestep constraint τ ≤ O(h M ) does not compromise (asymptotic) efficiency,
ifp ≥ 2 (p ˆ= degree of spatial Lagrangian finite elements).
✧
✦
Numerica
Methods
for PDEs
total discretization error = spatial error + temporal error
Rem. 5.3.45 still applies:
(6.2.43) does not give information about actual error, but only about the
trend of the error, when discretization parametersh M andτ are varied.
Nevertheless, as in the case of the a priori error estimates of Sect. 5.3.5, we can draw conclusions
about optimal y refinement strategies in order to achieve prescribed error reduction.
As in Sect. 5.3.5 we make the assumption that the estimates (6.2.43) are sharp for all contributions
to the total error and that the constants are the same (!)
contribution of spatial (energy) error ≈ Ch p M , h M ˆ= mesh width (→ Def. 5.2.3) ,
contribution of temporal error ≈ Cτ 2 , τ ˆ= timestep size .
(6.2.45)
This suggests the following change ofh M ,τ in order to achieve error reduction by a factor ofρ > 1:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
reduce mesh width by factor ρ 1/p
reduce timestep by factor ρ 1/2 (6.1.65)
=⇒ (energy) error reduction byρ > 1 . (6.2.46)
6.2
p. 702
6.2
p. 70

Numerical
Methods
for PDEs
7.1.1 Modelling fluid flow
Numerica
Methods
for PDEs
7 Convection-Diffusion Problems
Flow field:
v : Ω ↦→ R d
Assumption:
OPTIONAL CHAPTER
v is continuous,v ∈ (C 0 (Ω)) d
Dependencies:
• Theory and variational formulation of 2nd-order elliptic BVP, Ch. 2
• Basic concepts and algorithms for finite elements, Sects. 3.1, 3.3, 3.4
R. Hiptmair
C. Schwab,
H.
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V.
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DRAFT
In fact, we will require that v is uniformly Lipschitz
continuous, but this is a mere technical assumption.
Fig. 205
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• 1D finite differences/finite elements Sect. 1.5.1.2
• MATLAB FEM library [6], Sect. 3.5
• Method of lines, Sect. 6.1.3
Clearly:
v(x) ˆ= fluid velocity at pointx ∈ Ω
➣
v corresponds to a velocity field!
Supplementary and further reading:
[25] offers comprehensive about theory and algorithms for the numerical approximation of singularly
perturbed problems of the type treated in this chapter.
7.0
p. 705
Numerical
Methods
for PDEs
Given a flow fieldv ∈ (C 0 (Ω)) d we can consider the autonomous initial value problems
d
dt y = v(y) , y(0) = x 0 . (7.1.1)
Its solution t ↦→ y(t) defines the path travelled by a particle carried along by the fluid, a particle
trajectory, also called a streamline.
7.1
p. 70
Numerica
Methods
for PDEs
7.1 Heat conduction in a fluid
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✁ particle trajectories (streamlines) in flow field
of Fig. 205.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
(∗ ˆ= initial particle positions)
Ω ⊂ R d ˆ= bounded computational domain,d = 1,2,3
To begin with we want to develop a mathematical model for stationary fluid flow, for instance, the
Fig. 206
steady streaming of water.
7.1
7.1
p. 706
p. 70

A flow field induces a transformation (mapping) of space! to explain this, let us temporarily make the
assumption that
Numerical
Methods
for PDEs
Φ τ (V) ˆ= volume occupied at timet = τ by particles that occupiedV ⊂ Ω at timet = 0.
Numerica
Methods
for PDEs
7.1.2 Heat convection and diffusion
the flow does neither enter nor leaveΩ,
(this applies to fluid flow in a close container)
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u : Ω ↦→ R ˆ= stationary temperature distribution in fluid moving according to a stationary flow field
v : Ω ↦→ R d
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C. Schwa
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V.
Gradinar
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which can be modelled by
x 0
Φ t x 0
Fig. 207
DRAFT
We adapt the considerations of Sect. 2.5 that led to the stationary heat equation. Recall
DRAFT
v(x)·n(x) = 0 ∀x ∈ ∂Ω , (7.1.2)
that is, the flow is always parallel to the boundary ofΩ: all particle trajectories stay insideΩ.
Now we fix some “time of interest”t > 0.
{
➣ mapping Φ t Ω ↦→ Ω
: , t ↦→ y(t) solution of IVP (7.1.1) , (7.1.3)
x 0 ↦→ y(t)
7.1
p. 709
✬
✫
Conservation of energy
∫ ∫
j·ndS = f dx for all “control volumes”V . (2.5.2)
∂V V
power flux through surface ofV
heat production insideV
✩
✪
7.1
p. 71
is well-defined mapping ofΩto itself, the flow map. Obviously, it satisfies
Φ 0 x 0 = x 0 ∀x 0 ∈ Ω . (7.1.4)
In [18, Def. 12.1.27] the more general concept of an evolution operator was introduced, which agrees
with the flow map in the current setting.
Numerical
Methods
for PDEs
From 2.5.2 by Gauss’ theorem Thm. 2.4.5
∫ ∫
divj(x)dx = f(x)dx for all “control volumes” V ⊂ Ω .
V
V
Now appeal to another version of the fundamental lemma of the calculus of variations, see
Lemma 2.4.10, this time sporting piecewise constant test functions.
Numerica
Methods
for PDEs
x 2
11
10
9
8
7
q = α
11
10
9
8
7
t=0
t=0.5
t=1
t=2
t=3
t=5
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C. Schwab,
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DRAFT
local form of energy conservation:
divj = f inΩ . (2.5.5)
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H.
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V.
Gradinar
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DRAFT
6
6
5
4
−1.5 −1 −0.5 0 0.5 1 1.5
x 1
Fig. 208
5
4
−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
p
Fig. 209
flow fieldv : Ω ↦→ R 2 snapshots ofΦ t (V) for control volumeV
7.1
p. 710
However, in a moving fluid a power flux through a fixed surface is already caused by the sheer fluid
flow carrying along thermal energy. This is reflected in a modified Fourier’s law (2.5.3):
7.1
p. 71

✬
Fourier’s law in moving fluid
j(x) = −κgradu(x)+v(x)ρu(x) , x ∈ Ω . (7.1.5)
✩
Numerical
Methods
for PDEs
7.1.3 Incompressible fluids
Numerica
Methods
for PDEs
✫
diffusive heat flux
(due to spatial variation of temperature)
convective heat flux
(due to fluid flow)
✪
For the sake of simplicity we will mainly consider incompressible fluids.
κ > 0 ˆ= heat conductivity ([κ] = 1Km W ), ρ > 0 ˆ= heat capavity ([ρ] = J
Km3), both assumed to be
constant (in contrast to the models of Sect. 2.5 and Sect. 6.1.1).
Combine equations (2.5.5) & (7.1.5):
divj = f + j(x) = −κgradu(x)+v(x)ρu(x)
−div(κgradu)+div(ρv(x)u) = f in Ω . (7.1.6)
Linear scalar convection-diffusion equation (for unknown temperatureu)
Terminology :
−div(κgradu) + div(ρv(x)u) = f .
↓ ↓
diffusive term convective term
(2nd-order) (1st-order)
The 2nd-order elliptic PDE (7.1.6) has to be supplemented with exactly one boundary condition on
any part of ∂Ω, see Sect. 2.6, Ex. 2.6.7.
introduced in Sect. 2.6:
Dirichlet boundary conditions:
Neumann boundary conditions:
(non-linear) radiation boundary conditions:
flux, radiative heat flux).
This can be any of the (“elliptic”) boundary conditions
u = g ∈ C 0 (∂Ω) on∂Ω (fixed surface temperatur),
j·n = −h on∂Ω (fixed heat flux),
j · n = Ψ(u) on ∂Ω (temperature dependent heat
R. Hiptmair
C. Schwab,
H.
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A. Chernov
DRAFT
7.1
p. 713
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R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
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DRAFT
Definition 7.1.7 (Incompressible flow field).
A fluid flow is called incompressible, if the associated flow mapΦ t is volume preserving,
|Φ t (V)| = |Φ 0 (V)| for all sufficiently smallt > 0, for all control volumesV .
Can incompressibility be read off the velocity fieldvof the flow?
To investigate this issue, again assume the “no flow through the boundary condition” (7.1.2) and recall
that the flowmapΦ t from (7.1.3) satisfies
∂
Φ(t,x) = v(Φ(t,x)) , x ∈ Ω, t > 0 .
∂t (7.1.8)
Here, in order to make clear the dependence on independent variables, time occurs as an argument
ofΦin brackets, on par withx.
Next, formal differentiation w.r.t. x and change of order of differentiation yields a differential equation
for the JacobianD x Φ t ,
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DRAFT
7.1
(7.1.8) ⇒ ∂ ∂t ( D x Φ)(t,x) = Dv(Φ(t,x))(D x Φ)(t,x) . (7.1.9)
Jacobian ∈ R d,d
Jacobian ∈ R d,d
Second strand of thought: apply transformation formula for integrals (3.5.31), [29, Satz 8.5.2]: for
fixedt > 0
∫ ∫
|Φ(t,V)| = 1dx = |det(D x Φ)(t,̂x)|d̂x . (7.1.10)
Φ(t,V) V
Volume preservation by the flow map is equivalent to
t ↦→ |Φ(t,V)| = const. ⇐⇒ d dt |Φ(t,V)| = 0 ,
p. 71
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Guideline:
Required boundary conditions determined by highest-order term
7.1
p. 714
for any control volumeV ⊂ Ω.
(7.1.10) ⇒ d dt |Φ(t,V)| = ∫
V
∂
∂t |det(D xΦ)(t,̂x)|d̂x .
7.1
p. 71

✬
Theorem 7.1.11 (Differentiation formula for determinants).
LetS : I ⊂ R ↦→ R n,n be a smooth matrix-valued function. If S(t 0 ) is regular for somet 0 ∈ I,
then
✫
d
dt (det◦S)(t 0) = det(S(t 0 )) tr( dS
dt (t 0)S −1 (t 0 )) .
∂
∂t det(D xΦ)(t,̂x) (7.1.9)
= det(D x Φ)(t,̂x) tr(Dv(Φ(t,̂x))(D x Φ)(t,̂x)(D x Φ) −1 (t,̂x) )
} {{ }
=I
= det(D x Φ)(t,̂x) divv(Φ(t,̂x)) ,
because the divergence of a vector fieldvis just the trace of its JacobianDv! From (7.1.4) we know
that for small t > 0 the Jacobian D x Φ(t,̂x) will be close to I and, therefore, det(D x Φ)(t,̂x) ≠ 0
fort ≈ 0. Thus, for smallt > 0 we conclude
d
|Φ(t,V)| = 0 ⇔ divv(Φ(t,̂x)) = 0 ∀̂x ∈ V .
dt
Since this is to hold for any control volumeV, the final equivalence is
✬
R.
d
|Φ(t,V)| = 0 ∀ control volumesV ⇔ divv = 0 inΩ .
dt
Theorem 7.1.12 (Divergence-free velocity fields for incompressible flows).
A stationary fluid flow inΩis incompressible (→ Def. 7.1.7), if and only if its associated velocity
fieldvsatisfiesdivv = 0 everywhere inΩ.
✫
✩
Numerical
Methods
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✪
✩
✪
Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.1
p. 717
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Methods
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←divv = 0
−κ∆u+ρv·gradu = f in Ω . (7.1.14)
When carried along by the flow of an incompressible fluid, the temperature cannot be increased by
local compression, the effect that you can witness when pumping air. Hence, only sources/sinks can
lead to local extrema of the temperature.
Now recall the discussion of the physical intuition behind the maximum principle of Thm. 5.7.2. These
considerations still apply to stationary heat flow in a moving incompressible fluid.
✬
Theorem 7.1.15 (Maximum principle for scalar 2nd-order convection diffusion equations).
[12, 6.4.1, Thm. I]
Let v : Ω ↦→ R d be a continuously differentiable vector field. Then there holds the maximum
principle
✫
−∆u+v·gradu ≥ 0 =⇒ min u(x) = min
x∈∂Ω
−∆u+v·gradu ≤ 0 =⇒ max
x∈∂Ω
7.1.4 Transient heat conduction
u(x) , x∈Ω
u(x) = max u(x) . x∈Ω
In Sect. 6.1.1 we generalized the laws of stationary heat conduction derived in Sect. 2.5 to timedependent
temperature distributions u = u(x,t) sought on a space-time cylinder ˜Ω := Ω×]0,T[.
The same ideas apply to heat conduction in a fluid:
→
✩
✪
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
7.1
p. 71
Numerica
Methods
for PDEs
d∑ ∂v j
In the sequel we make the assumption: divv = = 0 .
∂x
j=1 j
(Note: ford = 1 this boils down to
dx dv = 0 and impliesv = const.)
R. Hiptmair
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V.
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DRAFT
• Start from energy balance law (6.1.1) and convert it into local form (6.1.2).
• Combine it with the extended Fourier’s law (7.1.5).
R. Hiptm
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DRAFT
Then we can use the product rule in higher dimensions of Lemma 2.4.3:
Lemma 2.4.3
div(ρvu) = ρ(u divv+v·gradu) divv=0 = ρv·gradu . (7.1.13)
∂
∂t (ρu)−div(κgradu)+div(ρv(x,t)u) = f(x,t) in ˜Ω := Ω×]0,T[ . (7.1.16)
Thus, we can rewrite the scalar convection-diffusion equation (7.1.6) for an incompressible flow field
For details and notations refer to Sect. 6.1.1.
−div(κgradu)+div(ρv(x)u) = f in
Ω
7.1
p. 718
This PDE has to be supplemented with
7.1
p. 72

• boundary conditions (as in the stationary case, see Sect. 2.6),
• initial conditions (same as for pure diffusion, see Sect. 6.1.1).
Under the assumption div x v(x,t) = 0 of incompressibilty (→ Def. 7.1.7 and Thm. 7.1.12) (7.1.16)
is equivalent to, cf. (7.1.13),
Numerical
Methods
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Remark 7.2.2 (Variational formulation for convection-diffusion BVP).
Standard “4-step approach” of Sect. 2.8 can be directly applied to BVP (7.2.1) with one new twist:
Do not use integration by parts (Green’s formula, Thm. 2.4.7) on convective terms!
Numerica
Methods
for PDEs
∂
∂t (ρu)−κ∆u+ρv(x,t)·gradu = f(x,t) in ˜Ω := Ω×]0,T[ . (7.1.17)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
variational formulation for BVP (7.2.1):
∫
∫ ∫
u ∈ H0 1 (Ω): ǫ gradu·gradvdx+ (v·gradu)vdx = f(x)dx
Ω
Ω
} {{ }
bilinear forma(u,v)
ˆ= a linear variational problem, see Sect. 2.3.1.
Ω
} {{ }
linear forml(v)
∀v ∈ H 1 0 (Ω) .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
7.2 Stationary convection-diffusion problems
Obvious: a is not symmetric, see (2.1.19).
➥ a does not induce an energy norm (→ Def. 2.1.27)
7.2
p. 721
As replacement for the energy norm useH 1 (Ω)-(semi)norm (→ Def. 2.2.10)
7.2
p. 72
Model problem, cf. (7.1.14), modelling stationary heat flow in an incompressible fluid with prescribed
temperature at “walls of the container” (↔ Dirichlet boundary conditions).
−κ∆u+ρv(x)·gradu = f inΩ , u = 0 on∂Ω .
Perform scaling ˆ= choice of physical units: makes equation non-dimensional by fixing “reference
length”, “reference time interval”. “reference temperature”, “reference power”.
Numerical
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In this case we have to make sure thatafits the chosen norm in the sense that
∃C > 0: |a(u,v)| ≤ C|u| H 1 (Ω) |v| H 1 (Ω) ∀u,v ∈ H0 1 (Ω) . (7.2.3)
←→ Terminology: (7.2.3) ˆ=ais continuous onH 1 (Ω), cf. (3.1.2).
By Cauchy-Schwarz inequality for integrals (2.1.28):
A suitable choice of physical units leads to rescaled physical constantsκ→ǫ,ρ→1,‖v‖ L ∞ (Ω) →1.
|a(u,v)| ≤ ‖v‖
R. Hiptmair L ∞ (Ω) |u| H 1 (Ω) ‖v‖ Thm. 2.2.16
L 2 (Ω) ≤ diam(Ω)‖v‖ L ∞ (Ω) |u| H 1 (Ω) |v| H 1 (Ω) ∀u,v ∈ H0 1 (Ω)
C. Schwab,
After scaling we deal with the non-dimensional boundary value problem
H.
Harbrecht which confirms (7.2.3) ]
V.
Gradinaru
A. Chernov
DRAFT
DRAFT
Surprise: a is positive definite (→ Def. 2.1.25), because
−ǫ∆u+v(x)·gradu = f inΩ , u = 0 on∂Ω , (7.2.1)
∫ ∫
(v·gradu)udx = (vu)·gradudx
diffusion term
convection term
Ω
Ω ∫ ∫
Green’s formula
(2nd-order term)
(1st-order term)
= − div(vu)udx+
}{{}
u 2 v·ndS
withǫ > 0,‖v‖ L ∞ (Ω) = 1, divv = 0 → incompressible fluid, see Def. 7.1.7 . Ω ∂Ω =0
∫
(2.4.4) & divv=0
= − (v·gradu)udx .
7.2
7.2
p. 722
Ω
p. 72
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern

∫
a(u,u) = ǫ ‖gradu‖ 2 dx > 0 ∀u ∈ H0 1 (Ω)\{0} . (7.2.4)
Ω
From this and (7.2.3) we conclude existence and uniqueness of solutions of the BVP (7.2.1) in the
Sobolev spaceH 1 0 (Ω).
△
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Cased = 1 (Ω =]0,1[,v = ±1)
(7.2.6)
What about this constantC?
d=1
± du
dx (x) = f(x) ⇒ u(x) = ∫
f dx+C . (7.2.7)
Ifv = 1 ↔ fluid flows “from left to right”, so we should integrate the source from0tox:
∫x ∫x
u(x) = u(0)+ f(s)ds = f(s)ds , (7.2.8)
0
because u(0) = 0 by the boundary condition u = 0 on ∂Ω. If v = −1 we start the integration at
x = 1. Note that this makes the maximum principle of Thm. 7.1.15 hold.
0
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
7.2.1 Singular perturbation
Ford > 1 we can solve (7.2.6) by
the method of characteristics:
Setting: fast-moving fluid↔convection dominates diffusion↔ǫ≪1in (7.2.1)
.
Example 7.2.5 (1D convection-diffusion boundary value problem).
Forǫ ≪ 1:
Pointwise limit:
−ǫ d2 u ǫ
dx 2 + du ǫ
dx = 1 inΩ ,
u ǫ (0) = 0 , u ǫ (1) = 0 ,
u ǫ (x) = x+ exp(−x/ǫ)−1
1−exp(−1/ǫ) .
boundary layer atx = 1
lim u ǫ(x) → x ∀0 < x < 1 .
ǫ→0
u(x)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
ε = 1
ε = 1.000000e−01
ε = 1.000000e−02
ε = 1.000000e−03
0
0 0.2 0.4 0.6 0.8 1
x
Fig. 210
✸
7.2
p. 725
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
To motivate it, be aware that (7.2.6) describes pure transport of a temperature distribution in the
velocity field v, that is, the heat/temperature is just carried along particle trajectories and changes
only under the influence of heat sources/sinks along that trajectory.
Denote byuthe solution of (7.2.6) and recall the differential equation (7.1.1) for a particle trajectory
➣
dy
dt (t) = v(y(t)) , y(0) = x 0 . (7.1.1)
d
d
(7.2.6)
u(y(t)) = gradu(y(t))· y(t) = gradu·v(y(t)) = f(y(t)) .
dt dt
Computeu(y(t)) by integrating sourcef along particle trajectory!
∫t
u(y(t)) = u(x 0 )+
0
f(y(s))ds (7.2.9)
Taking the cue fromd = 1 we choosex 0 as “the point on the boundary where the particle entersΩ”.
These points form the part of the boundary through which the flow entersΩ, the inflow boundary
Its complement in∂Ω contains the outflow boundary
Γ in := {x ∈ ∂Ω: v(x)·n(x) < 0} . (7.2.10)
Γ out := {x ∈ ∂Ω: v(x)·n(x) > 0} . (7.2.11)
7.2
p. 72
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
“Limit problem”: ignore diffusion ➣ set ǫ = 0
ǫ=0
(7.2.1) v(x)·gradu = f(x) inΩ . (7.2.6)
7.2
p. 726
Remark 7.2.12 (Streamlines).
7.2
p. 72

→ velocity field
11
10
9
Numerical
Methods
for PDEs
In mathematical terms, singular perturbation for boundary values for PDEs is defined as a change
of type of the PDE for ǫ = 0: in the case of (7.2.1) the type changes from elliptic to hyperbolic, see
Rem. 2.0.1.
Numerica
Methods
for PDEs
—: Streamline connectingΓ in andΓ out
8
x 2
—: Closed streamline
(recirculating flow)
7
6
7.2.2 Upwinding
5
4
−1.5 −1 −0.5 0 0.5 1 1.5
x 1
Fig. 211
In the case of closed streamlines the stationary pure transport problem fails to have a unique solution:
on a closed streamlineucan attain “any” value, because there is no boundary value to fixu.
△
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.2
p. 729
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
R. Hiptm
Focus: linear finite element Galerkin discretization for 1D model problem, cf. Ex. 7.2.5
DRAFT
−ǫ d2 u
dx 2 + du = f(x) inΩ , u(0) = 0 , u(1) = 0 . (7.2.14)
dx
Variational formulation, see Rem. 7.2.2:
∫1
∫1
∫1
u ∈ H0 1 (]0,1[): ǫ du dv
(x)
dx dx (x)dx+ du
dx (x)v(x)dx = f(x)v(x)dx ∀v ∈ H0 1 (]0,1[) .
0 0 0
} {{ } } {{ }
=:a(u,v)
=:l(v)
7.2
p. 73
Return to case d = 1. In general solution u(x) from (7.2.7) will not satisfy the boundary condition
u(1) = 0! Also for u(x) from (7.2.9) the homogeneos boundary conditions may be violated where
the particle trajectory leavesΩ!
Numerical
Methods
for PDEs
As in Sect. 1.5.1.2: use equidistant meshM(mesh widthh > 0), composite trapezoidal rule (1.5.59)
for right hand side linear form, standard “tent function basis”, see (1.5.53).
Numerica
Methods
for PDEs
In the limit caseǫ = 0 not all boundary conditions of (7.2.1) can be satisfied.
Notion 7.2.13 (Singularly perturbed problem).
A boundary value problem depending on parameterǫ ≈ ǫ 0 is called singularly perturbed, if the
limit problem forǫ → ǫ 0 is not compatible with the boundary conditions.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
linear system of equations for coefficients µ i , i = 1,...,M − 1, providing approximations for
point valuesu(ih) of exact solutionu.
(
− ǫ h − 1 )
µ
2 i−1 + 2ǫ
h µ i +
(
− ǫ h + 1 2
)
µ i+1 = hf(ih) , i = 1,...,M −1 , (7.2.15)
where the homogeneous Dirichlet boundary conditions are taken into account by settingµ 0 = µ M =
0.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Especially in the case of 2nd-order elliptic boundary value problems:
Singular perturbation = 1st-order terms become dominant forǫ → ǫ 0
Remark 7.2.16 (Finite differences for convection-diffusion equation in 1D).
As in Sect. 1.5.3 on the finite difference in 1D, we can also obtain (7.2.15) by replacing the derivatives
7.2
p. 730
7.2
p. 73

y suitable difference quotients:
−ǫ d2 u du
dx 2 + = f(x)
dx
↕ ↕ ↕
ǫ −µ i+1 +2µ i −µ i−1
}
h
{{ 2 + µ i+1 −µ i−1
} }
2h
{{ }
difference quotient for d2 u
dx 2 symmetric d.q. for du
dx
= f(ih) .
Example 7.2.17 (Linear FE discretization of 1D convection-diffusion problem).
(7.2.15)
△
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
In order to understand this observation, study the linear finite element Galerkin discretization in the
limit caseǫ = 0
(7.2.15)
ǫ=0
µ i+1 −µ i−1 = 2hf(ih) , i = 1,...,M . (7.2.18)
(7.2.18) ˆ= Linear system of equations with singular system matrix!
Forǫ > 0 the Galerkin matrix will always be regular due to (7.2.4), but the linear relationship (7.2.18)
will become more and more dominant as ǫ > 0 becomes smaller and smaller. In particular, (7.2.18)
sends the message that values at even and odd numbered nodes will become decoupled, which
accounts for the oscillations.
Desired: robust discretization of (7.2.14)
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
= discretization that produces qualitatively correct(∗) solutions for anyǫ > 0
Model boundary value problem (7.2.14)
(∗): “qualitatively correct”, e.g., satisfaction of maximum principle, Thm. 7.1.15]
linear finite element Galerkin discretization as described above
As in Ex. 7.2.5: f ≡ 1
7.2
p. 733
Guideline:
7.2
p. 73
u(x)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
ε = 1
ε = 1.000000e−01
ε = 1.000000e−02
ε = 1.000000e−03
0
0 0.2 0.4 0.6 0.8 1
x
Fig. 212
exact solutions
u N
(x)
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
ε = 1
ε = 1.000000e−01
ε = 1.000000e−02
ε = 1.000000e−03
For very smallǫ: spurious oscillations of linear FE Galerkin solution.
Linear finite elements, M=30
0
0 0.2 0.4 0.6 0.8 1
x
Fig. 213
FE solutions
✸
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Numerical methods for singularly perturbed problems must “work” for the limit problem
What is a meaningful scheme for limit problemu ′ = f on an equidistant mesh ofΩ :=]0,1[?
Explicit Euler method: µ i+1 −µ i = hf(ξ i ) i = 0,...,N ,
Implicit Euler method: µ i+1 −µ i = hf(ξ i+1 ) i = 0,...,N .
Use one-sided difference quotients for discretization of convective term !
Which type ? (Explicit or implicit Euler ?)
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
du
Linear system arising from use of backward difference quotient = µ DRAFT
i−µ i−1
:
dx|x=x i h
(− ǫ ) ( ) 2ǫ
h −1 µ i−1 +
h +1 µ i +− ǫ h µ i+1 = hf(ih) , i = 1,...,M −1 , (7.2.19)
du
Linear system arising from use of forward difference quotient = µ i+1 −µ i
:
dx|x=x i h
− ǫ ( ) 2ǫ
7.2 h µ i−1+
h −1 µ i +
(− ǫ )
h +1 µ i+1 = hf(ih) , i = 1,...,M −1 , (7.2.20)
7.2
p. 734
p. 73

Example 7.2.21 (One-sided difference approximation of convective terms).
Model problem of Ex. 7.2.17, discretizations (7.2.19) and (7.2.20).
Numerical
Methods
for PDEs
Nodal finite element Galerkin discretization ˆ= basis expansion coefficients µ i of Galerkin solution
u N ∈ V N double as point values of u N at interpolation nodes. This is satisfied for Lagrangian finite
element methods (→ Sect. 3.4) when standard nodal basis functions according to (3.4.3) are used.
Numerica
Methods
for PDEs
1
0.9
0.8
ε = 1
ε = 1.000000e−01
ε = 1.000000e−02
ε = 1.000000e−03
Upwind discretization, M=30
1
0.5
ε = 1
ε = 1.000000e−01
ε = 1.000000e−02
ε = 1.000000e−03
Downwind discretization, M=30
Recall the sign-conditions (5.7.9)–(5.7.11) for the system matrix A arising from nodal finite element
Galerkin discretization or finite difference discretization:
u N
(x)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1
x
Fig. 214
backward difference quotient
u N
(x)
0
−0.5
−1
−1.5
0 0.2 0.4 0.6 0.8 1
x
Fig. 215
forward difference quotient
Only the discretization of du
dx
based on the backward difference quotient generates qualitatively correct
(piecewise linear) discrete solutions (a “good method”).
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.2
p. 737
(5.7.9): positive diagonal entries, (A) ii > 0 ,
(5.7.10): non-positive off-diagonal entries, (A) ij ≤ 0, ifi ≠ j ,
“(5.7.11)”: diagonal dominance,
∑
These conditions are met for equidistant meshes in 1D
j (A) ij ≥ 0 .
for the standardS 0 1 (M)-Galerkin discretization (7.2.15), provided that |ǫh−1 | ≥ 1 2 ,
when using backward difference quotients for the convective term (7.2.19) for any choice ofǫ ≥ 0,
h > 0,
when using forward difference quotients for the convective term (7.2.20), provided that
|ǫh −1 | ≥ 1.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
7.2
p. 73
If the forward difference quotient is used, the discrete solutions may violate the maximum principle of
Thm. 7.1.15 (a “bad method”).
Numerical
Methods
for PDEs
Only the use of a backward difference quotient for the convective term guarantees the (discrete)
maximum principle in anǫ → 0-robust fashion!
Numerica
Methods
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How can we tell a good method from a bad method by merely examining the system matrix?
Terminology:
Approximation of du
dx
by backward difference quotients
ˆ= upwinding
Example 7.2.22 (Spurious Galerkin solution for 2D convection-diffusion BVP).
Heuristic criterion forǫ → 0-robust stability of nodal finite element Galerkin discretization/finite
difference discretization of singularly perturbed scalar linear convection-diffusion BVP (7.2.1)
(with Dirichlet b.c.):
(Linearly interpolated) discrete solution satisfies maximum principle (5.7.3).
⇕
System matrix complies with sign-conditions (5.7.9)–(5.7.11).
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Triangle domain Ω = {(x,y) : 0 ≤ x ≤ 1,−x ≤ y ≤ x}.
Velocityv(x) = ( 1)
0 ➣ (7.2.1) becomes−ǫ∆u+u x = 1.
Exact solution: u ǫ (x 1 ,x 2 ) = x − 1
1−e −1/ǫ(e−(1−x 1)/ǫ − e −1/ǫ ), Dirichlet boundary conditions
set accordingly
Standard Galerkin discretization by means of linear finite elements on sequence of triangular
mesh created by regular refinement.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
7.2
p. 738
7.2
p. 74

0 0.2 0.4 0.6 0.8 1
1
0.8
0.6
0.4
2D triangular mesh
800
600
400
Numerical
Methods
for PDEs
We opt for the composite trapezoidal rule
∫ 1
0
M−1 ∑
ψ(x)dx ≈ h ψ(jh) , for ψ ∈ C 0 (]0,1[), ψ(0) = ψ(1) = 0 .
j=1
Numerica
Methods
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0.2
0
200
for evaluation of convective term in bilinear forma:
−0.2
−0.4
−0.6
−0.8
−1
Fig. 216
# Vertices : 15, # Elements : 16, # Edges : 30
Coarse initial mesh
Exact solution
(ǫ = 10 −10 )
Fig. 217
0
−200
−400
−600
Fig. 218
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Standard Galerkin solution onx 2 = 0-line
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
∫1
0
du N
dx (x)v N(x)dx ≈ h
M−1 ∑
j=1
du N
dx (jh)v(hj) , v N ∈ S1,0 0 (M) . (7.2.23)
Note: this is not a valid formula, because du N
dx (jh) is ambiguous, since du N
dx
is discontinuous at
nodes of the mesh for u N ∈ S1,0 0 (M)!
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
As expected:
spurious oscillations mar Galerkin solution
➣ Difficulty observed in 1D also haunts discretization in higher dimensions.
Up to now we resolved this ambiguity by the policy of local quadrature, see Sect. 3.5.4: quadrature
rule applied locally on each cell with all information taken from that cell.
✸
7.2
p. 741
However:
✗
✖
Convection transports information in the direction ofv!
✔
✕
7.2
p. 74
Issue: extension of upwinding idea tod > 1
Numerical
Methods
for PDEs
Idea:
Use upstream/upwind information to evaluate du N
dx (jh) in (7.2.23)
Numerica
Methods
for PDEs
du N du
(jh) := lim N
dx δ→0 dx (jh−δ) = du N
.
dx |]x j−1 ,x j [
7.2.2.1 Upwind quadrature
ˆ= upwind quadrature
v
Revisit 1D model problem
−ǫ d2 u
dx 2 + du = f(x) inΩ , u(0) = 0 , u(1) = 0 , (7.2.14)
dx
with variational formulation, see Rem. 7.2.2:
convective term
∫1
u ∈ H0 1 (]0,1[): ǫ
∫1
du
dx (x)du dx (x)dx+ du
dx (x)v(x)dx
0 0
} {{ }
=:a(u,v)
∫1
=
f(x)v(x)dx
0
} {{ }
=:l(v)
∀v ∈ H 1 0 (]0,1[) .
Linear finite element Galerkin discretization on equidistant meshMwithM cells, meshwidthh = 1 M ,
cf. Sect. 1.5.1.2.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.2
p. 742
x 0 x 1 x 2 x 3 x 4 x 5 x 6
x7
x
Fig. 219
Upwind quadrature yields the following contribution of the discretized convective term to the linear
system using the basis expansion u N = ∑ M−1
l=1 µ l b l N
into locally supported nodal basis functions
(“tent functions”)
where we used
∫1
0
M−1 ∑
l=1
db l µ N l
dx (x)bi (7.2.23)
N (x)dx ≈ h µ i −µ i−1
,
h
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
7.2
p. 74

•b i N (jh) = δ ij, see (1.5.54),
• du N
dx = µ i −µ i−1
from (1.5.55).
|]x j−1 ,x j [ h
Numerical
Methods
for PDEs
Idea: Use upstream/upwind information to evaluategradu N (p) in (7.2.24)
v(p)·gradu N (p) := lim δ→0
v(p)·gradu N (p−δv(p)) . (7.2.25)
ˆ= general upwind quadrature
Numerica
Methods
for PDEs
Linear system from upwind quadrature:
(− ǫ ) ( ) 2ǫ
h −1 µ i−1 +
h +1 µ i +− ǫ h µ i+1 = hf(ih) , i = 1,...,M −1 , (7.2.19)
which is the same as that obtained from a backward finite difference discretization of du
dx !
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Note: By (7.1.1) the vectorv(p) supplies the direction of the streamline throughp. Hence,−v(p) is
the direction from which information is “carried intop” by the flow.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
The idea of upwind quadrature can be generalized tod > 1: we considerd = 2 and linear Lagrangian
finite element Galerkin discretization on triangular meshes, see Sect. 3.2.
➊
Approximation of contribution of convective terms to bilinear form by means of global trapezoidal
rule:
7.2
p. 745
7.2
p. 74
∫
(v·gradu)vdx ≈ ∑ ( ∑ ) 13 |K| · v(p)·gradu(p)v(p) . (7.2.24)
Ω
p∈N(M) K∈U p
ambiguous foru ∈ S1 0(M) !
✎ notation: U p := {K ∈ M:p ∈ K}
v(p)
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Contribution of convective term to thei-th linear of
the final linear system of equations (test function =
tent functionb i N )
( 13
∑
)
|K| v(x i )· gradu N|Ku ,
K∈U i
} {{ }
=:U i
whereK u is the upstream triangle ofp.
✄
x j
v(x i )
n k x i
K u
n j
n i x k
Fig. 221
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
➋ Fix the ambiguous value of v(p) ·
gradu N (p), u N ∈ S1 0 (M), by taking
the gradient from the triangle upstream to
the nodep:
p
upstream triangle
Fig. 220
7.2
p. 746
Using the expressions for the gradients of barycentric coordinate functions from Sect. 3.2.5 and the
nodal basis expansion of u N , we obtain for the convective contribution to the i-th line of the final
linear system
⎛
⎞
U ∥ i
∥∥x
⎜
2|K u | ⎝ − j −x k∥ ∥ ∥ni ·v(x i ∥
) µ i −∥x i −x j∥ ∥ ∥nk ·v(x i ∥
)µ k −∥x i −x k∥ ∥ ∥nj ·v(x i )µ j ⎟
} {{ }
⎠
↔ diagonal entry
7.2
p. 74

By the very definition of the upstream triangleK u we find
n i·v(x i ) ≤ 0 , n k ·v(x i ) ≥ 0 , n j ·v(x i ) ≥ 0 .
➣ sign conditions (5.7.9), (5.7.10) are satisfied, (5.7.11) is obvious.
Example 7.2.26 (Upwind quadrature discretization).
Numerical
Methods
for PDEs
Upwind quadrature scheme respects maximum principle, whereas the standard Galerkin solution is
rendered useless by spurious oscillations.
✸
Numerica
Methods
for PDEs
All the numerical examples can be found in the folder LehrFEM/Examples/DiffConv/. The
necessary data is produced by running main_exp2.m and running plot_exp2.m produces the
plots and saves everything as *.eps files.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.2.2.2 Streamline diffusion
We take another look at the 1D upwind discretization of (7.2.14) and view it from a different perspective.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Ω = [0,1] 2
−ǫ∆u+ ( 1
1
)·gradu = 0
Dirichlet boundary conditions: u(x,y) = 1 forx > y andu(x,y) = 0 for x ≤ y
Limiting case (ǫ → 0): u(x,y) = 1 for x > y andu(x,y) = 0 forx ≤ y
layer along the diagonal from ( 0
1
)
to
( 10 )
in the limitǫ → 0
linear finite element upwind quadrature discretization on triangular mesh.
Monitored: discrete solutions along diagonal from ( 0
1
)
to
( 10 )
forǫ = 10 −10 .
7.2
p. 749
Numerical
Methods
for PDEs
1D upwind (finite difference) discretization of (7.2.14):
(− ǫ ) ( ) 2ǫ
h −1 µ i−1 +
h +1 µ i +− ǫ h µ i+1 = hf(ih) .i = 1,...,M −1 . (7.2.19)
fori = 1,...,M −1.
(ǫ+h/2) −µ i−1+2µ i −µ i+1
h 2 } {{ }
ˆ= difference quotient for d2 u
⇕
dx 2 +
−µ i−1 +µ i+1
= f(ih) ,
}
2h
{{ }
ˆ= difference quotient for du
dx
7.2
p. 75
Numerica
Methods
for PDEs
1.4
1.2
1
0.8
0.6
0.4
10000
8000
6000
4000
2000
0
−2000
−4000
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✬
✫
Upwinding = h-dependent enhancement of diffusive term
artificial diffusion/viscosity
We also observe that the upwinding strategy just ads the minimal amount of diffusion to make the
resulting system matrix comply with the conditions (5.7.9)–(5.7.11), which ensure that the discrete
solution satisfies the maximum principle.
✩
✪R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
0.2
0
Fig. 222
0 0.5 1 1.5
upwind quadrature solution
−6000
Fig. 223
−8000
0 0.5 1 1.5
standard Galerkin solution
7.2
p. 750
Issue: How to extend the trick of adding artificial diffusion tod > 1 ?
Well, just add an extrah-dependent multiple of−∆! Let’s try.
7.2
p. 75

x 1
x 1
x 1
x 1
Example 7.2.27 (Effect of added diffusion).
Convection-diffusion boundary value problem ((7.2.1) withv = ( 1
0
)
)
−ǫ∆u+ ∂u
∂x 1
= 0 inΩ =]0,1[ 2 , u = g on∂Ω .
Numerical
Methods
for PDEs
u = 1
u = 1
u ≈ 1
v
u = 0
Pure transport problem:
v·gradu = 0 inΩ ,
whereΩ =]0,1[ 2 ,v = ( 2
1
) ,ǫ = 10 −4 ,
Numerica
Methods
for PDEs
Here, Dirichlet datag(x) = 1−2|x 2 − 1 2 |.
Thus, for ǫ ≈ 0 we expect u ≈ g, because the Dirichlet data are just transported in x 1 -direction and
there are no boundary layers.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Internal layer
u = 0
u ≈ 0
Fig. 224
Dirchlet b.c. that can only partly be fulfilled: u = 1
on {x 1 = 0} ∪ {x 2 = 1}, u = 0 on {x 1 = 1} ∪
{x 2 = 0}
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1
Grid with 300 x/y−cells, ε = 1.000000
1
Grid with 300 x/y−cells, ε = 0.100000
1
Grid with 300 x/y−cells, ε = 0.010000
1
Grid with 300 x/y−cells, ε = 0.001000
0.9
0.9
0.9
0.9
0.8
0.7
0.6
0.8
0.7
0.6
0.8
0.7
0.6
0.8
0.7
0.6
Solution of pure transport problem with discontinuous boundary data
0.5
0.5
0.5
0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 2
ǫ = 1
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 2
ǫ = 0.1
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 2
ǫ = 0.01
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 2
ǫ = 0.001
Stronger diffusion leads to “smearing” of features that the flow field transports into the interior of the
domain.
Remark 7.2.28 (Internal layers).
(Too much) artificial diffusion ➣ smearing of internal layers
(We are no longer solving the right problem!)
✸
7.2
p. 753
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
• displays a discontinuity across the streamline emanating from the point of discontinuity on∂Ω,
• is smooth along streamlines.
Qualitative solution of
−δ∆+v·gradu = 0 inΩ ,
withδ > 0, the same boundary data
➣ Smearing of internal layer !
We would also find a boundary layer which is omitted
in the figure.
✄
u = 1
u = 1
u ≈ 1
v
Internal layer
u = 0
u ≈ 0
Fig. 225
u = 0
△
7.2
p. 75
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Heuristics:
If the solution is smooth along streamlines, then adding diffusion in the direction of
streamlines cannot do much harm.
7.2
p. 754
What does “diffusion in a direction” mean?
7.2
p. 75

☞
Think of a generalized Fourier’s law (2.5.3) for d = 2, e.g„
( ) 1 0
j(x) = − gradu(x) .
0 0
This means, only a temperature variation inx 1 -direction triggers a heat flow.
Numerical
Methods
for PDEs
Note: the variational crimes investigated in Sect. 5.5 represent non-consistent modifications.
Ensuring consistency for streamline-upwind variational problem:
Numerica
Methods
for PDEs
➥ diffusion in a directionv ∈ R 2 j(x) = −vv T gradu(x) (7.2.29)
Such an extended Fourier’s law is an example of anisotropic diffusion.
Anisotropic diffusion can simply be taken into account in variational formulations and Galerkin discretization
by replacing the heat conductivity κ/stiffness σ with a symmetric, positive (semi-)definite
matrix, the diffusion tensor.
Idea:
Anisotropic artificial diffusion in streamline direction
On cellK replace: ǫ ← ǫI+δ K v K vK
T ∈ R 2,2 .
} {{ }
new diffusion tensor
v K ˆ= local velocity (e.g., obtained by averaging)
δ K > 0 ˆ= method parameter controlling the streng of anisotropic diffusion
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.2
p. 757
∫
Idea:
Add anistropic diffusion through a residual term that vanishes for the exact
solutionu
streamline-upwind variational problem: given meshMseeku ∈ H 1 0 (Ω)∩H2 (M)
ǫgradu·gradv +v(x)·graduvdx
Ω
+ ∑ ∫
∫
δ K (−ǫ∆u+v·gradu−f)·(v·gradv)dx =
K∈M
}
K
{{ }
Ω
stabilization term
☞
fvdx ∀v ∈ H0 1 (Ω) . (7.2.32)
Note that enhanced smoothness ofu, namely in additionu ∈ H 2 (K) for allK ∈ M, is required
to render (7.2.32) meaningful (→ Sobolev spaceH 2 (M)).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
7.2
p. 75
This idea underlies the so-called
streamline-diffusion method.
Numerical
Methods
for PDEs
Note: in the case of Galerkin discretization based on V N,0 = S 0 1 (M), we find ∆u N = 0 in each
K ∈ M.
Numerica
Methods
for PDEs
Thus, (for the model problem) Galerkin discretization may target the variational problem
∫
∫
( ǫI+δK v K vK
T ) gradu·gradv +v(x)·graduvdx = fvdx ∀v ∈ H0 1 (Ω) . (7.2.30)
Ω
!
Desirable:
This tampering affects the solutionu
(solution of (7.2.30)≠solution of (7.2.1))
Maintain consistency of variational problem!
Ω
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
For Galerkin discretization of (7.2.32) by means of linear Lagrangian finite elements, the local control
parametersδ K are usually chosen according to the rule
⎧
⎪⎨ ǫ −1 h 2 ‖v‖ K,∞ h K
K
δ K :=
, if ≤ 1 ,
2ǫ
‖v‖
⎪⎩ K,∞ h K
h K , if > 1 .
2ǫ
which is suggested by theoretical investigations and practical experience.
Example 7.2.33 (Streamline-diffusion discretization).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Exactly the same setting as in Ex. 7.2.26 with the upwind quadrature approach replaced with the
Definition 7.2.31 (Consistent modifications of variational problems).
streamline diffusion method.
A variational problem is called a consistent modification of another, if both possess the same
(unique) solution(s).
7.2
p. 758
Code 7.2.34: Assembling SUPG stabilization part of element matrix in LehrFEM
1 f u n c t i o n Aloc = STIMASUPGLFE(Vertices, f l a g , QuadRule, VHandle,
a,d1,d2, varargin)
7.2
p. 76

2 % ALOC = STIMA_SUPG_LFE(VERTICES) provides the extra terms for SUPG
stabilization to be
3 % added to the Galerkin element matrix for linear finite elements
4 %
5 % VERTICES is 3-by-2 matrix specifying the vertices of the current element
6 % in a row wise orientation.
7 %
8 % a: diffusivity
9 % d1 d2: apriori chosen constants for SUPG-modification
10 %
11 % Flag not used, needed for interface to assemMat_LFE
12 %
13 % QUADRULE is a struct, which specifies the Gauss qaudrature that is used
14 % to do the numerical integration:
15 % W Weights of the Gauss quadrature.
16 % X Abscissae of the Gauss quadrature.e:
17 %
18 % VHANDLE is function handle for velocity field
19
20 % Preallocate memory for element matrix
21 Aloc = zeros(3,3);
22
23 % Analytic computation of entries of element matrix using barycentric
24 % coordinates, see Sect. 3.2.5
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
48 % Entries of anisotropic diffusion tensor
49 FHandle=[c(:,1).*c(:,1) c(:,1).*c(:,2) c(:,2).*c(:,1)
c(:,2).*c(:,2)];
50
51 % Compute local PecletNumber for SUPG control parameter
52 hK=max([norm(P2-P1),norm(P3-P1),norm(P2-P3)]);
53 v_infK=max(abs(c(:))); PK=v_infK*hK/(2*a);
54 % Apply quadrature rule and fix constant part
55 w = QuadRule.w; e = sum((FHandle.*[w w w w]), 1);
56 te = (reshape(e,2,2)’)/det_BK;
57
58 % Compute Aloc values
59 Aloc(1,1) = (te*[l1x l1y]’)’*[l1x l1y]’;
60 Aloc(1,2) = (te*[l1x l1y]’)’*[l2x l2y]’;
61 Aloc(1,3) = (te*[l1x l1y]’)’*[l3x l3y]’;
62 Aloc(2,2) = (te*[l2x l2y]’)’*[l2x l2y]’;
63 Aloc(2,3) = (te*[l2x l2y]’)’*[l3x l3y]’;
64 Aloc(3,3) = (te*[l3x l3y]’)’*[l3x l3y]’;
65 Aloc(2,1) = (te*[l2x l2y]’)’*[l1x l1y]’;
66 Aloc(3,1) = (te*[l3x l3y]’)’*[l1x l1y]’;
67 Aloc(3,2) = (te*[l3x l3y]’)’*[l2x l2y]’;
68
7.2 69 i f (PK

Observations:
• The streamline upwind method does not exactly respect the maximum principle, but offers a better
resolution of the internal layer compared with upwind quadrature (Parlance: streamline diffusion
method is “less diffusive”).
Numerical
Methods
for PDEs
Ich haenge noch 2 Referenzen zu diesem Thema an.
7.3 Transient convection-diffusion BVP
✸
Numerica
Methods
for PDEs
Example 7.2.35 (Convergence of SUPG and upwind quadrature FEM).
Ω =]0,1[ 2 , model problem (7.2.1),v(x) = ( 2
3
)
, right hand sidef such that
u ε (x,y) = xy 2 −y 2 e 2x−1 ǫ −xe 3y−1 ǫ +e 2x−1 ǫ +3 y−1
ǫ .
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Sect. 7.1.4 introduced the transient heat conduction model in a fluid, whose motion is described by a
non-stationary velocity field (→ Sect. 7.1.1)v : Ω×]0,T[↦→ R d
∂
∂t (ρu)−div(κgradu)+div(ρv(x,t)u) = f(x,t) in ˜Ω := Ω×]0,T[ , (7.1.16)
whereu = u(x,t) : ˜Ω ↦→ R is the unknown temperature.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Finite element discretization,V 0,N = S1 0 (M) und sequence of unstructured triangular “uniform”
meshes, with
• upwind quadrature stabilization from Sect. 7.2.2.1,
• SUPG stabilization according to (7.2.32).
Monitored: (Approximate) L 2 (Ω)-norm of discretization error (computed with high-order local
quadrature)
7.2
p. 765
Numerical
Methods
for PDEs
Assuming divv(x,t) = 0, as in Sect. 7.2, by scaling we arrive at the model equation for transient
convection-diffusion
supplemented with
∂u
∂t −ǫ∆u+v(x,t)·gradu = f in ˜Ω := Ω×]0,T[ , (7.3.1)
Dirichlet boundary conditions: u(x,t) = g(x,t) ∀x ∈ ∂Ω , 0 < t < T ,
7.3
p. 76
Numerica
Methods
for PDEs
1
10 −2
10 −1 h
L 2 −error u (Up)
L 2 −error u (FEM)
L 2 −error u (SUPG)
initial conditions: u(x,0) = u 0 (x) ∀x ∈ Ω.
0.8
0.6
0.4
0.2
0
0
0.5
0.5
x
1 0
y Fig. 228
u ǫ forǫ = 1
1
error
10 −3
10 −4
10 −5
p = 2.02
p = 2.02
p = 1.09
10 −6
10 −2 10 −1 10 0
Fig. 229
Convergence for ǫ = 1
Observation: SUPG stabilization does not affect O(h 2 M )-convergence of ‖u−u N‖ L 2 (Ω) for h-
refinement and h M → 0, whereas upwind quadrature leads to worse O(h M ) convergence of the
L 2 -error norm.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.2
p. 766
7.3.1 Method of lines
For the solution of IBVP (7.3.1) follow the general policy introduced in Sect. 6.1.3:
➊ Discretization in space on a fixed mesh ➣ initial value problem for ODE
➋ Discretization in time (by suitable numerical integrator = timestepping)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
7.3
p. 76

u
u
u
u
u
u
u
u
For instance, in the case of Dirichlet boundary conditions,
⎧
⎨ ∂u
∂t −ǫ∆u+v(x,t)·gradu = f in ˜Ω := Ω×]0,T[ ,
⎩
u(x,t) = g(x,t) ∀x ∈ ∂Ω, 0 < t < T , u(x,0) = u 0 (x) ∀x ∈ Ω .
(7.3.2)
Numerical
Methods
for PDEs
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
t=0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
central scheme
upwinding
h = 0.010000, tau = 0.001250, t=0.126250
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
central scheme
upwinding
h = 0.010000, tau = 0.001250, t=0.251250
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
central scheme
upwinding
h = 0.010000, tau = 0.001250, t=0.376250
Numerica
Methods
for PDEs
0.2
0.2
0.2
0.2
← spatial discretization
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0.1
Fig. 230
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
M d⃗µ (t)+ǫA⃗µ(t)+B⃗µ(t) = ⃗ϕ(t) , (7.3.3)
dt
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Observation:
• Central finite differences display spurious oscillations as in Ex. 7.2.17.
• Upwinding suppresses spurious oscillations, but introduces spurious damping.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
where
⃗µ = ⃗µ(t) :]0,T[↦→ R N ˆ= coefficient vector describing approximationu N (t) ofu(·,t),
A ∈ R N,N ˆ= s.p.d. matrix of discretized−∆, e.g., (finite element) Galerkin matrix,
M ∈ R N,N ˆ= (lumped→Rem. 6.2.34) mass matrix
Computations withǫ = 10 −5 , spatial upwind discretization,h = 0.01,τ = 0.005:
B ∈ R N,N ˆ= matrix for discretized convective term, e.g., Galerkin matrix, upwind quadrature
matrix (→ Sect. 7.2.2.1), streamline diffusion matrix (→ Sect. 7.2.2.2).
Example 7.3.4 (Implicit Euler method of lines for transient convection-diffusion).
1D convection-diffusion IBVP:
∂u u
∂t −ǫ∂2 ∂x 2 + ∂u
∂x = 0 , u(x,0) = max(1−3|x− 1 3 |,0) , u(0) = u(1) = 0 . (7.3.5)
7.3
p. 769
Numerical
Methods
for PDEs
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
t=0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.010000, τ = 0.005000, t=0.130000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.010000, τ = 0.005000, t=0.255000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.010000, τ = 0.005000, t=0.380000
Fig. 231
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
7.3
p. 77
Numerica
Methods
for PDEs
Spatial discretization on equidistant mesh with meshwidthh = 1/N:
1. central finite difference scheme, see (7.2.15),
2. upwind finite difference discretization, see (7.2.19),
M = hI (“lumped” mass matrix, see Rem. 6.2.34),
Temporal discretization with uniform timestepτ > 0:
1. implicit Euler method, see (6.1.23),
2. explicit Euler method, see (6.1.22),
R. Hiptmair
C. Schwab,
H.
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Gradinaru
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DRAFT
Observation:
timestepping.
implicit Euler timestepping causes stronger spurious damping than explicit Euler
However, explicit Euler subject to tight stability induced timestep constraint for larger values ofǫ, see
Sect. 6.1.4.2.
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Computations withǫ = 10 −5 , implicit Euler discretization,h = 0.01,τ = 0.00125:
7.3
p. 770
7.3
p. 77

Advice for spatial discretization for method of lines approach
✗
✖
Useǫ-robustly stable spatial discretization of convective term.
Remark 7.3.6 (Choice of timestepping for m.o.l. for transient convection-diffusion).
If ǫ-robustness for all ǫ > 0 (including ǫ > 1) desired ➣ Arguments of Sect. 6.1.4.2 stipulate
use of L(π)-stable (→ Def. 6.1.56) timestepping methods (implicit Euler (6.1.23), RADAU-3 (6.1.58),
SDIRK-2 (6.1.59))
In the singularly perturbed case0 < ǫ ≪ 1 conditionally stable explicit timestepping is an option, due
to a timestep constraint of the form “τ < O(h M )”, which does not interfere with efficiency, cf. the
discussion in Sect. 6.1.5.
✔
✕
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DRAFT
Now: focus on case f ≡ 0 (no sources)
Letu = u(x,t) be aC 1 -solution of
Recall:
➣
∂u
∂t +v(x,t)·gradu = 0 in ˜Ω := Ω×]0,T[ . (7.3.8)
for the stationary pure tranport problem (7.2.6) we found solutions by integrating the source
term along streamlines (following the flow direction).
study the behavior ofu“as seen from a moving fluid particle”
By the chain rule
t ↦→ u(y(t),t) , where
d
dt
y(t) solves
dy
(t) = v(y(t),t) , see (7.1.1) .
dt
dy
u(y(t),t) = gradu(y(t),t)·
dt
= gradu(y(t),t)·v(y(t),t)+ ∂u (7.3.8)
(y(t),t) = 0 .
∂t
(t)+
∂u
∂t (y(t),t)
A fluid particle “sees” a constant temperature!
(7.3.9)
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7.3.2 Transport equation
△
7.3
p. 773
Numerical
Methods
for PDEs
Remark 7.3.10 (Solution formula for sourceless transport).
Situation: no inflow/outflow (e.g., fluid in a container)
v(x,t)·n(x) = 0 ∀x ∈ ∂Ω , 0 < t < T . (7.1.2)
7.3
p. 77
Numerica
Methods
for PDEs
Focus on the situation of singular perturbation (→ Def. 7.2.13): 0 < ǫ ≪ 1
➣ study limit problem (as in Sect. 7.2.1)
➣ all streamlines will “stay insideΩ”, flow mapΦ t (7.1.3) defined for all timest ∈ R.
Initial value problem:
∂u
∂t −ǫ∆u+v(x,t)·gradu = f in ˜Ω := Ω×]0,T[ ,
←ǫ = 0
∂u
∂t +v(x,t)·gradu = f in ˜Ω := Ω×]0,T[ . (7.3.7)
= transport equation
R. Hiptmair
C. Schwab,
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DRAFT
Exact solution
v(x,t)·gradu = 0 in ˜Ω , u(x,0) = u 0 (x) ∀x ∈ Ω .
u(x,t) = u 0 (x 0 (x,t)) , (7.3.11)
wherex 0 (x,t) is the position at time 0 of the fluid
particle that is located atxat timet.
x
R. Hiptm
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7.3
p. 774
x 0 (x,t)
Fig. 232
△
7.3
p. 77

This solution formula can be generalized to any divergence free velocity fieldv : Ω ↦→ R d andf ≠ 0.
The new aspect is that streamlines can enter and leave the domainΩ. In the former case the solution
value is given by a “transported boundary value”:
d
u(y(t)) = f(y(t),t)
⎧ dt
∫
⎪⎨ u 0 (x 0 )+ t
f(y(s),s)ds , if y(s) ∈ Ω ∀0 < s < t ,
u(x,t) = 0
∫
⎪⎩ g(y(s 0 ),s 0 )+ t
f(y(s),s)ds , if y(s 0 ) ∈ ∂Ω, y(s) ∈ Ω ∀s 0 < s < t ,
s 0
where we have assumed Dirichlet boundary conditions on the inflow boundary
(7.3.12)
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DRAFT
Strang splitting single step method for (7.3.13), timestep τ := t j −t j−1 > 0: compute y (j) ≈
y(t j ) fromy (j−1) ≈ y(t j−1 ) according to
ỹ := z(t j−1 + 1 2 τ) , where z(t) solves ż = g(t,z) , z(t j−1) = y (j−1) , (7.3.14)
ŷ := w(t j ) where w(t) solves ẇ = h(t,w) , w(t j−1 ) = ỹ , (7.3.15)
Numerica
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y (j) := z(t j ) , where z(t) solves ż = g(t,z) , z(t j−1 + 1 2 τ) = ŷ . (7.3.16) R. Hiptm
C. Schwa
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u(x,t) = g(x,t) for x ∈ Γ in := {x ∈ ∂Ω : v(x)·n(x) < 0} , cf. (7.2.10).
7.3.3 Lagrangian split-step method
Lagrangian discretization schemes for the IBVP (7.3.2) are inspired by insight into the traits of solutions
of pure transport problems.
The variant that we are going to study separates the transient convection-diffusion problem into a
pure diffusion problem (heat equation → Sect. 6.1.1) and a pure transport problem (7.3.7). This is
achieved by means of a particular approach to timestepping.
7.3.3.1 Split-step timestepping
Abstract perspective: consider ODE, whose right hand side is the sum of two (smooth) functions
7.3
p. 777
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Methods
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DRAFT
One timestep involves three sub-steps:
➀ Solveż = g(t,z) over time[t j−1 ,t j−1 + 2 1 τ] using the result
of the previous timestep as initial value ↔ (7.3.14).
➁ Solve ẇ = h(t,w) over time τ using the result of ➀ as
initial value ↔ (7.3.15).
➂ Solveż = g(t,z) over time[t j−1 + 2 1τ,t j] using the result
of ➁ as initial value ↔ (7.3.16). y (j−1)
✬
Theorem 7.3.17 (Order of Strang splitting single step method).
ż = g(z)
ż = g(z)
ẇ = h(w)
Assuming exact solution of the initial value problems of the sub-steps, the Strang splitting single
step method for (7.3.13) is of second order.
✫
Fig. 233
y (j)
✩
✪
7.3
p. 77
Numerica
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H.
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ẏ = g(t,y)+h(t,y) , g,h : R m ↦→ R m . (7.3.13)
There is an abstract timestepping scheme that offers great benefits if one commands efficient methods
to solve initial value problems for bothż = g(z) andẇ = h(w).
7.3
p. 778
This applies to Strang splitting timestepping for initial value problems for ODEs. Now we boldly regard
(7.3.2) as an “ODE in function space” for the unknown “function space valued function” u = u(t) :
7.3
p. 78

[0,T] ↦→ H 1 (Ω).
du
= ǫ∆u + f −v·gradu
dt
↕ ↕ ↕
ẏ = g(y) + h(y)
Formally, we arrive at the following “timestepping scheme in function space” on a temporal mesh
0 = t 0 < t 1 < ··· < t M := T for (7.3.1):
Given approximation u (j−1) ≈ u(t j−1 ),
➀ Solve (autonomous) IBVP for pure diffusion fromt j−1 to t j−1 + 1 2 τ
(7.3.14) ↔
∂w
∂t −ǫ∆w = 0 inΩ×]t j−1,t j−1 + 1 2 τ[ ,
w(x,t) = g(x,t j−1 ) ∀x ∈ ∂Ω, t j−1 < t < t j−1 + 1 2 τ ,
w(x,t j−1 ) = u (j−1) (x) ∀x ∈ Ω .
(7.3.18)
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C. Schwab,
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DRAFT
t 0 t 1 t 2 t 3 t 4
Fig. 234
—: sub-step ➁, —: sub-step ➀ & ➂
Remark 7.3.21 (Approximate sub-steps for Strang splitting time).
The solutions of the initial value problems in the sub-steps of Strang splitting timestepping may be
computed only approximately.
If this is done by one step of a 2nd-order timestepping method in each case, then the resulting
approximate Strang splitting timestepping will still be of second order, cf. Thm. 7.3.17.
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➁ Solve IBVP for pure transport (= advection), see Sect. 7.3.2,
(7.3.15) ↔
∂z
∂t +v(x,t)·gradz = f(x,t) inΩ×]t j−1,t j [ ,
z(x,t) = g(x,t) on inflow boundaryΓ in , t j−1 < t < t j , (7.3.19)
z(x,t j−1 ) = w(x,t j−1 + 1 2 τ) ∀x ∈ Ω .
➂ Solve IBVP for pure diffusion fromt j−1 + 1 2 τ tot j
(7.3.16) ↔
Then set u (j) (x) := w(x,t j ),x ∈ Ω.
∂w
∂t −ǫ∆w = 0 inΩ×]t j−1+ 1 2 τ,t j[ ,
w(x,t) = g(x,t j ) ∀x ∈ ∂Ω , t j−1 + 1 2 τ < t < t j ,
w(x,t j−1 + 1 2 τ) = z(x,t j) ∀x ∈ Ω .
(7.3.20)
7.3
p. 781
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Gradinaru
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DRAFT
7.3.3.2 Particle method for advection
Recall the discussion of the IBVP for the pure transport (= advection) equation from Sect. 7.3.2
with inflow boundary
∂u
∂t +v(x,t)·gradu = f in ˜Ω := Ω×]0,T[ ,
u(x,t) = g(x,t) on Γ in ×]0,T[ ,
u(x,0) = u 0 (x) inΩ ,
△
(7.3.22)
Γ in := {x ∈ ∂Ω: v(x)·n(x) < 0} . (7.2.10)
Case f ≡ 0: a travelling fluid particle sees a constant solution, see (7.3.9)
7.3
p. 78
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R. Hiptm
C. Schwa
H.
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DRAFT
Efficient “implementation” of Strang splitting timestepping, ifg = g(y):
combine last sub-step ➂ with first sub-step ➀ of the next timestep
u(x,t) =
where s ↦→ y(s) solves the initial value problem
trajectory”).
{
u0 (x 0 ) , if y(s) ∈ Ω ∀0 < s < t ,
g(y(s 0 ),s 0 ) , if y(s 0 ) ∈ ∂Ω, y(s) ∈ Ω ∀s 0 < s < t ,
(7.3.23)
dy
ds (s) = v(y(s),s), y(t) = x (“backward particle
7.3
p. 782
7.3
p. 78

Case of generalf, see Rem. 7.3.10: Since
dt d u(y(t)) = f(y(t),t)
⎧
∫
⎪⎨ u 0 (x 0 )+ t
f(y(s),s)ds , if y(s) ∈ Ω ∀0 < s < t ,
u(x,t) = 0
∫
⎪⎩ g(y(s 0 ),s 0 )+ t
f(y(s),s)ds , if y(s 0 ) ∈ ∂Ω, y(s) ∈ Ω ∀s 0 < s < t .
s 0
(7.3.12)
Numerical
Methods
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For general velocity fieldv : Ω ↦→ R d :
Stop trackingi-th trajectory as soon as an interpolation nodesp (j)
i
lies outside spatial domainΩ.
In each timestep start new trajectories from fixed locations on inflow boundaryΓ in (“particle injection”).
These interpolation nodes will carry the boundary value.
Example 7.3.24 (Point particle method for pure advection).
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R. Hiptmair
The solution formula (7.3.12) suggests an approach for solving (7.3.22) approximately.
DRAFT
We first consider the simple situation of no inflow/outflow (e.g., fluid in a container, see Rem. 7.3.10)
v(x,t)·n(x) = 0 ∀x ∈ ∂Ω , 0 < t < T . (7.1.2)
➀ Pick suitable interpolation nodes {p i } N i=1 ⊂ Ω (initial ‘particle positions”)
7.3
p. 785
IBVP (7.3.22) onΩ =]0,1[ 2 ,T = 2, withf ≡ 0,g ≡ 0.
∥
Initial locally supported bump u 0 (x) = max{0,1−4∥x− ( 1/2) ∥ 1/4 ∥}.
Two stationary divergence-free velocity fields
( ) −sin(πx1 )cos(πx
•v 1 (x) = 2 )
satisfying (7.1.2),
cos(πx 1 )sin(πx 2 )
( ) −x2
•v 2 (x) = .
x 1
R. Hiptm
C. Schwa
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7.3
p. 78
➁ Solve initial value problems (cf. ODE (7.1.1) for particle trajectories)
ẏ(t) = v(y(t),t) , y(0) = p i , i = 1,...,N ,
by means of a suitable single-step method with uniform timestepτ := T/M,M ∈ N.
➣ sequencies of solution points p (j)
i ,j = 0,...,M,i = 1,...,N
➂ Reconstruct approximationu (j)
N ≈ u(·,t j),t j := jτ, by interpolation:
u (j)
j−1 ∑
N (p(j) i ) := u 0 (p i )+τ f( 2 1(p(l)
i +p (l−1)
i ), 1 2 (t l +t l−1 )) , i = 1,...,N
l=1
where the composite midpoint quadrature rule was used to approximate the source integral in
(7.3.12).
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V.
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Initial positions of interpolation points on regular tensor product grid with meshwidthh = 1 40 .
Approximation of trajectories by means of explicit trapezoidal rule [18, Eq. 12.4.6] (method of
Heun).
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This method falls into the class of
• particle methods, because the interpolation nodes can be regarded fluid particles tracked by the
method,
velocity fieldv 1 (circvel)
Fig. 235
velocity fieldv 2 (rotvel)
Fig. 236
• Lagrangian methods, which treat the IBVP in coordinate systems moving with the flow,
• characteristic methods, which reconstruct the solution from knowledge about its behavior along
streamlines.
7.3
p. 786
7.3
p. 78

Code 7.3.25: Confined velocity field
1 f u n c t i o n V = circvel(P)
2 % Circular velocity (divergence free, zero normal component on unit square).
3 % P: 2xN matrix of point coordinates
4 % return value: velocity vectors at points in P
5
6 v = @(p) [-sin ( p i *p(1))*cos( p i *p(2));sin ( p i *p(2))*cos( p i *p(1))];
7
8 V = [];
9 f o r p=P
10 V = [V, v(p)];
11 end
Code 7.3.26: Pass-through velocity field
1 f u n c t i o n V = rotvel(P)
2 % Circular velocity
3
4 v = @(p) [-p(2); p(1)];
5
6 V = [];
7 f o r p=P
8 V = [V, v(p)];
9 end
Code 7.3.27: Point particle method for pure advection
1 f u n c t i o n partadv(v,u0,g,n,tau,m)
2 % Point particle method for pure advection problem
3 % on the unit sqaure
4 % v: handle to a function returning the velocity field for (an array) of
points
5 % u0: handle to a function returning the initial value u 0 for (an array)
6 % of points
7 % g: handle to a function g = g(x) returning the Dirichlet boundary values
8 % n: h = 1/n is the grid spacing of the inintial point distribution
9 % tau: timestep size, m: number of timesteps, that is, T = mτ
10
11 % Initialize points
12 h = 1/n; [Xp,Yp] = meshgrid(0:h:1,0:h:1);
13 P = [reshape(Xp,1,(n+1)^2);reshape(Yp,1,(n+1)^2)];
14 % Initialize points on the boundary
15 BP = [[(0:h:1);zeros(1,n+1)],[ones(1,n+1);(0:h:1)],...
16 [(0:h:1);ones(1,n+1)],[zeros(1,n+1);(0:h:1)]];
17 U = u0(P); % Initial values
18
19 % Plot velocity field
20 hp = 1/10; [Xp,Yp] = meshgrid(0:hp:1,0:hp:1);
21 Up = zeros( s i z e (Xp)); Vp = zeros( s i z e(Xp));
22 f o r i=0:10, f o r j=0:10
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43
7.3 44 t = 0;
p. 789
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7.3
p. 790
23 x = v([Xp(i+1,j+1);Yp(i+1,j+1)]);
24 Up(i+1,j+1) = x(1); Vp(i+1,j+1) = x(2);
25 end; end
26 f i g u r e (’name’,’velocity field’,’renderer’,’painters’);
27 q u i v e r(Xp,Yp,Up,Vp,’b-’); s e t(gca,’fontsize’,14); hold on;
28 p l o t ([0 1 1 0 0],[0 0 1 1 0],’k-’);
29 a x i s([-0.1 1.1 -0.1 1.1]);
30 x l a b e l (’{\bf x_1}’); y l a b e l (’{\bf x_2}’);
31 a x i s off;
32
33 fp = f i g u r e (’name’,’particles’,’renderer’,’painters’);
34 fs = f i g u r e (’name’,’solution’,’renderer’,’painter’);
35
36 % Visualize points (interior points in red, boundary points in blue)
37 f i g u r e (fp); p l o t (P(1,:),P(2,:),’r+’,BP(1,:),BP(2,:),’b*’);
38 t i t l e ( s p r i n t f (’n = %i, t = %f, \\tau = %f, %i
points’,n,0,tau, s i z e (P,2)));
39 drawnow; pause;
40
41 % Visualize solution
42 f i g u r e (fs); plotpartsol(P,U); drawnow;
45 f o r l=1:m
46 % Advect points (explicit trapezoidal rule)
47 P1 = P + tau/2*v(P); P = P + tau*v(P1);
48
49 % Remove points on the boundary or outside the domain
50 Pnew = []; Unew = []; l = 1;
51 f o r p=P
52 i f ((p(1) > eps) (p(1) < 1-eps) (p(2) > eps) (p(2) <
1-eps))
53 Pnew = [Pnew,p]; Unew = [Unew; U(l)];
54 end
55 l = l+1;
56 end
57
58 % Add points on the boundary (particle injection)
59 P = [Pnew, BP]; U = [Unew; g(BP)];
60
61 % Visualize points
62 f i g u r e (fp); p l o t (P(1,:),P(2,:),’r+’,BP(1,:),BP(2,:),’b*’);
63 t i t l e ( s p r i n t f (’n = %i, t = %f, \\tau = %f, %i
points’,n,t,tau, s i z e (P,2)));
64 drawnow;
65 % Visualize solution
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7.3
p. 79
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7.3
p. 79

66 f i g u r e (fs); plotpartsol(P,U); drawnow;
67
68 t = t+tau;
69 end
Numerical
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➊ Given triangular meshM (j−1) ofΩ,
u (j−1)
N ∈ S1,0 0 (M(j−1) )↔coefficient vector ⃗µ (j−1) ∈ R Nj−1 ,
approximately solve (7.3.18) by a single step of implicit Euler (6.1.23) (size 1 2 τ)
⃗ν = (M+ 1 2 τǫA)−1 ⃗µ (j−1) ,
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Start animations:
partadv(@circvel,@initvals,@bdvals,40,0.025,80);
partadv(@rotvel,@initvals,@bdvals,40,0.025,80);
7.3.3.3 Particle mesh method
✸
R. Hiptmair
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7.3
p. 793
where A ∈ R N j−1,N j−1 ˆ= S 0
1,0 (M)-Galerkin matrix for −∆, M ˆ= (possibly lumped) S 0 1,0 (M)-
mass matrix.
C. Schwa
H.
Harbrech
V.
Gradinar
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R. Hiptm
More advisable to maintain 2nd-order timestepping: 2nd-order L(π)-stable single step
method,e.g., SDIRK-2 (6.1.59).
DRAFT
➋ Lagrangian advection step (of sizeτ) for (7.3.19) with
initial “particle positions”p i given by nodes ofM (j−1) ,i = 1,...,N j ,
initial “particle temperatures” given by corresponding coefficientsν i .
➌ Remeshing: advection step has moved nodes to new positions ˜p i (and, maybe, introduced new
nodes by “particle injection”, deleted nodes by “particle removal”).
➣ Create new triangular meshM (j) with nodes ˜p i (+ boundary nodes),i = 1,...,N j
7.3
p. 79
The method introduced in the previous section, can be used to tackle the pure advection problem
(7.3.19) in the 2nd sub-step of the Strang splitting timestepping.
Issue: How to combine Lagrangian advection with a method for the pure diffusion problem (7.3.18)
faced in the other sub-steps of the Strang splitting timestepping?
Numerical
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➍ Repeat diffusion step ➊ starting with w N ∈ S 0 1,0 (M(j) ) = linear interpolant (→ Def. 5.3.13) of
“particle temperatures” onM (j) .
➣
new approximate solutionu (j)
N
Example 7.3.28 (Delaunay-remeshing in 2D).
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Idea:
two views
“particle temperatures”
↕
u(p (j)
i )
Nodal values of finite element functionu (j)
N ∈ S0 1 (M)
Outline: algorithm for one step of size τ > 0 of Strang splitting timestepping for transient
convection-diffusion problem
⎧
⎨ ∂u
∂t −ǫ∆u+v(x,t)·gradu = f in ˜Ω := Ω×]0,T[ ,
⎩
u(x,t) = 0 ∀x ∈ ∂Ω, 0 < t < T , u(x,0) = u 0 (x) ∀x ∈ Ω .
(7.3.2)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.3
p. 794
Delaunay algorithm for creating a 2D triangular
mesh with prescribed nodes:
1 Compute Voronoi cells, see (4.2.3) &
http://www.qhull.org/.
2 Connect two nodes, if their associated Voronoi
dual cells have an edge in common.
➥ MATLAB TRI = delaunay(x,y)
Fig. 237
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
7.3
p. 79

Code 7.3.29: Demonstration of Delaunay-remeshing
1 f u n c t i o n meshadv(v,n,tau,m)
2 % Point advaction and remeshing for Lagrangian method
3 % v: handle to a function returning the velocity field for (an array) of
points
4 % n: h = 1/n is the grid spacing of the inintial point distribution
5
6 % Initialize points
7 h = 1/n; [Xp,Yp] = meshgrid(0:h:1,0:h:1);
8 P = [reshape(Xp,1,(n+1)^2);reshape(Yp,1,(n+1)^2)];
9 % Initialize points on the boundary
10 BP = [[(0:h:1);zeros(1,n+1)],[ones(1,n+1);(0:h:1)],...
11 [(0:h:1);ones(1,n+1)],[zeros(1,n+1);(0:h:1)]];
12
13 % Plot triangulation
14 fp = f i g u r e (’name’,’evolving meshes’,’renderer’,’painters’);
15 TRI = delaunay(P(1,:),P(2,:));
16 p l o t (P(1,:),P(2,:),’r+’); hold on;
triplot(TRI,P(1,:),P(2,:),’blue’); hold off;
17 t i t l e ( s p r i n t f (’n = %i, t = %f, \\tau = %f, %i
points’,n,0,tau, s i z e (P,2)));
18 drawnow; pause;
19
20 t = 0;
21 f o r l=1:m
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DRAFT
7.3
p. 797
43 end
Ω =]0,1[ 2 , velocity fields like in Ex. 7.3.24. Advection of interpolation nodes by means of explicit
trapezoidal rule.
Start animations:
meshadv(@circvel,20,0.05,40);
meshadv(@rotvel,20,0.05,40);
Example 7.3.30 (Lagrangian method for convection-diffusion in 1D).
✸
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DRAFT
7.3
p. 79
22 % Advect points (explicit trapezoidal rule)
23 P1 = P + tau/2*v(P); P = P + tau*v(P1);
24
25 % Remove points on the boundary or outside the domain
26 Pnew = []; l = 1;
27 f o r p=P
28 i f ((p(1) > eps) (p(1) < 1-eps) (p(2) > eps) (p(2) <
1-eps))
29 Pnew = [Pnew,p];
30 end
31 l = l+1;
32 end
33
34 P = [Pnew, BP]; % Add points on the boundary (particle injection)
35
36 % Plot triangulation
37 TRI = delaunay(P(1,:),P(2,:));
38 p l o t (P(1,:),P(2,:),’r+’); hold on;
triplot(TRI,P(1,:),P(2,:),’blue’); hold off;
39 t i t l e ( s p r i n t f (’n = %i, t = %f, \\tau = %f, %i
points’,n,t,tau, s i z e (P,2)));
40 drawnow;
41
42 t = t+tau;
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C. Schwab,
H.
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V.
Gradinaru
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DRAFT
7.3
p. 798
Same IBVP as in Ex. 7.3.4
Linear finite element Galerkin discretization
with mass lumping in space
Strang splitting applied to diffusive and convective
terms
Implicit Euler timestepping for diffusive partial
timestep
u
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
t=0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Fig. 238
Code 7.3.31: Lagrangian method for (7.3.5)
1 f u n c t i o n lagr(epsilon,N,M)
2 % This function implements a simple Lagrangian advection scheme for the 1D
convection-diffusion
3 % IBVP −ǫ d2 u
+ du
dx 2 dx = 0, u(x,0) = max(1−3|x− 1 3 |,0),
4 % and homogeneous Dirichlet boundary conditions u(0) = u(1) = 0. Timestepping
employs Strang splitting
5 % applied to diffusive and convective spatial operators.
6 % epsilon: strength of diffusion
7 % N: number of cells of spatial mesh
8 % M: number of timesteps
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H.
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Gradinar
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DRAFT
7.3
p. 80

u
u
u
u
u
u
u
u
u
u
u
u
9
10 T = 0.5; tau = T/M; %
timestep size
11 h = 1/N; x = 0:h:1; u = max(1-3*abs(x(2:end-1)-1/3),0)’;
value
% Initial
12
13 [Amat,Mmat] = getdeltamat(x); % Obtain stiffness and mass
matrix
14 u = (Mmat+0.5*tau*epsilon*Amat)\(Mmat*u); % Implicit Euler timestep
15
16 f o r j=1:M+1
17 % Advection step: shift meshpoints, drop those travelling out of Ω =]0,1[,
insert
18 % new meshpoints from the left. Solution values are just copied.
19 xm = x(2:end-1)+tau; % Transport of meshpoints (here: explicit
Euler)
20 idx = f i n d (xm < 1); % Drop meshpoints beyond x = 1
21 x = [0,tau,xm(idx),1]; % Insert new meshpoint at left end of Ω
22 u = [0;u(idx)]; % Copy nodal values and feed 0 from left
23
24 % Diffusion partial timestep
25 [Amat,Mmat] = getdeltamat(x); % Obtain stiffness and mass
matrix on new mesh
26 u = (Mmat+tau*epsilon*Amat)\(Mmat*u); % Implicit Euler step
27 end
28 end
ǫ = 10 −5 :
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Lagrangian advection: M = 113.000000, τ = 0.005000, t=0.130000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
ǫ = 0.1:
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Lagrangian advection: M = 113.000000, τ = 0.005000, t=0.130000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Lagrangian advection: M = 126.000000, τ = 0.005000, t=0.255000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Lagrangian advection: M = 126.000000, τ = 0.005000, t=0.255000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Lagrangian advection: M = 138.000000, τ = 0.005000, t=0.380000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
Lagrangian advection: M = 138.000000, τ = 0.005000, t=0.380000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Lagrangian advection: M = 151.000000, τ = 0.005000, t=0.505000
Fig. 239
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
Lagrangian advection: M = 151.000000, τ = 0.005000, t=0.505000
Fig. 240
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
“Reference solution” computed by method of lines, see Ex. 7.3.4, withh = 10 −3 ,τ = 5·10 −5 :
Numerical
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C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.3
p. 801
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
17
H.
Harbrecht
V. 19
Gradinaru
A. Chernov
DRAFT
7.3
p. 802
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.001000, τ = 0.000050, t=0.125050
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.001000, τ = 0.000050, t=0.250050
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.001000, τ = 0.000050, t=0.375050
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Example 7.3.32 (Lagrangian method for convection-diffusion in 2D).
IBVP (7.3.2) onΩ =]0,1[ 2 ,T = 1,
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.001000, τ = 0.000050, t=0.500050
Fig. 241
✸
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Particle mesh method based on Delaunay remeshing, see Ex. 7.3.28, and linear finite element
Galerkin discretizatin for diffusion step.
Code 7.3.33: Particle mesh method in 2D
1 f u n c t i o n ConvDiffLagr(v,epsilon,u0,n,tau,m)
2 % Point particle method for convection-diffusion problem on the unit sqaure
3 % v: handle to a function returning the velocity field for (an array) of
points
4 % u0: handle to a function returning the initial value u 0 for (an array)
5 % of points
6 % n: h = 1/n is the grid spacing of the inintial point distribution
7 % tau: timestep size, m: number of timesteps, that is, T = mτ
8
9 % Initialize points
10 h = 1/n; [Xp,Yp] = meshgrid(0:h:1,0:h:1);
11 P = [reshape(Xp,1,(n+1)^2);reshape(Yp,1,(n+1)^2)];
12 % Initialize points on the boundary
13 BP = [[(0:h:1);zeros(1,n+1)],[ones(1,n+1);(0:h:1)],...
14 [(0:h:1);ones(1,n+1)],[zeros(1,n+1);(0:h:1)]];
15 % Construct initial mesh by Delaunay algorithm
16 TRI = delaunay(P(1,:),P(2,:));
18 U = u0(P); % Initial values
20 fp = f i g u r e (’name’,’particles’,’renderer’,’painters’);
21 fs = f i g u r e (’name’,’solution’,’renderer’,’painters’);
22
23 % Visualize mesh, points (interior points in red, boundary points in blue)
24 % the piecewise linear approximate solution
25 f i g u r e (fp); p l o t (P(1,:),P(2,:),’r+’,BP(1,:),BP(2,:),’m*’); hold on;
26 triplot(TRI,P(1,:),P(2,:),’blue’); hold off;
27 t i t l e ( s p r i n t f (’n = %i, t = %f, \\tau = %f, %i points’,n,0,tau, s i z e (P,2)));
28 drawnow;
29
30 f i g u r e (fs); trisurf(TRI,P(1,:),P(2,:),U’);
31 a x i s([0 1 0 1 0 1]); x l a b e l (’{\bf x_1}’);
32 y l a b e l (’{\bf x_2}’); z l a b e l (’{\bf u}’);
33 t i t l e ( s p r i n t f (’n = %i, t = %f, \\tau = %f, %i points’,n,0,tau, s i z e (P,2)));
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DRAFT
7.3
p. 80
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C. Schwa
H.
Harbrech
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Gradinar
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DRAFT
7.3
p. 80

34 pause;
35
36 % Initial diffusion half step (implicit Euler)
37 [Amat,Mmat] = getGalerkinMatrices(TRI,P(1,:),P(2,:)); % Compute Galerkin
matrices
38 % Isolate indices of interior points
39 j = 1; intidx = [];
40 f o r p=P
41 i f ((p(1) > eps) (p(1) < 1-eps) (p(2) > eps) (p(2) < 1-eps))
42 intidx = [intidx,j];
43 end
44 j = j+1;
45 end
46 Amat = Amat(intidx,intidx); Mmat = Mmat(intidx,intidx);
47 U(intidx) = (Mmat+0.5*epsilon*tau*Amat)\(Mmat*U(intidx));
48
49 % full(Amat), full(Mmat), return;
50
51 t = 0;
52 f o r l=1:m
53 % Advect points (explicit trapezoidal rule)
54 P1 = P + tau/2*v(P); P = P + tau*v(P1);
55
56 % Remove points on the boundary or outside the domain
57 Pnew = []; Unew = []; l = 1; j = 0;
58 f o r p=P
59 i f ((p(1) > eps) (p(1) < 1-eps) (p(2) > eps) (p(2) < 1-eps))
60 Pnew = [Pnew,p]; Unew = [Unew; U(l)];
61 j = j+1; % Counter for interior points
62 end
63 l = l+1;
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H.
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V.
Gradinaru
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DRAFT
7.3
p. 805
Invocation: ConvDiffLagr(@circvel,0.001,@initvals,1/40,0.01,100)
Advantage of Lagrangian (particle) methods for convection diffusion:
No artificial diffusion required (no “smearing”)
No stability induced timestep constraint
Drawback of Lagrangian (particle) methods for convection diffusion:
7.3.4 Semi-Lagrangian method
Remeshing (may be) expensive and difficult.
Point advection may produce “voids” in point set.
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64 end
65
66 % Add points on the boundary (particle injection)
67 P = [Pnew, BP];
68
69 % Delaunay algorithm for building triangulation
70 TRI = delaunay(P(1,:),P(2,:));
71 [Amat,Mmat] = getGalerkinMatrices(TRI,P(1,:),P(2,:)); % Compute Galerkin
matrices
72 Amat = Amat(1:j,1:j); Mmat = Mmat(1:j,1:j);
73 U = (Mmat+epsilon*tau*Amat)\(Mmat*Unew); % implicit Euler step
74 U = [U; zeros( s i z e (BP,2),1)]; % zero padding for boundary nodes
75
76 % Visualize mesh, points (interior points in red, boundary points in blue)
77 % the piecewise linear approximate solution
78 f i g u r e (fp); p l o t (P(1,:),P(2,:),’r+’,BP(1,:),BP(2,:),’m*’); hold on;
79 triplot(TRI,P(1,:),P(2,:),’blue’); hold off;
80 t i t l e ( s p r i n t f (’n = %i, t = %f, \\tau = %f, %i points’,n,t,tau, s i z e (P,2)));
81 drawnow;
82
83 f i g u r e (fs); trisurf(TRI,P(1,:),P(2,:),U’);
84 a x i s ([0 1 0 1 0 1]); x l a b e l (’{\bf x_1}’);
85 y l a b e l (’{\bf x_2}’); z l a b e l (’{\bf u}’);
86 t i t l e ( s p r i n t f (’n = %i, t = %f, \\tau = %f, %i points’,n,t,tau, s i z e (P,2)));
87 t = t+tau;
88 end
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DRAFT
Now we study a family of methods for transient convection-diffusion that takes into account transport
along streamlines, but, in constrast to genuine Lagrangian methods, relies on a fixed mesh.
Definition 7.3.34 (Material derivative).
Given a velocity field v : Ω×]0,T[↦→ R d , the material derivative of a function f = f(x,t) at
(x,t) is
Df
Dv (x,t f(x,t 0 )−f(Φ −τ
t
0) = lim 0
x,t 0 −τ)
τ→0 τ
, x ∈ Ω, 0 < t 0 < T ,
withΦ t t 0
the flow map (at timet 0 ) associated withv, that is, cf. (7.1.3), (7.1.4),
The material derivative Df
Dv
is the
dΦ t t 0
x
= v(Φ t dt 0
x,t−t 0 ) , Φ 0 x = x .
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p. 806
rate of change off experienced by a particle carried along by the flow
7.3
p. 80

ecauseΦ t t 0
x describes the trajectory of a particle located atxat timet 0 (↔t = 0).
Numerical
Methods
for PDEs
Cast (7.3.40) into variational form according to the recipe of Sect. 2.8 and apply Galerkin discretization
(here discussed for linear finite elements, homogeneous Dirichlet boundary conditionsu = 0 on∂Ω).
This yields one timestep (size τ) for the semi-Lagrangian method: the approximation u (j)
N
for u(jτ)
(equidistant timesteps) is computed from the previous timestep according to
Numerica
Methods
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By a straightforward application of the chain rule for smoothf
Df
Dv (x,t) = grad xf(x,t)·v(x,t)+ ∂f
∂t (x,t) . (7.3.37)
➣ The transient convection-diffusion equation can be rewritten as (7.3.1)
Idea:
∂u
∂t −ǫ∆u+v(x,t)·gradu = f in ˜Ω := Ω×]0,T[ ,
← (7.3.37)
Du
Dv −ǫ∆u = f in ˜Ω := Ω×]0,T[ . (7.3.38)
Backward difference (“implicit Euler”) discretization of material derivative
Du
≈ u(x,t 0)−u(Φ −τ
t 0
x,t 0 −τ)
,
Dv|(x,t)=(x,t 0 ) τ
with timestepτ > 0, wheret ↦→ Φ t x solves the initial value problem
dΦ t t 0
x
(t) = v(Φ t t
dt 0
x,t 0 +t) , Φ 0 t 0
x = x .
Semi-discretization of (7.3.38) in time (with fixed timestepτ > 0)
u (j) (x)−u (j−1) (Φ −τ
t j
x)
−ǫ∆u (j) (x) = f(x,t
τ
j ) in Ω ,
+ boundary conditions att = t j ,
(7.3.40)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.3
p. 809
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DRAFT
u (j)
N ∈ S0 1,0 (M): ∫
Ω
u (j)
N (x)−u(j−1) N (Φ −τ
t j
x)
τ
∫
=
Here,Mis supposed to be a fixed triangular mesh ofΩ.
∫
v N (x)dx+ǫ
Ω
Ω
gradu (j)
N ·gradv N dx
f(x,t j )v N (x)dx ∀v N ∈ S1,0 0 (M) . (7.3.41)
However, (7.3.41) cannot be implemented: x ↦→ u (j−1)
N (Φ −τ
t j
x) is a finite element function that
has been “transported with the (reversed) flow” (in the sense of pullback, see Def. 3.6.2)
Fig. 242
✁ - - - ˆ= image ofM(—) underΦ −τ
t j
The pullback x ↦→ v N (Φ −τ
t j
x) of v N ∈ S1,0 0 (M)
is piecewise smooth w.r.t. the mapped mesh
drawn with - - -.
the cells ofM.
Hence, it is not smooth inside
R. Hiptm
C. Schwa
H.
Harbrech
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Gradinar
A. Chern
DRAFT
7.3
p. 81
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DRAFT
whereu (j) : Ω ↦→ R is an approximation for u(·,t j ),t j := jτ,j ∈ N.
Note the difference to the method of lines (→ Sects. 6.1.3, 6.2.3, 7.3.1): in (7.3.40) semidiscretization
in time was carried out first, now followed by discretization in space, which reverses the order adopted
in the method of lines.
7.3
p. 810
➣ the transported function may not be a finite element function onM,
➣ the transported function may not even be piecewise smooth onM
7.3
p. 81

➤
✬
local quadrature for the approximate computation of the integral in (7.3.41) that
involves u (j−1)
N (Φ −τ
t j
x) is not possible, because accurate numerical quadrature
requires a (locally) smooth integrand.
✫
Idea:
replaceu (j−1)
N (y x (−τ)) with linear interpolant (→ Def. 5.3.13)
approximate Φ −τ
t j
x by
(“streamline backtracking”)
( (j−1)
I 1 u N ◦Φ −τ )
t j ∈ S
0
1,0 (M),
x−τv(x,t j ) (explicit Euler).
Implementable version of (7.3.41) (using mass lumping, see Rem. 6.2.34)
∫
u (j)
N ∈ S0 1,0 (M): 1
3 |U p |(µ (j)
p −u (j−1)
N (p−τv(p,t j ))+τ gradu (j)
N ·gradbp N dx
Ω
= 3 1|U p|f(p) , p ∈ N(M)∩Ω , (7.3.42)
where µ (j)
p are the nodal values of u (j)
N ∈ S0 1,0 (M) associated with the interior nodes of the mesh
M,b p N is the “tent function” belonging to nodep,|U p| is the sum of the areas of all triangles adjacent
top.
Example 7.3.43 (Semi-Lagrangian method for convection-diffusion in 1D).
Same IBVP as in Ex. 7.3.30
Linear finite element Galerkin discretization
with mass lumping in space
Semi-Lagrangian method:
(7.3.41)
1D version of
Explicit Euler streamline backtracking
u
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
t=0
✩
✪
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.3
p. 813
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Code 7.3.44: Lagrangian method for (7.3.5)
1 f u n c t i o n semilagr(epsilon,N,M,filename)
2 % Semi-Lagrangian linear finite element method for transient
convection-diffusion problem in 1D
3 i f (nargin < 4), filename = ’semilagr’; end
4
5 T = 0.5; tau = T/M; tstop = T/4; % timestep size
6 % Initial value u(x,0)
7 h = 1/N; x = 0:h:1; u = max(1-3*abs(x(2:end-1)-1/3),0)’;
8 % Obtain stiffness and mass matrix, see Sect. 1.5.1.2
9 [Amat,Mmat] = getdeltamat(x);
10
11 f i g u r e ; p l o t (x,[0;u;0],’r+’); a x i s([0 1 0 1]);
12 x l a b e l (’{\bf x}’,’fontsize’,14);
13 y l a b e l (’{\bf u}’,’fontsize’,14);
14 t i t l e (’t=0’);
15 p r i n t (’-depsc2’, s p r i n t f (’../../../../rw/Slides/NPDEPics/%s%i.eps’,
filename,0));
16
17 f o r j=1:M+1
18 xs = x(2:end-1) - tau; % Shifted gridpoints
19 uv = [0;u;0]; % Nodal values padded with zero
20 % Find intervals to which shifted points belong and perform linear
interpolation
21 xsp = xs/h; xsp( f i n d (xsp < 0)) = 0;
22 xw = (xsp - f l o o r (xsp))’;
23 uxs = (1-xw).*uv( f l o o r (xsp)+1) + xw.*uv( c e i l (xsp)+1);
24
25 %Timestep for material derivative
26 u = (Mmat+tau*epsilon*Amat)\(Mmat*uxs);
27
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
7.3
p. 81
Numerica
Methods
for PDEs
28 i f (j*tau > tstop)
29 f i g u r e ;
30 p l o t (x,[0;u;0],’r+’); a x i s([0 1 0 1]);
31 x l a b e l (’{\bf x}’,’fontsize’,14);
32 y l a b e l (’{\bf u}’,’fontsize’,14);
33 t i t l e ( s p r i n t f (’Semi-Lagrangian method: M = %f, \\tau = %f,
t=%f’, l e n g t h(x)-1,tau,j*tau));
DRAFT
34 tstop = tstop + T/4;
35 p r i n t (’-depsc2’, s p r i n t f (’../../../../rw/Slides/NPDEPics/%s%i.eps’,
filename,j));
36 end
37 end
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Fig. 243
7.3
p. 814
ǫ = 10 −5 :
7.3
p. 81

u
u
u
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
u
u
u
u
u
u
u
u
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
t = 0.500000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.9
0.8
Semi−Lagrangian method: M = 100.000000, τ = 0.005000, t=0.130000
1
0.9
0.8
Semi−Lagrangian method: M = 100.000000, τ = 0.005000, t=0.255000
1
0.9
0.8
Semi−Lagrangian method: M = 100.000000, τ = 0.005000, t=0.380000
1
0.9
0.8
Semi−Lagrangian method: M = 100.000000, τ = 0.005000, t=0.505000
Numerical
Methods
for PDEs
Example withǫ = 0:
t = 0.150000
1
0.9
1
t = 0.300000
0.8
1
t = 1.000000
0.7
Numerica
Methods
for PDEs
0.7
0.6
0.5
0.7
0.6
0.5
0.7
0.6
0.5
0.7
0.6
0.5
0.9
0.8
0.7
0.8
0.7
0.6
0.9
0.8
0.7
0.7
0.6
0.9
0.8
0.7
0.6
0.5
0.4
0.4
0.4
0.4
0.6
0.5
0.6
0.5
0.6
0.4
0.3
0.2
0.1
0
x
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0.3
0.2
0.1
Fig. 244
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0.5
0.4
0.3
0.2
0.1
0.4
0.3
0.2
0.1
0.5
0.4
0.3
0.2
0.1
0.4
0.3
0.2
0.1
0.5
0.4
0.3
0.2
0.1
0.3
0.2
0.1
0
0
0
0
0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ǫ = 0.1:
1
t = 1.450000
0.6
1
t = 2.000000
0.5
1
t = 2.500000
0.4
1
t = 3.000000
0.35
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Semi−Lagrangian method: M = 100.000000, τ = 0.005000, t=0.130000
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Semi−Lagrangian method: M = 100.000000, τ = 0.005000, t=0.255000
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Semi−Lagrangian method: M = 100.000000, τ = 0.005000, t=0.380000
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Semi−Lagrangian method: M = 100.000000, τ = 0.005000, t=0.505000
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.5
0.4
0.3
0.2
0.1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.3
0.25
0.2
0.15
0.1
0.05
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
0
0
0
0
0
0
0
0
0.2
0.2
0.2
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1
0.1
0.1
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Fig. 245
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
“Reference solution” computed by method of lines, see Ex. 7.3.4, withh = 10 −3 ,τ = 5·10 −5 :
We observe smearing of initial data due to numerical diffusion inherent in the interpolation step of the
semi-Lagrangian method.
7.3
p. 817
✸
7.3
p. 81
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.001000, τ = 0.000050, t=0.125050
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.001000, τ = 0.000050, t=0.250050
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.001000, τ = 0.000050, t=0.375050
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
implicit Euler
explicit Euler
Expl. Euler: h = 0.001000, τ = 0.000050, t=0.500050
Fig. 246
✸
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Numerical
Methods
for PDEs
8
Numerical Methods for Conservation
Laws
Numerica
Methods
for PDEs
Example 7.3.45 (Semi-Lagrangian method for convection-diffusion in 2D).
2nd-order scalar convection diffusion problem (7.3.2),Ω :=]0,1[ 2 ,f = 0,g = 0,
velocity field
v(x) :=
Initial condition: “compactly supported cone shape”
( ) −sin(πx1 )cos(πx 2 )
.
sin(πx 2 )cos(πx 1 )
u0(x) = max(0,1-4*sqrt((x(:,1)-0.5).^2+(x(:,2)-0.25).^2));
semi-Lagrangian finite element Galerkin discretization according to (7.3.41) on regual triangular
meshes of square domainΩ, see Fig. 123.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
7.3
p. 818
KEY CHAPTER
Dependencies:
• Discussion of transport equation, Sect. 7.3.2
• Idea of 1D finite differences, Sect. 1.5.3
• Spurious oscillations introduced by central differencing for transient convection problem, Ex. 7.3.4
• Upwinding and artificial diffusion, Sect. 7.2.2
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.0
p. 82

Conservation laws describe physical phenomena governed by
conservation laws for certain physical quantities (e.g., mass momentum, energy, etc.),
transport of conserved physical quantities.
Numerical
Methods
for PDEs
Focus:
Cauchy problems
Spatial domain Ω = R d (unbounded!)
➤ Cauchy problems are pure initial value problems (no boundary values).
Numerica
Methods
for PDEs
We have already examined problems of this type in connection with transient heat conduction in
Sect. 7.1.4. There thermal energy was the conserved quantity and a prescribed external velocity
fieldv determined the transport.
A new aspect emerging for general conservation laws is that the transport velocity itself may depend
on the conserved quantities themselves, which gives rise to non-linear models.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Rationale: ➊
Finite speed of propagation typical of conservation laws
(Potential spatial boundaries will not affect the solution for some time in the case of
compactly supported initial data, cf. situation for wave equation, where we also examined
the Cauchy problem, see (6.2.15).)
➋ No spatial boundary ➣ need not worry about (spatial) boundary conditions!
(Issue of spatial boundary conditions can be very intricate for conservation laws)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Supplementary and further reading:
[22]: Comprehensive monograph and textbook about so-called finite volume method providing de-
tailed explanations.
[20]: concisely written textbook adopting a mathematical perspective and delving into technical details.
[9]: mathematical monograph about the theory of initial value problems for conservation laws.
8.0
p. 821
Numerical
Methods
for PDEs
8.1.1 Linear advection
Cauchy problem for linear transport equation (advection equation) → Sect. 7.1.4, (7.1.16):
∂
∂t (ρu)+div(v(x,t)(ρu)) = f(x,t) in ˜Ω := R d ×]0,T[ , (8.1.1)
u(x,0) = u 0 (x) for all x ∈ R d (initial conditions) . (8.1.2)
8.1
p. 82
Numerica
Methods
for PDEs
8.1 Conservation laws: Examples
R. Hiptmair
C. Schwab,
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V.
Gradinaru
A. Chernov
DRAFT
u = u(x,t) ˆ= temperature,ρ > 0 ˆ= heat capacity,v = v(x,t) ˆ= prescibed velocity field.
(8.1.1) = linear scalar conservation law
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H.
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Gradinar
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DRAFT
OPTIONAL SECTION
➤ Conserved quantity: thermal energy (density)ρu
(Recall the derivation of (7.1.16) through conservation of energy, cf. (6.1.1).)
8.1
p. 822
8.1
p. 82

Simplfied problem: assume constant heat capacity ρ ≡ 1, no sources f ≡ 0, stationary velocity
fieldv = v(x) ➣ rescaled initial value problem written in conserved variables
Convention:
Special case:
∂u
∂t +div(v(x)u) = 0 in ˜Ω := R d ×]0,T[ ,
u(x,0) = u 0 (x) for all x ∈ R d (initial conditions) .
differential operatordiv acts on spatial independent variable only,
⎛
(divf)(x,t) := ∂f 1
+···+ ∂f f ⎞
1(x,t)
1
, f(x,t) = ⎝ . ⎠ .
∂x 1 ∂x d f d (x,t)
d = 1 ➣ Ω = R,
constant velocity v = const. .
Constant coefficient linear advection in 1D
(8.1.4)
∂u
∂t + ∂
∂x (vu) = 0 in ˜Ω = R×]0,T[ , u(x,0) = u 0 (x) ∀x ∈ R . (8.1.5)
Numerical
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for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.1
p. 825
made it possible to give a meaning to solutions ∈ H 1 (Ω) that are merely continuous and piecwise
differentiable, see Rem. 1.3.25.
Related to (8.1.6): d’Alembert solution formula (6.2.16) for 1D wave equation (6.2.15).
Remark 8.1.11 (Boundary conditions for linear advection).
Recall the discussion in Sects. 7.2.1, 7.3.2, cf. solution formula (7.3.12):
For the scalar linear advection initial boundary value problem
∂u
∂t +div(v(x,t)u) = f(x,t) in ˜Ω := Ω×]0,T[ , (8.1.12)
u(x,0) = u 0 (x) for all x ∈ Ω , (8.1.13)
on a bounded domainΩ ⊂ R d , boundary conditions (e.g., prescribed temperature)
u(x,t) = g(x,t) on Γ in (t)×]0,T[ ,
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.1
p. 82
This is the 1D version of the transport equation (7.3.7) ➣ solution given by (7.3.11)
Numerical
Methods
for PDEs
can be imposed on the inflow boundary
Γ in (t) := {x ∈ ∂Ω: v(x,t)·n(x) < 0} , 0 < t < T . (8.1.14)
Numerica
Methods
for PDEs
(7.3.11)
u(x,t) = u 0 (x−vt) , x ∈ R , 0 ≤ t < T . (8.1.6)
Note:
Bottom line:
Γ in can change with time!
Solutionu = u(x,t) = initial data “travelling” with velocityv.
Solution formula (8.1.6) makes perfect sense even for discontinuous initial datau 0 !
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
★
✧
Knowledge of local and current direction of transport
needed to impose meaningful boundary conditions!
✥
✦
△
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
➥
We should not expectu = u(x,t) to be differentiable in space or time.
A “weaker” concept of solution is required, see Sect. 8.2.3 below.
This consideration should be familiar: for second order elliptic boundary value problems, for which
classical solutions are to be twice continuously differentiable, the concept of a variational solution
8.1
p. 826
8.1
p. 82

8.1.2 Inviscid gas flow
Assumption:
Gas
Fig. 247x
Frictionless gas flow in (infinitely) long pipe
Terminology:
frictionless ˆ= inviscid
variation of gas density negligible (“near incompressibility”)
motion of fluid driven by inertia ↔ conservation of linear momentum
We derive a continuum model for inviscid, nearly incompressible fluid in a straight infinitely long pipe
↔Ω = R (Cauchy problem).
This simple model will be based on conservation of linear momentum, whereas conservation of mass
and energy will be neglected (and violated). Hence, the crucial conserved quantity will be the momentum.
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
A. Chernov
DRAFT
t 1 ∫
t 0
1
2 u 2 (x 1 ,t)− 1 2 u2 (x 0 ,t)dt = τ( 1 2 u2 (x 1 ,t 0 )− 1 2 u2 (x 0 ,t 0 ))+O(τ 2 ) for τ → 0 .
Then employ Taylor expansion for the differences:
u(x 0 ,t 1 )−u(x 0 ,t 0 ) = ∂u
∂t (x 0,t 0 )τ +O(τ 2 ) for τ → 0 ,
1
2 u2 (x 1 ,t 0 )− 1 2 u2 (x 0 ,t 0 ) = ∂
∂x (1 2 u2 )(x 0 ,t 0 )h+O(h 2 ) for h → 0 .
Finally, divide byhandτ and take the limitτ → 0,h → 0:
∂u
∂t + ∂ ( 12 u 2) = 0 inΩ×]0,T[ . (8.1.16)
∂x
(8.1.16) = Burgers equation: a one-dimensional scalar conservation law (without sources)
Remark 8.1.17 (Euler equations).
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Unkown:
by near incompressibility
u = u(x,t) = momentum density ∼ local velocityv = v(x,t) of fluid
8.1
p. 829
8.1
p. 83
✗
✖
Conserved quantity:
(linear) momentum of fluidu = u(x,t)
✔
Numerical
Methods
for PDEs
✕
The above gas model blatantly ignores the fundamental laws of conservation of mass and of energy.
These are taken into account in a famous more elaborate model of inviscid fluid flow:
Numerica
Methods
for PDEs
➣ flux of linear momentum f ∼ v ·u (after scaling: f(u) = 1 2 u·u)
(“momentumuadvected by velocityu”)
Conservation of linear momentum (∼ u): for all control volumesV :=]x 0 ,x 1 [⊂ Ω:
x 1
∫ ∫
u(x,t 1 )−u(x,t 0 )dx+
1
2 u 2 (x 1 ,t)− 1 2 u2 (x 0 ,t)dt = 0 ∀0 < t 0 < t 1 < T . (8.1.15)
x 0 t 0
} {{ }
change of momentum inV
t 1
} {{ }
outflow of momentum
Temporarily assume that u = u(x,t) is smooth in both x and t and set x 1 = x 0 + h, t 1 = t 0 + τ.
First approximate the integrals in (8.1.15).
x 1 ∫
x 0
u(x,t 1 )−u(x,t 0 )dx = h(u(x 0 ,t 1 )−u(x 0 ,t 0 ))+O(h 2 ) for h → 0 ,
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DRAFT
Euler equations [7], a more refined model for inviscid gas flow in an infinite pipe
⎛
∂
ρ ⎞ ⎛ ⎞
⎝ρu⎠+ ∂ ρu
⎝ ρu 2 +p ⎠ = 0 in R×]0,T[ , (8.1.18)
∂t
E
∂x
(E +p)u
u(x,0) = u 0 (x) , ρ(x,0) = ρ 0 (x) , E(x,0) = E 0 (x) forx ∈ R ,
where ρ = ρ(x,t) ˆ= fluid density,[ρ] = kgm −1 ,
u = u(x,t) ˆ= fluid velocity,[u] = ms −1 ,
p = p(x,t) ˆ= fluid pressure,[p] = N,
E = E(x,t) ˆ= total energy density,[E] = Jm −1 .
+ state equation (material specific constitutive equations), e.g., for ideal gas
Conserved quantities (densities):
p = (γ −1)(E − 1 2 ρu2 ) , with adiabatic index 0 < γ < 1 .
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
8.1
p. 830
ρ ↔ mass density , ρu ↔ momentum density , E ↔ energy density.
8.1
p. 83

Underlying physical conservation principles for individual densities:
Numerical
Methods
for PDEs
Our focus below: scalar case m = 1
Numerica
Methods
for PDEs
• First equation
• Second equation
• Third equation
∂ρ
∂t + ∂
∂x
∂(ρu)
(ρu) = 0 ↔ conservation of mass,
+ ∂
∂x (ρu2 +p) = 0 ↔ conservation of momentum,
∂t
∂E
∂t + ∂ ((E +p)u) = 0 ↔ conservation of energy.
∂x
Conservation law for transient state distribution u : ˜Ω ↦→ U: u = u(x,t), for 0 ≤ t ≤ T
∫ ∫ ∫
d
udx+ f(u,x)·ndS(x) = s(u,x,t)dx ∀ “control volumes”V ⊂ Ω . (8.2.1)
dt
V ∂V
V
change of amount inflow/outflow production term
Euler equations (8.1.18) = non-linear system of conservation laws (in 1D)
As is typical of non-linear systems of conservations laws, the analysis of the Euler equations is intrinsically
difficult: hitherto not even existence and uniqueness of solutions for general initial values could
be established. Moreover, solutions display a wealth of complicated structures. Therefore, this course
is confined to scalar conservation laws, for which there is only one unknown real-valued function of
space and time.
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
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DRAFT
Terminology: ✄ flux function f : U ×Ω ↦→ R d
✄ source function s : U ×Ω×]0,T[↦→ R (here usuallys = 0)
For Burgers equation (8.1.16): f(u,x) = 1 2 u2 , s = 0,
For linear advection (8.1.1): f(u,x) = v(x,t)u, s = f(x,t)
(Note: in this case the conserved quantity is actuallyρu, which was again denoted byu)
R. Hiptm
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H.
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V.
Gradinar
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DRAFT
8.2 Scalar conservation laws in 1D
8.2.1 Integral and differential form
What we have seen so far (except for Euler’s equations in Rem. 8.1.17)
∂u
Burgers equation:
∂t + ∂ ( 12 u 2) = 0 inΩ×]0,T[ , (8.1.16)
∂x
∂
linear advection:
∂t (ρu)+div(v(x,t)(ρu)) = f(x,t) in Rd ×]0,T[ . (8.1.1)
Now, we learn about a class of Cauchy problems to which these two belong. First some notations
and terminology:
Ω ⊂ R d ˆ= fixed (bounded/unbounded) spatial domain (Ω = R d = Cauchy problem)
△
8.2
p. 833
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
A. Chernov
DRAFT
☞
✬
✫
(8.2.1) has the same structure as the “conservation of energy law” (6.1.1) for heat conduction.
Conservation of energy:
∫ ∫ ∫
d
ρudx+ j·ndS = f dx for all “control volumes”V (6.1.1)
dt V ∂V V
energy stored inV power flux through∂V heat generation inV
In this case the heat flux was given by
Fourier’s law j(x) = −κ(x)gradu(x) , x ∈ Ω , (2.5.3)
or its extended version (7.1.5). In Fourier’s law the flux is a linear function of derivatives ofu.
Conversely, for the flux function
f : U ×Ω ↦→ R d in (8.2.1) we assume
f only depends on local stateu, not on derivatives ofu!
8.2
p. 83
Numerica
Methods
for PDEs
✩
✪
R. Hiptm
C. Schwa
H.
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V.
Gradinar
A. Chern
DRAFT
computational domain:
space-time cylinder ˜Ω := Ω×]0,T[,T > 0 final time
U ⊂ R m (m ∈ N) ˆ= phase space (state space) for conserved quantititiesu i (usuallyU = R m )
8.2
p. 834
On the other hand we go far beyond Fourier’s law, since
f will in general be a non-linear function ofu!
8.2
p. 83

Remark 8.2.3 (Diffusive flux).
Taking into account the relationship with heat “diffusion”, a flux function of the form of Fourier’s law
(2.5.3)
is called a diffusive flux.
f(u) = −κ(x)gradu ,
Now, integrate (8.2.1) over time period[t 0 ,t 1 ] ⊂ [0,T] and use fundamental theorem of calculus:
△
Numerical
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 8.2.7 (Boundary values for conservation laws).
Suitable boundary values on∂Ω×]0,T[ ?
→ usually tricky question (highlyf-dependent)
Reason: remember discussion in Rem. 8.1.11, meaningful boundary conditions hinge on knowledge
of local (in space and time) transport direction, which, in a non-linear conservation law, will usually
depend on the unknown solutionu = u(x,t).
8.2.2 Characteristics
△
Numerica
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R. Hiptm
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H.
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DRAFT
Space-time integral form of (8.2.1), cf. (8.1.15),
We consider Cauchy problem (Ω = R) for one-dimensional scalar conservation law (8.2.6):
∫
V
for all
∫
u(x,t 1 )dx−
V ∂V
V ⊂ Ω,0 < t 0 < t 1 < T ,n ˆ= exterior unit normal at∂V
t 1
∫ ∫
∫ ∫
u(x,t 0 )dx+ f(u,x)·ndS(x)dt =
t 0 t 0
[Gauss theorem Thm. 2.4.5] (local) differential form of (8.2.1):
t 1
V
s(u,x,t)dxdt (8.2.4)
8.2
p. 837
Numerical
Methods
for PDEs
∂u
∂t + ∂ f(u) = 0 in R×]0,T[ ,
∂x
u(x,0) = u 0 (x) in R .
Assumption: flux functionf : R ↦→ R smooth (f ∈ C 2 ) and convex [29, Def. 5.5.2]
(8.2.9)
8.2
p. 83
Numerica
Methods
for PDEs
∂
∂t u+div xf(u,x) = s(u,x,t) in ˜Ω . (8.2.5)
Recall [29, Thm. 5.5.2]: f convex ⇒ derivativef ′ increasing
div acting on spatial variablexonly
+ initial condition u(x,0) = u 0 (x), x ∈ Ω
C. Schwab,
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R. Hiptmair
Special cased = 1 ↔ (8.2.5) = one-dimensional scalar conservation law for “density”u : ˜Ω
DRAFT
↦→ R
∂u
∂t (x,t)+ ∂
∂x (f(u(x,t),x)) = s(u(x,t),x,t) in]α,β[×]0,T[, α,β ∈ R∪{±∞} . (8.2.6)
Definition 8.2.10 (Characteristic curve for one-dimensional scalar conservation law).
A curve Γ := (γ(τ),τ) : [0,T] ↦→ R×]0,T[ in the (x,t)-plane is a characteristic curve of
(8.2.9), if
d
dτ γ(τ) = f′ (u(γ(τ),τ)) , 0 ≤ τ ≤ T , (8.2.11)
whereuis a continuously differentiable solution of (8.2.9).
R. Hiptm
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H.
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Gradinar
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DRAFT
8.2
p. 838
8.2
p. 84

t
γ
δt
fast→
← slow
✁ x−t-diagram
d
γ(τ) = speed of interfaceγ.
dτ
Numerical
Methods
for PDEs
So,
u is constant on a characteristic curve.
➣ f ′ (u) is constant on a characteristic curve.
(8.2.11) ⇒ slope of characteristic curve is constant!
Characteristic curve through(x 0 ,0) = straight line(x 0 +f ′ (u 0 (x 0 ))τ,τ),0 ≤ τ ≤ T !
Numerica
Methods
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δx
Example 8.2.12 (Characteristics for advection).
Constant linear advection (8.1.5): f(u) = vu
x
t
R. Hiptmair
C. Schwab,
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V.
Gradinaru
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!? implicit solution formula for (8.2.9) (f ′ monotone !):
u(x,t) = u 0 (x−f ′ (u(x,t))t) . (8.2.14)
R. Hiptm
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➸ characteristicsγ(τ) = vτ +c,c ∈ R.
✄
v
DRAFT
Example 8.2.15 (Breakdown of characteristic solution formula).
DRAFT
solution (8.1.6)
u(x,t) = u 0 (x−vt)
1
meaningful for any u 0 ! (cf. Sect. 7.3.2)
✸
x
8.2
p. 841
8.2
p. 84
This example reveals a close relationship between streamlines (→ Sect. 7.1.1) and characteristic
curves. That the latter are a true generalization of the former is also reflected by the following simple
observation, which generlizes the considerations in Sect. 7.3.2, (7.3.9).
Numerical
Methods
for PDEs
1
0.8
for Burger’s equation (8.1.16):
(f(u) = 1 2 u2 smooth and strictly convex)
Numerica
Methods
for PDEs
✬
Lemma 8.2.13 (Classical solutions and characteristic curves).
✫
Proof.
curve:
Smooth solutions of (8.2.9) are constant along characteristic curves.
Apply chain rule twice, cf. (7.3.9), and use the defining equation (8.2.11) for a characteristic
d
dτ
chain rule
u(γ(τ),τ) = ∂u
∂x (γ(τ),τ) d
dτ
γ(τ)+
∂u
∂t (γ(τ),τ)
(8.2.11)
= ∂u
∂x (γ(τ),τ)·f′ (u(γ(τ),τ))+ ∂u
∂t (γ(τ),τ)
chain rule
= ( ∂
∂x f(u)) (γ(τ),τ)+ ∂u
∂t (γ(τ),τ) = 0 .
✎ notation: f ′ ˆ= derivative of flux functionf : U ⊂ R ↦→ R
✩
✪
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.2
p. 842
t,u(x)
0.6
0.4
0.2
0
u0
−1.5 −1 −0.5 0 0.5 1 1.5
x
Fig. 248
✄ f ′ (u) = u (increasing)
✁ ifu 0 smooth and decreasing
➤ characteristic curves intersect !
➤ solution formula (8.2.14) becomes invalid
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
8.2
p. 84

1
1
Numerical
Methods
for PDEs
Still, (8.2.18) encounters problems, if a discontinuity of u coincides with an edge of the space-time
rectangle.
Numerica
Methods
for PDEs
0.8
0.8
u(t,.)
0.6
0.4
u(t,.)
0.6
0.4
Idea:
Obtain weak form of (8.2.16) from (8.2.17) by integration by parts, that is,
application of Green’s first formula Thm. 2.4.7 in space-time!
0.2 t = 0.00
t = 0.50
t = 0.80
t = 1.00
0 t = 1.20
t = 1.30
−1.5 −1 −0.5 0
x
0.5 1 1.5
Fig. 249
t < 1.3: solution by (8.2.14)
0.2
0
t = 1.30
t = 1.35
t = 1.40
t = 1.50
t = 1.70
−1.5 −1 −0.5 0
x
0.5 1 1.5
Fig. 250
the wave breaks: “multivalued solution”
✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
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DRAFT
STEP I:
Test (8.2.17) with compactly supported smooth function Φ : ˜Ω ↦→ R, Φ(·,T) = 0, and
integrate over space-time cylinder ˜Ω = R×[0,T]:
∫ ( ) f(u)
(8.2.17) div (x,t) Φ(x,t)dxdt = 0 .
u
˜Ω
R. Hiptm
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Gradinar
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DRAFT
breakdown of classical solutions & Ex. 8.2.12 ➥ new concept of solution of (8.2.9)
8.2.3 Weak solutions
“Space-time Gaussian theorem”
t
8.2
p. 845
Numerical
Methods
for PDEs
STEP II: Perform integration by parts using Green’s first formula Thm. 2.4.7 on ˜Ω:
∫ ( ) f(u)
div (x,t) Φ(x,t)dxdt = 0
u
˜Ω
∫
Thm. 2.4.7
⇒
˜Ω
( ) ∫∞
f(u)
·grad
u (x,t) Φdxdt+ u(x,0)Φ(x,0)dx = 0 ,
−∞
because ∂˜Ω = R × {0} ∪ R × {T} with “normals” n = ( 0
−1
) (t = 0 boundary) and n =
( 01 )
(t = T boundary), which has to be taken into account in the boundary term in Green’s formula. The
“t = T boundary part” does not enter asΦ(·,T) = 0.
8.2
p. 84
Numerica
Methods
for PDEs
∂u
∂t + ∂ f(u) = 0 (8.2.16)
∂x
⇕
( ) f(u)
div (x,t) = 0 in
u
˜Ω . (8.2.17)
∀ “space-time control volumes”Ṽ ⊂ ˜Ω:
∫ ( ) ( ) f(u(˜x)) nx (˜x)
· dS(˜x) = 0 ,
u(˜x) n t (˜x)
∂Ṽ
ñ := (n x ,n t ) T ˆ= space-time unit normal
Ṽ
ñ
x
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C. Schwab,
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DRAFT
Note thatu(x,0) is fixed by the initial condition:
u(x,0) = u 0 (x).
Definition 8.2.19 (Weak solution of Cauchy problem for scalar conservation law).
Foru 0 ∈ L ∞ (R),u : R×]0,T[↦→ R is a weak solution of the Cauchy problem (8.2.9), if
u ∈ L ∞ (R×]0,T[) ∧
for allΦ ∈ C ∞ 0
∫∞ ∫T
−∞ 0
(R×[0,T[),Φ(·,T) = 0.
{
u ∂Φ } ∫∞
∂t +f(u)∂Φ dtdx+ u
∂x 0 (x)Φ(x,0)dx = 0 ,
−∞
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DRAFT
(8.2.17) for space-time rectangle Ṽ =]x 0,x 1 [×]t 0 ,t 1 [ ➤ integral form of (8.2.16), cf. (8.2.4):
x 1
x 1
t 1
∫ ∫ ∫ ∫
u(x,t 1 )dx− u(x,t 0 )dx = f(u(x 0 ,t))dt− f(u(x 1 ,t))dt . (8.2.18)
x 0 x 0 t 0 t 0
t 1
8.2
p. 846
Remark 8.2.21 (Properties of weak solutions).
8.2
p. 84

By reversing integration by parts, it is easy to see that
✗
✖
u weak solution of (8.2.9) &u ∈ C 1 ⇐⇒ u classical solution of (8.2.9).
✔
✕
Numerical
Methods
for PDEs
Apply this insight to vectorfield on space-time domain ˜Ω = R×]0,T[:
∂u
∂t + ∂
( ) f(u)
∂x f(u) = 0 ⇔ div (x,t) = 0 in
u
˜Ω . (8.2.17)
} {{ }
=:j
Numerica
Methods
for PDEs
Arguments from mathematical integration theory confirm
u ∈ L ∞ loc (R×]0,T[) weak solution of (8.2.9) ⇒ u satisfies integral form (8.2.18)
for “almost all”x 0 < x 1 ,0 < t 0 < t 1 < T .
△
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V.
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DRAFT
Normal atC 1 -curveΓ := τ ↦→ (γ(τ),τ) in(γ(τ),τ)
( )
1 1
ñ = √
1+|ṡ| 2 −ṡ
, ṡ := dγ (τ) “speed of curve” .
dτ
To see this, recall that the normal is orthogonal to the tangent vector ( ṡ) 1 and that in 2D the direction
orthogonal to ( x ) ( 1
x is given by −x2 )
2 x . 1
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DRAFT
8.2.4 Jump conditions
For piecewise smooth vectorfieldj : Ω ⊂ R 2 :
8.2
p. 849
Numerical
Methods
for PDEs
“normal continuity” of
piecewise smooth vectorfield(f(u),u) T
t
Γ := (γ(τ),τ)
8.2
p. 85
Numerica
Methods
for PDEs
“divj = 0”
⇕
∫
j·ndS = 0 ∀ control volumesV ⊂ Ω
∂V
Necessary condition:
★
✧
Continuity of normal components
across discontinuities
discontinuous divergence-free vectorfield
✄
✥
✦
Fig. 251
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
⇕
( ) ( ) 1 [f(u)]
− dγ · = 0 , (8.2.22)
[u]
dτ
where[·] ˆ= jump acrossΓ.
˜Ω ˜Ω r
l
( ) 1
ñ ‖
−ṡ
Fig. 252x
Terminology: (8.2.22) = Rankine-Hugoniot (jump) condition, shorthand notation:
ṡ(u l −u r ) = f l −f r
, ṡ := dγ
dτ
“propagation speed of discontinuity” (8.2.23)
R. Hiptm
C. Schwa
H.
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V.
Gradinar
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DRAFT
To see this, consider a slender tiny rectangle aligned with a line of discontinuity of j. In the absence
of normal continuity a net flux through its boundary will result, provided that the rectangle is small
enough (“pillbox argument”).
8.2
p. 850
Remark 8.2.26 (Discontinuity connecting constant states).
8.2
p. 85

The simplest situtation compliant with Rankine-Hugoniot jump condition: constant states to the left
and right of the curve of discontinuity (8.2.22):
Numerical
Methods
for PDEs
1
1
Numerica
Methods
for PDEs
t
0.8
0.8
u l
ṡ
u r
Fig. 253x
0
1
u(x,t) =
{
ul ∈ R , for x < ṡt ,
u r ∈ R , for x < ṡt ,
(8.2.27)
with constant speed ṡ of discontinuity, according
to (8.2.23) given by (foru l ≠ u r )
ṡ = f(u l)−f(u r )
u l −u r
.
△
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C. Schwab,
H.
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V.
Gradinaru
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DRAFT
u 0
(x)/ t
0.6
u 0
char.
0.4
0.2
0
−1.5 −1 −0.5 0 0.5 1 1.5
x
Fig. 254
intersecting characteristics
u 0
(x)/ t
0.6
u 0
char.
0.4
0.2
0
−1.5 −1 −0.5 0 0.5 1 1.5
x
Fig. 255
diverging characteristics
R. Hiptm
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DRAFT
8.2.5 Riemann problem
Definition 8.2.29 (Shock).
If Γ is a smooth curve in the (x,t)-plane and u a weak solution of (8.2.9), a discontinuity of u
acrossΓis called a shock.
Rem. 8.2.26 ➤ shock speeds ↔ Rankine-Hugoniot jump conditions:
8.2
p. 853
(x 0 ,t 0 ) ∈ Γ: ṡ = f(u l)−f(u r )
u l −u r
,
u l := lim ǫ→0 u(x 0 −ǫ,t 0 ) ,
u r := lim ǫ→0 u(x 0 +ǫ,t 0 ) .
(8.2.30)
8.2
p. 85
Rem. 8.2.26: situation of locally constant states in particularly easy.
Numerical
Methods
for PDEs
✬
Numerica
Methods
for PDEs
✩
Consider: Cauchy-problem (8.2.9) for piecewise constant initial datau 0 .
Definition 8.2.28 (Riemann problem).
{
ul ∈ R , ifx < 0 ,
u 0 (x) =
u r ∈ R , ifx > 0 .
ˆ= Riemann problem for (8.2.9)
Assumption, cf. Sect. 8.2.2: flux function f : R ↦→ R smooth & convex
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DRAFT
Lemma 8.2.31 (Shock solution of Riemann problem).
{
ul forx < ṡt ,
u(x,t) = ṡ := f(u l)−f(u r )
, x ∈ R, 0 < t < T ,
u r forx > ṡt , u l −u r
✫
u 0
(x)/ t
is weak solution of Riemann problem (→ Def. 8.2.28) for (8.2.9).
1
0.8
0.6
0.4
u 0
characteristic
shock curve
Riemann problem: Burger flux
u 0
(x)/ t
1
0.8
0.6
0.4
u 0
characteristic
shock curve
Riemann problem: Burger flux
u 0
(x)/ t
1
0.8
0.6
0.4
u 0
characteristic
shock curve
Riemann problem: Burger flux
✪
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
f ′ non-decreasing ➤ pattern of characteristic curves for Riemann problem:
0.2
0.2
0.2
0
0
0
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5 −1 −0.5 0 0.5 1 1.5
x
x
x
Burgers fluxf(u) = 1 2 u2 ,u l > u r : characteristic curves impinge on shock
Fig. 256
8.2
p. 854
8.2
p. 85

−1.5 −1 −0.5 0 0.5 1 1.5
1
u 0
characteristic
shock curve
Riemann problem: Burger flux
1
u 0
characteristic
shock curve
Riemann problem: Burger flux
1
u 0
characteristic
shock curve
Riemann problem: Burger flux
Numerical
Methods
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1
Numerica
Methods
for PDEs
u 0
(x)/ t
0.8
0.6
u 0
(x)/ t
0.8
0.6
u 0
(x)/ t
0.8
0.6
0.8
u ǫ (x,t) = classical solution of (8.2.33) for allt > 0,
x ∈ R (only foru l > u r !).
0.4
0.2
0.4
0.2
0.4
0.2
u(x,1)
0.6
✁ u l > u r , t = 0.5
0
x
Burgers fluxf(u) = 1 2 u2 ,u l < u r :
0
−1.5 −1 −0.5 0 0.5 1 1.5
x
Example 8.2.32 (Vanishing viscosity for Burgers equation).
0
−1.5 −1 −0.5 0 0.5 1 1.5
x
characteristic curves emanate from shock
(expansion shock)
Fig. 257
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C. Schwab,
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V.
Gradinaru
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0.4
0.2
ε = 0.1
ε = 0.05
ε = 0.01
ε = 0.001
ε = 0.0001
0
−1.5 −1 −0.5 0 0.5 1 1.5
x
Fig. 258
emerging shock forǫ → 0
u ǫ → u from Lemma 8.2.31 inL ∞ (R).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
DRAFT
There is no such material as an “invsicid” fluid in nature, because in any physical system there will be
a tiny amount of friction. This leads us to the very general understanding that conservation laws can
usually be regarded as limit problemsǫ = 0 for singularly perturbed transport-diffusion problems with
an “ǫ-amount” of diffusion.
In 1D, for anyǫ > 0 these transport-diffusion problems will possess a unique smooth solution. Studying
its behavior for ǫ → 0 will tell us, what are “physically meaningful” solutions for the conservation
8.2
p. 857
8.2
p. 85
law. This consideration is called the vanishing viscosity method to define solutions for conservation
laws.
Here we pursue this idea for Burgers equation, see Sect. 8.1.2.
Viscous Burgers equation:
∂u
∂t + ∂
∂x
( 12 u 2) =
ǫ ∂2 u
∂x 2 . (8.2.33)
dissipative (viscous) term
Travelling wave solution of Riemann problem for (8.2.33) via Cole-Hopf transform → [12,
Sect. 4.4.1]
u ǫ (x,t) = w(x−ṡt) , w(ξ) = u r + 2 1(u l −u r ) ( ( ) ξ(ul −u r ) )
1−tanh , ṡ = 1
4ǫ 2 (u l +u r ) .
Numerical
Methods
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Highly accurate numerical solution of
Riemann problem for (8.2.33)
u l < u r u ǫ (x,0.5)✄
no shock asǫ → 0 !
ε=0.0001
u ǫ → a piecewise linear function! −1 −0.5 0 0.5 1 1.5
Let us try to derive a (weak) solution of the homogeneous scalar conservation law (8.2.16) with the
structure observed in Ex. 8.2.32.
u(x,0.5)
1
0.8
0.6
0.4
0.2
0
x
ε = 0
ε = 0.1
ε = 0.01
ε = 0.001
✸
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Idea:
conservation law (8.2.16) homogeneous in spatial/temporal derivatives:
8.2
p. 858
∂u
∂t + ∂
∂x f(u) = 0 in R×R+ ⇒ ∂u λ
∂t + ∂
∂x f(u λ) = 0 in R×R + ,
8.2
p. 86

u λ (x,t) := u(λx,λt), λ > 0. This suggests that we look for solutions of the Riemann problem that
are constant on all straight lines in thex−t-plane that cross(0,0) T .
Numerical
Methods
for PDEs
because(f ′ ◦g)(x/t) = x/t and by fundamental theorem of calculus.
✷
Numerica
Methods
for PDEs
try similarity solution:
u(x,t) = ψ(x/t)
← insert in ∂u
∂t + ∂
∂x f(u) = 0
Terminology: solution of Lemma 8.2.34 = rarefaction wave: continuous solution !
1
1
f ′ (ψ(x/t))ψ ′ (x/t) = (x/t)ψ ′ (x/t) ∀x ∈ R, 0 < t < T .
ψ ′ ≡ 0 ∨ f ′ (ψ(w)) = w ⇔ ψ(w) = (f ′ ) −1 (w) .
f ′ strictly monotone !
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
A. Chernov
DRAFT
u(t,x)
0.5
0
−0.5
1
0.8
0.6
t
0.4
0
−0.5
1
0.8
0.6
1.5
1
0.4
0.5
0
0
0.2
−0.5
0.2
−0.5
−1
−1
0 −1.5
t
0 −1.5
x
x
Burger flux functionf(u) = 1 2 u2 ,u l < u r : rarefaction wave solutions
u(t,x)
0.5
0.5
1
1.5
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
t
ṡ = f ′ (u l ) ṡ = f ′ (u r )
Fig. 259
x
✬
Lemma 8.2.34. (Rarefaction solution of
Riemann problem)
Iff ∈ C 2 (R) strictly convex,u l < u r , then
⎧
⎪⎨ u l for x < f ′ (u l )t ,
u(x,t) := g( x t ⎪⎩
) for f′ (u l ) < x t < f′ (u r ) ,
u r for x > f ′ (u r )t ,
✫
g := (f ′ ) −1 , is a weak solution of the
Riemann problem (→ Def. 8.2.28).
✩
8.2
p. 861
R. Hiptmair
C. Schwab,
H.
✪Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
∫∞
⎫⎪
∂x dx+ ∂Φ
u r
∂t +F(u r) ∂Φ ⎬
∂x dx dt
⎪
f(u r )t ⎭
g ′ ( x t )x t 2Φ−f′ (g( x t ))1 t g′ ( x t )Φdxdt = 0 , 8.2
Proof. We show that the rarefaction solution is a weak solution according to Def. 8.2.19 ➣ for
Φ ∈ C0 ∞(R×]0,T[)
⎧
∫T f ⎪⎨ ∫
′ (u l )t
∂Φ
u l
∂t
0
⎪⎩
+f(u l) ∂Φ
f ′ ∫(u r t)
∂x dx+ g( x t )∂Φ ∂t +f(g(x t ))∂Φ
−∞
f ′ (u l )t
∫T f ′ ∫(u r )t
=
0 f ′ (u l )t
Numerical
Methods
for PDEs
p. 862
8.2.6 Entropy condition
Sect 8.2.5 ➤ Non-uniqueness of weak solutions:
iff ′ is increasing andu l < u r both a shock and a rarefaction wave provide valid weak solutions.
u(t,x)
1
0.5
0
−0.5
1
0.8
0.6
t
0.4
0.2
0
−1.5
−1
−0.5
x
0
0.5
1
1.5
Fig. 260
Riemann solution (Burgers equation):
shock
?
←→
How to select “physically meaningful” = admissible solution ?
Vanishing viscosity technique
u(t,x)
1
0.5
0
−0.5
1
0.8
0.6
t
0.4
0.2
0
−1.5
−1
−0.5
x
0
0.5
1
1.5
Fig. 261
Riemann solution (Burgers equation):
rarefaction wave
(→ Ex. 8.2.32): add an “ǫ-amount” of diffusion (“friction”) and study
solution forǫ → 0.
8.2
p. 86
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.2
p. 86

t
t
However, desirable: simple selection criteria (entropy conditions)
Numerical
Methods
for PDEs
Analytical solution available for Burgers eqution (8.1.16) with intial data, see [12, Sect. 3.4, Ex. 3]
{
0 , ifx < 0 orx > 1 ,
u 0 (x) =
1 , if0 ≤ x ≤ 1 .
Numerica
Methods
for PDEs
Definition 8.2.35 (Lax entropy condition).
u ˆ= weak solution of (8.2.9), piecewise classical solution in a neigborhood of C 2 -curve Γ :=
(γ(τ),τ),0 ≤ τ ≤ T , discontinuous acrossΓ.
u satisfies the Lax entropy
condition in(x 0 ,t 0 ) ∈ Γ
:⇔ f ′ (u l ) > ṡ := f(u l)−f(u r )
u l −u r
> f ′ (u r ) .
⇕
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
✗
Gradinaru
A. Chernov
✔
DRAFT
Characteristic curves must not emanate from shock ↔ no “generation of information”
✖
✕
Parlance: shock satisfying Lax entropy condition = physical shock
u(t,x)
1
0.8
0.6
0.4
0.2
0
4
3.5
3
2.5
2
1.5
1
0.5
t
0
−0.5
0
0.5
1
x
1.5
2
2.5
3
3.5
Fig. 262
4
3.5
3
2.5
2
1.5
u = 0
1
0.5
u(x,t) = x/t
u = 0
u = 1
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
x
Fig. 263
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.2
p. 865
8.2
p. 86
Physically meaningful weak solution of conservation law = entropy solution
For scalar conservation laws with locally Lipschitz-continuous flux functionf:
Existence & uniqueness of entropy solutions
Numerical
Methods
for PDEs
Vector field inx−t-plane
( ) f(u(x,t))
u(x,t)
4.5
4
3.5
3
2.5
Numerica
Methods
for PDEs
for entropy solutionu = u(x,t) ✄.
2
Remark 8.2.36 (General entropy solution for 1D scalar Riemann problem). → [23]
Entropy solution of Riemann problem (→ Def. 8.2.28) for (8.2.9) with arbitraryf ∈ C 1 (R):
⎧
⎪⎨ argmin(f(u)−ξu) , ifu l < u r ,
u
u(x,t) = ψ(x/t) , ψ(ξ) := l ≤u≤u r
⎪⎩
argmax(f(u)−ξu) , ifu l ≥ u r .
u r ≤u≤u l
(8.2.37)
△
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Observe the normal continuity across the shock:
the vector field is tangential to the shock curve.
8.2.7 Properties of entropy solutions
1.5
1
0.5
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
x
Fig. 264
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Example 8.2.38 (Entropy solution of Burgers equation).
8.2
p. 866
8.2
p. 86

Setting:
with flux function
u ∈ L ∞ (R×]0,T[) weak (→ Def. 8.2.19) entropy solution of Cauchy problem
∂u
∂t + ∂
∂x f(u) = 0 in R×]0,T[ , u(x,0) = u 0(x) , x ∈ R . (8.2.9)
f ∈ C 1 (R) (not necessarily convex/concave).
Numerical
Methods
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t
(¯x,¯t)
ṡ
t
D + (I 0 )
Numerica
Methods
for PDEs
1
Notation: ū ∈ L ∞ (R×]0,T[) ˆ= entropy solution w.r.t. initial data ū 0 ∈ L ∞ (R).
1
✬
Theorem 8.2.39 (Comparison principle for scalar conservation laws).
Ifu 0 ≤ ū 0 a.e. on R ⇒ u ≤ ū a.e. on R×]0,T[
✫
✩R. Hiptmair
C. Schwab,
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Harbrecht
V.
Gradinaru
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DRAFT
✪
D − (¯x,¯t)
Fig. 265x
maximal domain of dependence of(¯x,¯t) ∈ ˜Ω
Analoguous to Thm. 6.2.18:
ṡ
I
Fig. 266 0 x
maximal domain of influence ofI 0 ⊂ R
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
With obvious consequences:
u 0 (x) ∈ [α,β] on R ⇒ u(x,t) ∈ [α,β] on R×]0,T[
✬
Corollary 8.2.43 (Domain of dependence for scalar conservation law). → [9, Cor. 6.2.2]
The value of the entropy solution at (¯x,¯t) ∈ ˜Ω depends only on the restriction of the initial data
to{x ∈ R:|x− ¯x| < ṡ¯t}.
✩
8.2
p. 869
✫
✪
8.2
p. 87
L ∞ -stability (➣ no blow-up can occur!)
∀0 ≤ t ≤ T: ‖u(·,t)‖ L ∞ (R) ≤ ‖u 0‖ L ∞ (R) . (8.2.40)
Numerical
Methods
for PDEs
Another strand of theoretical results asserts that the solution of a 1D scalar conservation law cannot
develop oscillations:
Numerica
Methods
for PDEs
✗
✔
u solves (8.2.9) ➤ No. of local extrema (in space) ofu(·,t) decreasing with time
✬
✩
✖
✕
Theorem 8.2.41 (L 1 -contractivity of evolution for scalar conservation law).
∫ ∫
∀t ∈]0,T[, R > 0: |u(x,t)|dx ≤ |u 0 (x)|dx ,
|x|

↔ related to advection with velocityv(x,t) = u(x,t):
∂u
(x,t) + u(x,t)∂u
∂t ∂x (x,t) = 0 in R×]0,T[ .
↕ ↕
∂u
(x,t) + v(x,t)∂u
∂t ∂x (x,t) = 0 in R×]0,T[ .
Ifu 0 (x) ≥ 0, then, by Thm. 8.2.39,u(x,t) ≥ 0 for all0 < t < T , that is, positive direction of transport
throughout.
Heeding guideline from Sect. 7.3.1:
use upwind discretization (backward differences) in space!
on (infinite) equidistant grid, meshwidthh > 0,x j = hj,j ∈ Z, obtain semi-discrete problem
for nodal values µ j = µ j (t) ≈ u(x j ,t)
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
MATLAB demo: run Numcourses/numhyp/matlab/1DScalarConsLaws/Burger_backward/burger_naive.m
burger_naive(50);
8.3.1 Semi-discrete conservation form
✸
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H.
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V.
Gradinar
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DRAFT
∂u
(x,t) + u(x,t)∂u
∂t ∂x (x,t) = 0 in R×]0,T[ .
↕ ↕
(8.3.3)
µ j −µ j−1
˙µ j (t) + µ j = 0 , j ∈ Z , 0 < t < T .
h
Numerical experiment with Cauchy problem from Ex. 8.2.38, h = 0.08, integration of (8.3.3) with
MATLAB ode45.
8.3
p. 873
Objective: spatial semi-discretization of Cauchy problem
∂u
∂t + ∂
∂x f(u) = 0 in R×]0,T[ , u(x,0) = u 0(x) , x ∈ R . (8.2.9)
on (infinite) equidistant spatial mesh with mesh widthh > 0
8.3
p. 87
h = 0.080000, t=1.011954
h = 0.080000, t=2.050846
h = 0.080000, t=3.073055
h = 0.080000, t=4.000000
1
0.8
0.6
finite difference solution
exact solution
1
0.8
0.6
finite difference solution
exact solution
1
0.8
0.6
finite difference solution
exact solution
1
0.8
0.6
finite difference solution
exact solution
Numerical
Methods
for PDEs
M := {]x j−1 ,x j [: x j := jh, j ∈ Z} . (8.3.4)
mesh cells and dual cells
✄
x j−1 x j x j+1 xFig. j+2 268
Numerica
Methods
for PDEs
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
Fig. 267
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
Finite volume interpretation of nodal unknownsµ j (↔x j ,j ∈ Z):
= conserved quantities in dual cells ]x j−1/2 ,x j+1/2 [, midpointsx j−1/2 := 1 2 (x j +x j−1 ):
Observation from numerical experiment:
of shock correctly!
OK for rarefaction wave, but scheme cannot capture speed
{
1 , ifj < 0 ,
Analysis: consider µ j (0) =
0 , ifj ≥ 0 .
↔ Riemann problem withu 0 (x) = 1 forx < 0−ǫ,u 0 (x) = 0 forx > 0−ǫ,ǫ ≪ 1.
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⃗µ(t) := ( µ j (t) ) j∈Z ∈ RZ ←→
x ∫j+1/2
µ j (t) ≈ 1 u(x,t)dx . (8.3.5)
h
x j−1/2
u N (x,t) =
j∈Zµ ∑ j (t)χ ]xj−1/2 ,x j+1/2 [(x) . (8.3.6)
a function !
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Entropy solution (for thisu 0 ) = travelling
shock (→ Lemma 8.2.31), speed
ṡ = 1 2 > 0
✁✄
Numerical solution:
⃗µ(t) = ⃗µ 0 for allt > 0 !
✎ notation: characteristic function χ ]xj−1/2 ,x j+1/2 [ (x) = {
1 , ifxj−1/2 < x ≤ x j+1/2 ,
0 elsewhere.
➤ 3-point FDM (8.3.3) “converges” to wrong solution !
8.3
p. 874
➥
(
µj (t) ) j∈Z ←→ piecewise constant approximationu N(t) ≈ u(·,t)
8.3
p. 87

Note: u N (t) discontinuous at dual cell boundariesx j+1/2 !
By spatial integration over dual cells, which now play the role of the control volumes in (8.2.1):
x ∫j+1/2
d
u(x,t)dx+f(u(x
dt
j+1/2 ,t))−f(u(x j−1/2 ,t)) = 0 , j ∈ Z , (8.3.9)
x j−1/2
(
)
f(u N (x j+1/2 ,t)) −f(u N (x j−1/2 ,t)) = 0 , j ∈ Z . (8.3.10)
} {{ } } {{ }
? ?
(8.3.5) dµ j
dt (t)+ 1 h
Problem: jump of u N (t) ➣ ambiguity of values u N (x j+1/2 ,t), u N (x j−1/2 ,t)), as we encountered
it in the context of upwind quadrature in Sect. 7.2.2.1.
Abstract “solution”:
Approximation f(u N (x j+1/2 ,t)) ≈ f j+1/2 (t) := F(µ j−ml +1(t),...,µ j+mr (t)) , j ∈ Z ,
with numerical flux function F : R m l+m r
↦→ R, m l ,m r ∈ N 0 .
Note:
the same numerical flux function is used for all dual cells!
Finite volume semi-discrete evolution for (8.2.9) in conservation form:
dµ (
)
j
dt (t) = −1 F(µ
h j−ml +1(t),...,µ j+mr (t))−F(µ j−ml (t),...,µ j+mr −1(t)) , j ∈ Z .
(8.3.11)
numerical flux (function)F : R m l+m r ↦→ R
Special case: 2-point numerical flux (m l = m r = 1): F = F(v,w)
(v ˆ= left state,w ˆ= right state)
dµ j
(8.3.11)
dt (t) = −1 ( F(µj (t),µ
h j+1 (t))−F(µ j−1 (t),µ j (t)) ) , j ∈ Z . (8.3.12)
Assumption on numerical flux functions: F Lipschitz-continuous in each argument.
Code 8.3.13: Wrapper code for finite volume evolution with 2-point flux
1 f u n c t i o n ufinal = consformevl(a,b,N,u0,T,F)
2 % finite volume discrete evolution in conservation form with 2-point flux, see
(8.3.12)
3 % Cauchy problem over time [0,T] restricted to finite interval [a,b],
4 % equidistant mesh with meshwidth N cells, meshwidth h := b−a /N.
5 % 2-point numerical flux function F = F(v,w) passed in handle F
6 h = (b-a)/N; x = a+0.5*h:h:b-0.5*h; % centers of dual cells
7 mu0 = h*u0(x)’; % vector ⃗µ 0 of initial cell averages (column vector)
8 % right hand side function for MATLAB ode solvers
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R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
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DRAFT
8.3
p. 877
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8.3
p. 878
9 odefun = @(t,mu) (-1/h*fluxdiff(mu,F));
10 % timestepping by explicit Runge-Kutta method of order 5
11 options = odeset(’abstol’,1E-8,’reltol’,1E-6,’stats’,’on’);
12 [t,MU] = ode45(odefun,[0 T],mu0,options);
13 % 3D graphical output of u(x,t) over space-time plane
14 [X,T] = meshgrid(x,t);
15 f i g u r e ; s u r f(X,T,MU/h); colormap(copper);
16 x l a b e l (’{\bf x}’,’fontsize’,14);
17 y l a b e l (’{\bf t}’,’fontsize’,14);
18 z l a b e l (’{\bf u}’,’fontsize’,14);
19 ufinal = MU(:,end);
20 end
21
22 f u n c t i o n fd = fluxdiff(mu,F)
23 n = l e n g t h(mu); fd = zeros(n,1);
24 % constant continuation of data outside [a,b]
25 fd(1) = F(mu(1),mu(2)) - F(mu(1),mu(1));
26 f o r j=2:n-1
27 fd(j) = F(mu(j),mu(j+1)) - F(mu(j-1),mu(j)); % see (8.3.12)
28 end
29 fd(n) = F(mu(n),mu(n)) - F(mu(n-1),mu(n));
30 end
8.3.2 Discrete conservation property
An evident first property of finite volume methods in conservation form:
µ j (0) = µ 0 ∈ R ∀j ∈ Z ⇒ µ j (t) = µ 0 ∀j ∈ Z, ∀t > 0 . (8.3.15)
that is, constant solutions are preserved by the method.
A “telescopic sum argument” combined with the interpretation (8.3.6) shows that the conservation
form (8.3.11) of the semi-discrete conservation law involves
x m+1/2 ∫
d
m∑ dµ j
u
dt N (x,t)dx = h
dt (t) = −( f m+1/2 (t)−f k−1/2 (t) ) ∀k,m ∈ Z .
x k−1/2
l=k
⇕
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C. Schwa
H.
Harbrech
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DRAFT
8.3
p. 87
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.3
p. 88

x m+1/2 ∫
d
u(x,t)dx = − ( f(u(x
dt
j+1/2 ,t))−f(u(x k−1/2 ,t)) ) ,
x k−1/2
With respect to unions of dual cells and numerical fluxes, the semidiscrete solution u N (t)
satisfies a balance law of the same structure as a (weak) solution of (8.2.9).
Numerical
Methods
for PDEs
approximate location of shock at timet):
x ∗ (t) ∈ R:
x∫
∗ (t) ∫∞
u l −u N (x,t)dx = u N (x,t)−u r dx
−∞ x ∗ (t)
u
u l
Numerica
Methods
for PDEs
Of course, the numerical flux functionF has to fit the flux functionf of the conservation law:
Definition 8.3.16 (Consistent numerical flux function).
A numerical flux functionF : R m l+m r ↦→ R is consistent with the flux functionf : R ↦→ R, if
F(u,...,u) = f(u) ∀u ∈ R .
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Gradinaru
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DRAFT
equality of yellow areas
(8.3.18)
=⇒
∫ xm+1/2
✄
u r
Fig. 269
x ∗ x
x −m−1/2
u N (x,t)dx = (x ∗ (t)+x −m−1/2 )u l +(x m+1/2 −x ∗ (t))u r .
dx ∗
dt (t) = 1
u l −u r
∑
j∈Z
dµ j
dt (t) = f(u l)−f(u r )
u l −u r
(8.2.23)
= ṡ .
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Harbrech
V.
Gradinar
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DRAFT
Focus:
(8.3.11):
solution of Riemann problem (→ Def. 8.2.28) by finite volume method in conservation form
Initial data “constant at±∞”: µ −j (0) = u l , µ j (0) = u r for largej.
Consistency of the numerical flux function implies for largem ≫ 1
8.3
p. 881
Numerical
Methods
for PDEs
★
✧
Conservation form with consistent numerical flux yields correct “discrete shock speed”
(not liable to effect of Ex. 8.3.1)
8.3.3 Numerical flux functions
8.3.3.1 Central flux
✥
✦
8.3
p. 88
Numerica
Methods
for PDEs
x m+1/2 ∫
d
u
dt N (x,t)dt = − ( F(u r ,...,u r )−F(u l ,...,u l ) ) = −(f(u r )−f(u l )) . (8.3.18)
−x −m−1/2
Exactly the same balance law holds for any weak solutions of the Riemann problem!
Situation:
discrete solutionu N (t) decreasing & supposed to approximate a shock
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Example 8.3.19 (Central flux for Burgers equation).
Cauchy problem for Burgers equation (8.1.16) (flux functionf(u) = 1 2 u2 ) from Ex. 8.2.38 (“box”
intial data)
Spatial finite volume discretization in conservvation form (8.3.11) with central numerical fluxes
F 1 (v,w) := 1 2( f(v)+f(w)) , F2 (v,w) := f ( 1
2 (v +w) ) . (8.3.20)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.3
p. 882
Obviously the 2-point numerical fluxes F 1 and F 2 are consistent according to Def. 8.3.16. The
resulting spatially semi-discrete scheme is given by, see (8.3.12)
F 1 :
F 2 :
dµ j
dt (t) = − 1
2h (f(µ j+1(t))−f(µ j−1 (t))) ,
dµ j
dt (t) = −1 h (f(1 2 (µ j(t)+µ j+1 (t)))−f( 1 2 (µ j(t)+µ j−1 (t)))) .
8.3
p. 88

t
t
timestepping based on adaptive Runge-Kutta method ode45 of MATLAB
(opts = odeset(’abstol’,1E-7,’reltol’,1E-6);).
Numerical
Methods
for PDEs
1
4
3.5
Numerica
Methods
for PDEs
0.8
Fully discrete evolution for central numerical fluxF 1 : h = 0.03
u(t,x)
0.6
0.4
0.2
3
2.5
0
1
h = 0.133333, t=1.005873
finite volume solution
exact solution
1
h = 0.133333, t=2.000926
finite volume solution
exact solution
1
h = 0.133333, t=3.000865
finite volume solution
exact solution
1
h = 0.133333, t=4.000000
finite volume solution
exact solution
4
3.5
2
0.8
0.8
0.8
0.8
3
1.5
2.5
0.6
0.6
0.6
0.6
2
1
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
Fig. 270
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
t
1.5
1
0.5
0
−0.5
0
0.5
1
x
1.5
2
2.5
3
3.5
0.5
0
0 0.5 1 1.5 2 2.5 3
x
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Observation: massive spurious oscillations utterly pollute numerical solution
✸
8.3
p. 885
Example 8.3.21 (Central flux for linear advection).
8.3
p. 88
u(t,x)
1
0.8
0.6
0.4
0.2
4
3.5
3
2.5
Numerical
Methods
for PDEs
Cauchy problem (8.1.5): constant valocity scalar linear advection,v = 1, flux functionf(u) = vu
∂u
∂t +v∂u ∂x = 0 in ˜Ω = R×]0,T[ , u(x,0) = u 0 (x) ∀x ∈ R . (8.1.5)
= Cauchy problem for 1D transport equation (7.3.7)!
Numerica
Methods
for PDEs
0
4
2
3.5
3
1.5
2.5
2
1
1.5
1
3.5 0.5
3
0.5
2.5
2
t
1.5
1
0
0.5
0
−0.5
0
x
0 0.5 1 1.5 2 2.5 3
x
Fully discrete evolution for central numerical fluxF 2 : h = 0.017
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Finite volume spatial discretization in conservation form (8.3.11) with central numerical fluxes from
(8.3.20):
F 1 (v,w) := 1 2( f(v)+f(w))
F 2 (v,w) := f ( 1
2 (v +w) )
⇒ dµ j
dt (t) = − v
2h (µ j+1(t)−µ j−1 (t)) , j ∈ Z .
= spatial semi-discretization using linear finite element Galerkin discretization of convective term,
see (7.2.15).
Sect. 7.3.1: this method is prone to spurious oscillations, see Ex. 7.3.4.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
1
h = 0.133333, t=1.001531
finite volume solution
exact solution
1
h = 0.133333, t=2.004984
finite volume solution
exact solution
1
h = 0.133333, t=3.000587
finite volume solution
exact solution
1
h = 0.133333, t=4.000000
finite volume solution
exact solution
This offers an explanation also for its failure for Burgers equation, see Ex. 8.3.19
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
✸
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.2
Fig. 271
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
8.3
p. 886
8.3
p. 88

t
8.3.3.2 Lax-Friedrichs flux
Numerical
Methods
for PDEs
1
0.8
h = 0.066667, t=1.002336
finite volume solution
exact solution
1
0.8
h = 0.066667, t=2.000915
finite volume solution
exact solution
1
0.8
h = 0.066667, t=3.000435
finite volume solution
exact solution
1
0.8
h = 0.066667, t=4.000000
finite volume solution
exact solution
Numerica
Methods
for PDEs
0.6
0.6
0.6
0.6
Sect. 7.2.2.2: artificial diffusion cures instability of central difference quotient
∂u
∂t
↕
∂u
∂t + ( ch/2) −µ j−1 +2µ j −µ j+1
}
h
{{ 2 }
ˆ= difference quotient for d2 u
dx 2 +
+ c ∂u
∂x = 0 in R×]0,T[ ,
↕
c µ j+1 −µ j−1
2h
} {{ }
ˆ= difference quotient forc du
dx
= 0 , j ∈ Z .
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
u(t,x)
4
1
0.8
0.6
0.4
0.2
0
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
4
3.5
3
2.5
2
0.4
0.2
Fig. 272
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
3.5
3
1.5
Can this be rewritten in conservation form (8.3.11)?
YES!
2.5
2
1.5
1
(ch/2) −µ j−1 +2µ j −µ j+1
h 2
central numerical flux
+c µ j+1 −µ j−1
2h
= 1 h( F(µj ,µ j+1 )−F(µ j−1 ,µ j )) ,
with F(v,w) := 2 c(v +w)− 2 c (w−v) . (8.3.24)
diffusive/viscous numerical flux
8.3
p. 889
t
1
0.5
0
−0.5
0
0.5
1
x
1.5
2
2.5
3
3.5
0.5
0
0 0.5 1 1.5 2 2.5 3
x
8.3
p. 89
Recall from Rem. 8.2.3: the flux function f(u) = − ∂u
∂x
models diffusion. Hence, the diffusive numerical
flux amounts to a central finite difference discretization of the partial derivative in space:
− ∂u
∂x (x,t) ≈ − 1 (
u(xj+1 ,t)−u(x
|x=x j+1/2
h j ,t) ) .
Numerical
Methods
for PDEs
Observation: spurious completely suppressed, qualitatively good resolution of both shock and rarefaction.
Effect of artificial diffusion: smearing of shock, cf. discussion in Ex. 7.2.27.
Numerica
Methods
for PDEs
How to adapt this to general scalar conservation laws?
(local) Lax-Friedrichs flux
∂u
∂t + ∂ ∂u
f(u) =
∂x ∂t +f′ (u) ∂u
∂x = 0 (8.3.25)
local speed of transport↔c
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.3.3.3 Upwind flux
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
F LF (v,w) = 1 2 (f(v)+f(w))− 2
1 max
min{v,w}≤u≤max{v,w} |f′ (u)|(w−v) . (8.3.26)
Another idea for stable spatial discretization of stationary transport in Sect. 7.2.2.1:
Example 8.3.27 (Lax-Friedrichs flux for Burgers equation).
“upwinding” = obtain information from where transport brings it
☞ same setting and conservative discretization as in Ex. 8.3.19
☞ Numerical flux function: Lax-Friedrichs flux (8.3.26)
8.3
p. 890
☞
remedy for ambiguity of evaluation of discontinuous gradient in upwind quadrature
8.3
p. 89

t
t
Ambiguity also faced in the evaluation of f(u N (x j+1/2 ),t),f(u N (x j−1/2 ),t), see (8.3.10), which
forced us to introduce numerical flux functions in (8.3.11).
Numerical
Methods
for PDEs
1
0.8
4
3.5
Numerica
Methods
for PDEs
(8.3.25): local velocity of transport at(x,t) ∈ ˜Ω is given by f ′ (u(x,t))
u(t,x)
0.6
0.4
0.2
3
2.5
0
➣ ambiguous local velocity of transport at discontinuity ofu N !
4
3.5
2
3
1.5
Idea:
deduce local velocity of transport from Rankine-Hugoniot jump condition
local velocity of transport =
(8.2.23)
{
f ′ (u) for unique state, u = u l = u r
f(u r )−f(u l )
u r −u l
(u l ,u r ˆ= states to left and right of discontinuity)
at discontinuity.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
2.5
t
2
1.5
1
0.5
0
−0.5
0
0.5
1
x
Example 8.3.30 (Upwind flux and transsonic rarefaction).
1.5
2
2.5
3
3.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3
x
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
upwind flux for scalar conservation law with flux functionf:
Cauchy problem (8.2.9) for Burgers equation (8.1.16), i.e.,f(u) = 1 2 u2 and initial data
F uw (v,w) =
{
f(v) , if ṡ ≥ 0 ,
f(w) , if ṡ < 0 ,
ṡ :=
{ f(w)−f(v)
w−v forv ≠ w ,
f ′ (v) forv = w .
(8.3.28)
u 0 (x) =
{
−1 for x < 0 or x > 1 ,
1 for 0 < x < 1 .
Example 8.3.29 (Upwind flux for Burgers equation).
8.3
p. 893
8.3
p. 89
☞ same setting and conservative discretization as in Ex. 8.3.19
☞ Numerical flux function: upwind flux (8.3.28)
Numerical
Methods
for PDEs
4
3.5
3
u(x,t) = x/t
u = −1
Numerica
Methods
for PDEs
1
h = 0.066667, t=1.003076
finite volume solution
exact solution
1
h = 0.066667, t=2.000768
finite volume solution
exact solution
1
h = 0.066667, t=3.001334
finite volume solution
exact solution
1
h = 0.066667, t=4.000000
finite volume solution
exact solution
4
2.5
3.5
0.8
0.8
0.8
0.8
3
2
0.6
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.6
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.6
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.6
0.4
0.2
Fig. 273
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
u(t,x)
2.5
1
2
0.5
1.5
0
−0.5
1
−1
0.5 t
−2.5 −2 −1.5 −1 −0.5 0
0 0.5 1 1.5
x
Fig. 274
1.5
1
0.5
u = −1
u = 1
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
x
Fig. 275
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
DRAFT
The entropy solution (→ Sect. 8.2.6) of this Cauchy problem features a transsonic rarefaction fan at
x = 0: this is a rarefaction solution (→ Lemma 8.2.34) whose “edges” move in opposite directions.
8.3
p. 894
8.3
p. 89

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
t
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
h = 0.066667, t=1.000000
finite volume solution
exact solution
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
h = 0.066667, t=2.000000
finite volume solution
exact solution
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
h = 0.066667, t=3.000000
finite volume solution
exact solution
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
4
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
h = 0.066667, t=4.000000
finite volume solution
exact solution
Fig. 276
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
Numerical
Methods
for PDEs
➤ initial nodal values µ j (0) =
➤
Semi-discrete evolution equation:
dµ j
dt (t) = − 1
2h ·
{
−1 forj < 0 ,
1 forj ≥ 0 .
{
µ 2
j+1 (t)−µ 2 j (t) for j ≥ 0 ,
µ 2 j (t)−µ2 j−1 (t) for j < 0 .
µ j (t) = µ j (0) for allt ➤ forh → 0, convergence to entropy violating expansion shock !
Numerica
Methods
for PDEs
u(t,x)
1
0.5
0
−0.5
3.5
3
2.5
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
finite volume method may converge to non-physical weak solutions !
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
−1
DRAFT
DRAFT
4
3.5
2
✸
3
1.5
2.5
2
1
t
1.5
1
0.5
0
−2.5
−2
−1.5
−1
x
−0.5
0
0.5
1
1.5
0.5
0
−2 −1.5 −1 −0.5 0 0.5 1
x
8.3.3.4 Godunov flux
Conservative finite volume discretization with upwind flux produces (stationary) expansion shock instead
of transonic rarefaction!
Sect. 8.2.6: this is a weak solution, but it violates the entropy condition, “non-physical shock”.
8.3
p. 897
Numerical
Methods
for PDEs
The upwind flux (8.3.28) is a numerical flux of the form
F(v,w) = f(u ↓ (v,w)) with an intermediate state u ↓ (v,w) ∈ R .
For the upwind flux the intermediate state is not really “intermediate”, but coincides with one of the
statesv,w depending on the sign of the “local shock speed”ṡ := f(w)−f(v)
w−v .
8.3
p. 89
Numerica
Methods
for PDEs
✸
Idea:
obtain suitable intermediate state as
R. Hiptmair
Example 8.3.31 (Upwind flux: Convergence to expansion shock).
C. Schwab,
u ↓ (v,w) = ψ(0) , (8.3.32)
H.
Harbrecht
V.
whereu(x,t) = ψ(x/t) solves the Riemann problem (→ Def. 8.2.28)
Cauchy problem (8.2.9) for Burgers equation (8.1.16), i.e.,f(u) = 2 1 Gradinaru
u2
A. Chernov
DRAFT
∂u
∂t + ∂
{
v , forx < 0 ,
∂x f(u) = 0 , u(x,0) = (8.3.33)
w , forx ≥ 0 .
u 0 (x) = 1 forx > 0, u 0 (x) = −1 forx < 0
➤ entropy solution = rarefaction wave (→ Lemma 8.2.34)
We focus on f : R ↦→ R strictly convex & smooth (e.g. Burgers equations (8.1.16))
FV in conservation form, upwind flux (8.3.28), on equidistant grid,x j = (j+ 2 1)h, meshwidthh > 0 ➤ Riemann problem (8.3.33) (→ Def. 8.2.28) has the entropy solution (→ Sect. 8.2.6):
8.3
p. 898
➊ Ifv > w ➤ discontinuous solution, shock (→ Lemma 8.2.31)
{
v ifx < ṡt ,
u(t,x) = ṡ = f(v)−f(w)
w ifx > ṡt , v −w
. (8.3.34)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.3
p. 90

t/u
➋ Ifv ≤ w ➤ continuous solution, rarefaction wave (→ Lemma 8.2.34)
⎧
⎪⎨ v ifx < f ′ (v)t ,
u(t,x) = g(x/t) iff
⎪⎩
(v) ≤ x/t ≤ f ′ (w) , g := (f ′ ) −1 . (8.3.35)
w ifx > f ′ (w)t ,
Numerical
Methods
for PDEs
t/u
replacements
v
w
➊: subsonic shock
x
t/u
w
v
➋: subsonic
rarefaction
x
t/u
v
w
➌: supersonic
shock
x
t/u
w
v
➍: supersonic
rarefaction
x
w
v
➎:
transonic rarefaction
x
Numerica
Methods
for PDEs
➣
All weak solutions of a Riemann problem are of the formu(x,t) = ψ(x/t) with a suitable function
ψ, which is
piecewise constant with a jump atṡ := f(w)−f(v)
w−v for a shock solution (8.3.34),
the continuous function (in the case of strictly convex flux functionf)
⎧
⎪⎨ v , ifξ < f ′ (v) ,
ψ(ξ) := (f
⎪⎩
′ ) −1 (ξ) , iff ′ (v) < ξ < f ′ (w) ,
w , ifξ > f ′ (v) ,
provided thatw > v = situation of a rarefaction solution (8.3.35), see Lemma 8.2.34.
t
u(x,t)/t
x j−2 x j−1 x j x j+1 x j+2 x j+3
Fig. 277
−− ˆ= piecewise constant functionu(0,t)
−− ˆ= shock in(t,x)-plane
−− ˆ= rarefaction wave in(t,x)-plane
for convex flux functionf
⎧
v > w ∧ ṡ < 0 (shock ➊) ,
w , if
⎪⎨ v < w ∧ f ′ (w) < 0 (rarefaction ➋) ,
u ↓ (v,w) = v > w ∧ ṡ > 0 (shock ➌),
v , if
v < w ∧ f ′ (v) > 0 (rarefaction ➍) ,
⎪⎩ (f ′ ) −1 (0) , ifv < w ∧ f ′ (v) ≤ 0 ≤ f ′ (w) (rarefaction ➎).
(8.3.36)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.3
p. 901
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Detailed analysis of (8.3.36):
{
v > w (shock case): f(u ↓ f(w)−f(v) f(v) , if
(v,w)) = w−v > 0 ⇔ f(w) < f(v) ,
f(w)−f(v)
f(w) , if w−v ≤ 0 ⇔ f(w) ≥ f(v) .
f(u ↓ (v,w)) = max{f(v),f(w)} .
For a convex flux functionf:
v < w ⇒ f ′ (v) ≤ f(w)−f(v) ≤ f ′ (w) .
w−v
Forv < w (rarefaction case)
⎧
⎪⎨ f(v) , iff ′ (v) > 0 ,
f(u ↓ (v,w)) = f(z) , iff
⎪⎩
′ (v) < 0 < f ′ (w) ,
f(w) , iff ′ (w) < 0 ,
wheref ′ (z) = 0⇔f has a global minimum inz.
2-point numerical flux function according to (8.3.32) and (8.3.33):
f
v
Godunov numerical flux
w
Fig. 278
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.3
p. 90
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.3
p. 902
8.3
p. 90

t
Using general Riemann solution (8.2.37): for any
flux function
Godunov numerical flux function
⎧
⎪⎨ min f(u) , if v < w ,
v≤u≤w
F GD (v,w) =
⎪⎩ max f(u) , if w ≤ v .
w≤u≤v
(8.3.37)
for Burgers equation (8.1.16) ✄
F(v,w)
0.5
0.4
0.3
0.2
0.1
0
1
0.8
0.6
Godunov numerical flux function for Burgers equation
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
w
−1
1
0.5
0
−0.5
−1 v
Fig. 279
Numerical
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
u(t,x)
1
0.5
0
−0.5
h = 0.066667, t=1.004154
finite volume solution
exact solution
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
h = 0.066667, t=2.001356
finite volume solution
exact solution
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
4
3.5
3
2.5
h = 0.066667, t=3.001176
finite volume solution
exact solution
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
h = 0.066667, t=4.000000
finite volume solution
exact solution
Fig. 280
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
Remark 8.3.38 (Upwind flux and expansion shocks).
DRAFT
−1
4
2
DRAFT
3.5
3
1.5
2.5
2
1
1.5
F uw (v,w) = F GD (v,w), except for the case of transsonic rarefaction!
t
1
0.5
0
−2.5
−2
−1.5
−1
x
−0.5
0
0.5
1
1.5
0.5
0
−2 −1.5 −1 −0.5 0 0.5 1
x
(transsonic rarefaction = rarefaction fan with edges moving in opposite direction, see Ex. 8.3.30)
What does the upwind fluxF uw (v,w) from (8.3.28) yield in the case of transsonic rarefaction?
Iff convex,v < w,f ′ (v) < 0 < f ′ (w),
F uw (v,w) = f(ψ(0)) ,
whereu(x,t) = ψ(x/t) is a non-physical entropy-condition violating (→ Def. 8.2.35) expansion shock
weak solution of (8.3.33).
8.3
p. 905
Numerical
Methods
for PDEs
Observation: Transonic rarefaction captured by discretization, but small remnants of an expansion
shock still observed.
✸
8.3
p. 90
Numerica
Methods
for PDEs
Upwind flux treats transsonic rarefaction as expansion shock!
➣ Explanation for observation made in Ex. 8.3.30.
Example 8.3.39 (Godunov flux for Burgers equation).
△
R. Hiptmair
C. Schwab,
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V.
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DRAFT
8.3.4 Montone schemes
Observations made for some piecewise constant solutions u N (t) of semi-discrete evolutions arising
from spatial finite volume discretization in conservation form (8.3.12):
Ex. 8.3.27 (Lax-Friedrichs numerical flux (8.3.26))
Ex. 8.3.39 (Godunov numerical flux (8.3.37))
:
min u 0(x) ≤ u N (x,t) ≤ max u 0(x)
x∈R x∈R
no new local extrema in numerical solution
R. Hiptm
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DRAFT
☞ same setting and conservative discretization as in Ex. 8.3.30
☞ Numerical flux function: Godunov numerical flux (8.3.37)
8.3
p. 906
In these respects the conservative finite volume discretizations based on either the Lax-Friedrichs
numerical flux or the Godunov numerical flux inherit crucial structural properties of the exact solution,
8.3
p. 90

see Sect. 8.2.7, in particular, Thm. 8.2.39 and the final remark: they display structure preservation,
cf. (5.7).
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Methods
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Assumption:
⃗µ = ⃗µ(t) and⃗η = ⃗η(t) solve (8.3.40) and satisfy for somet ∈ [0,T]
η k (t) ≤ µ k (t) ∀k ∈ Z , ξ := η j (t) = µ j (t) for some j ∈ Z .
Numerica
Methods
for PDEs
Is this coincidence for the special settings exmained in Ex. 8.3.27 and Ex. 8.3.39?
Focus: semi-discrete evolution (8.3.12) resulting from finite volume discretization in conservation form
on an equidistant infinite mesh
(8.3.11)
for Cauchy problem
dµ j
dt (t) = −1 ( F(µj (t),µ
h j+1 (t))−F(µ j−1 (t),µ j (t)) ) , j ∈ Z , (8.3.12)
∂u
∂t + ∂
∂x f(u) = 0 in R×]0,T[ , u(x,0) = u 0(x) , x ∈ R , (8.2.9)
induced by Lax-Friedrichs numerical flux (8.3.26)
F LF (v,w) = 1 2 (f(v)+f(w))− 2
1 max
min{v,w}≤u≤max{v,w} |f′ (u)|(w−v) . (8.3.26)
dµ j
dt = − 1 (
f(µ
2h j+1 )−f(µ j−1 )−
)
max
u∈[µ j ,µ j+1 ] |f′ (u)|(µ j+1 −µ j )+ max
u∈[µ j−1 ,µ j ] |f′ (u)|(µ j −µ j−1 ) .
(8.3.40)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.3
p. 909
Canη j raise aboveµ j ?
d µj −η j ) = −
dt( 1 (
)
F
h LF (ξ,µ j+1 )−F LF (ξ,η j+1 )+F LF (η j−1 ,ξ)−F LF (µ j−1 ,ξ) .
d( To show: µj −η
dt j ) ≥ 0 ➣ µ j (t) will stay aboveη j (t).
This can be concluded, if
F LF (ξ,µ j+1 )−F LF (ξ,η j+1 ) ≤ 0 and F LF (η j−1 ,ξ)−F LF (µ j−1 ,ξ) ≤ 0 . (8.3.42)
The only piece of information we are allowed to use is
µ j+1 ≥ η j+1 and µ j−1 ≥ η j−1 .
This would imply (8.3.42), if F LF was increasing in the first argument and decreasing in the second
argument.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.3
p. 91
Numerical
Methods
for PDEs
Definition 8.3.43 (Monotone numerical flux function).
A 2-point numerical flux functionF = F(v,w) is called monotone, if
F is an increasing function of its first argument
Numerica
Methods
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Goal: show thatu N (t) linked to⃗µ(t) from (8.3.40) through piecewise constant reconstruction (8.3.6)
satisfies
and
F is a decreasing function of its second argument.
min u N(x,0) ≤ u N (x,t) ≤ max u N(x,0) ∀x ∈ R , ∀t ∈ [0,T] . (8.3.41)
x∈R x∈R
Recall from Sect. 8.2.7: estimate (8.3.41) for the exact solution u(x,t) of (8.2.9) is a consequence
of the comparison principle of Thm. 8.2.39 and the fact that constant initial data are preserved during
the evolution. The latter property is straightforward for conservative finite volume spatial semidiscretization,
see (8.3.15).
R. Hiptmair
C. Schwab,
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V.
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DRAFT
✬
Simple criterion: A continuously differentiable 2-point numerical flux function F = F(v,w) is
montone, if and only if
✫
∂F
(v,w) ≥ 0 and
∂F
∂v
(v,w) ≤ 0 ∀(v,w) . (8.3.44)
∂w
✩R. Hiptm
C. Schwa
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V.
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✪
DRAFT
➣ Goal: Establish comparison principle for finite volume semi-discrete solutions based on Lax-
Friedrichs numerical flux:
{ }
⃗µ(t),⃗η(t) solve (8.3.40) ,
η j (0) ≤ µ j (0) ∀j ∈ Z
⇒ η j (t) ≤ µ j (t) ∀j ∈ Z , ∀0 ≤ t ≤ T .
8.3
p. 910
✬
Lemma 8.3.45 (Monotonicity of Lax-Friedrichs numerical flux and Godunov flux).
For any continuously differentiable flux function f the associated Lax-Friedrichs flux (8.3.26)
and Godunov flux (8.3.37) are monotone.
✫
✩
✪
8.3
p. 91

Proof.
➊
Lax-Friedrichs numerical flux:
Numerical
Methods
for PDEs
Proof (of Lemma 8.3.46, following the above considerations for the Lax-Friedrichs flux).
The two sequences of nodal values satisfy (8.3.12)
Numerica
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F LF (v,w) = 1 2 (f(v)+f(w))− 2
1 max
min{v,w}≤u≤max{v,w} |f′ (u)|(w−v) . (8.3.26)
Application of the criterion (8.3.44) is straightforward:
➋
∂F LF
∂v (v,w) = f′ (v)+ max
min{v,w}≤u≤max{v,w} |f′ (u)| ≥ 0 ,
∂F LF
∂w (v,w) = f′ (w)− max
min{v,w}≤u≤max{v,w} |f′ (u)| ≤ 0 .
Godunov numerical flux
⎧
⎪⎨ min f(u) , if v < w ,
v≤u≤w
F GD (v,w) =
⎪⎩ max f(u) w≤u≤v , if w ≤ v . (8.3.37)
v < w: If v increases, then the range of values over which the minimum is taken will shrink, which
makesF GD (v,w) increase.
If w is raised, then the minimum is taken over a larger interval, which causes F GD (v,w) to
become smaller.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.3
p. 913
dµ j
dt (t) = −1 ( F(µj (t),µ
h j+1 (t))−F(µ j−1 (t),µ j (t)) ) , j ∈ Z , (8.3.47)
dη j
dt (t) = −1 ( F(ηj (t),η
h j+1 (t))−F(η j−1 (t),η j (t)) ) , j ∈ Z . (8.3.48)
Let t 0 be the earliest time, at which ⃗η “catches up” with ⃗µ in at least one node x j , j ∈ Z, of the
mesh, that is
η k (t 0 ) ≤ µ k (t 0 ) ∀k ∈ Z , ξ := η j (t 0 ) = µ j (t 0 ) .
By subtracting (8.3.47) and (8.3.48) we get
d
dt (µ j −η j )(t 0 ) = − 1 (
)
F(ξ,µ
h j+1 (t 0 ))−F(ξ,η j+1 (t 0 ))+F(η j−1 (t 0 ),ξ)−F(µ j−1 (t 0 ),ξ) ≥ 0 ,
because for a monotone numerical flux function (→ Def. 8.3.43)
η j−1 (t 0 ) ≤ µ j−1 (t 0 )
η j+1 (t 0 ) ≤ µ j+1 (t 0 )
increasing in first argument
⇒ F(η j−1 (t 0 ),ξ)−F(µ j−1 (t 0 ),ξ) ≤ 0 ,
decreasing in second argument
⇒ F(ξ,µ j+1 (t 0 ))−F(ξ,η j+1 (t 0 )) ≤ 0 .
This means that “η j cannot overtakeµ j ”: no valueη j can ever raise aboveµ j .
✷
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.3
p. 91
v ≥ w: If v increases, then the range of values over which the maximum is taken will grow, which
makesF GD (v,w) increase.
If w is raised, then the maximum is taken over a smaller interval, which causes F GD (v,w)
to decrease.
✷
Numerical
Methods
for PDEs
Now we want to study the “preservation of the number of local extrema” during a semi-discrete evolution,
another structural property of exact solutions of conservations laws, see Sect. 8.2.7.
Numerica
Methods
for PDEs
✬
Lemma 8.3.46 (Comparison principle for monotone semi-discrete conservative evolutions).
Let the 2-point numerical flux function F = F(v,w) be monotone (→ Def. 8.3.43) and ⃗µ =
⃗µ(t),⃗η = ⃗η(t) solve (8.3.12). Then
✫
η k (0) ≤ µ k (0) ∀k ∈ Z ⇒ η k (t) ≤ µ k (t) ∀k ∈ Z , ∀ 0 ≤ t ≤ T .
✩
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✪
Intuitive terminology: ⃗µ has a local maximumu m ∈ R, if
∃j ∈ Z: µ j = u m and ∃k l < j < k r ∈ N: max
k l

local maximum
local maximum
Numerical
Methods
for PDEs
8.4 Timestepping
Numerica
Methods
for PDEs
Counting local extrema of ⃗µ
and the associated piecewise
constant reconstruction.
Focus: Explicit Runge-Kutta timestepping methods (→ Def. 6.1.26)
✬
Lemma 8.3.49 (Non-oscillatory monotone semi-discrete evolutions).
If ⃗µ = ⃗µ(t) solves (8.3.12) with a monotone numerical flux functionF = F(v,w) and ⃗µ(0) has
finitely many local extrema, then the number of local extrema of ⃗µ(t) cannot be larger than that
of ⃗µ(0).
j /u N
R. Hiptmair
local minimumµ
local minimum
local minimum Fig. 281
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
✩
DRAFT
x
Recall [18, Def. 12.4.9]: for explicit s-stage Runge-Kutta single step methods the coefficients a ij
vanish for j ≥ i, 1 ≤ i,j ≤ s ➣ the incrementsK i can be computed in turns (without solving a
non-linear system of equations).
Initial value problem for abstract semi-discrete evolution in R Z :
d⃗µ
dt (t) = L h(⃗µ(t)) , 0 ≤ t ≤ T , ⃗µ(0) = ⃗µ 0 ∈ R Z . (8.4.1)
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DRAFT
✫
✪
Here: L h : R Z ↦→ R Z ˆ= (non-linear) finite difference operator, e.g. for finite volume semidiscretization
in conservation form with 2-point numerical flux:
8.3
p. 917
(8.3.12) ➣ (L h ⃗µ) j := − 1 h( F(µj ,µ j+1 )−F(µ j−1 ,µ j ) ) . (8.4.2)
8.4
p. 91
Proof.
i ˆ= index of local maximum of ⃗µ(t),tfixed
µ i−1 (t) ≤ µ i (t) ,
µ i+1 (t) ≤ µ i (t)
monotone flux
=⇒ F(µ i ,µ i+1 ) ≥ F(µ i ,µ i ) ≥ F(µ i−1 ,µ i ) ,
⇒ d dt µ i(t) = − 1 h( F(µi ,µ i+1 )−F(µ i−1 ,µ i ) ) ≤ 0 .
➣ maxima of⃗µ subside, (minima of⃗µ rise !)
Idea of proof:
No new (local) extrema can arise !
Adjacent values cannot “overtake”:
local maximum: cannot move up
local minmum: cannot move down
u
µ j−2
µ j−1 µ j
µ j+1
x j− 3/2 x j− 1/2 x j+ 1/2 x j+ 3/2 x j+
Fig. 282x
5/2
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
L h is local: (L h (⃗µ)) j depends only on “neighboring values”µ j−nl ,...,µ j+nr .
Explicits-stage Runge-Kutta single step method for (8.4.1), timestepτ > 0:
⃗κ 1 = L h (⃗µ (k) ) ,
⃗κ 2 = L h (⃗µ (k) +τa 21 ⃗κ 1 ) ,
⃗κ 3 = L h (⃗µ (k) +τa 31 ⃗κ 1 +τa 32 ⃗κ 2 ) ,
.
⃗κ s = L h (⃗µ (k) s−1 ∑
+τ a sj ⃗κ j ) ,
j=1
s∑
⃗µ (k+1) = ⃗µ (k) +τ b l ⃗κ j . (8.4.3)
l=1
Here, a ij ∈ R and b l ∈ R are the coefficients from the Butcher scheme (6.1.27). For explicit RKmethods
the coefficient matrixAis strictly lower triangular.
Numerica
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.4
p. 918
8.4
p. 92

t
Setting:
equidistant spatial mesh M, meshwidth
h > 0, nodesx j := hj,j ∈ Z,
Numerical
Methods
for PDEs
8.4.1 CFL-condition
Numerica
Methods
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t 5
0 x 1 x 2 x 3 x 4 x 5 x 6 x 7
x
t 4
t 3
t 2
t 1
uniform timestep τ > 0, t k := τk, k ∈
N 0 .
Single step ( timestepping for (8.4.1) produces a sequence
⃗µ (k)) k∈N 0
µ (k)
j ≈ u(x j ,t k ) , j ∈ Z, k ∈ N 0 .
Fully discrete evolution
⃗µ (k+1) = H h (⃗µ (k−1) ) , k ∈ N 0 .
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C. Schwab,
H.
Harbrecht
V.
Gradinaru
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DRAFT
Remark 8.4.7 (Difference stencils).
Stencil notation: Visualization of flow of information in fully discrete explicit evolution (action ofH h ),
cf. Fig. 201.
t
t
t
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C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
t k−1
t k
Fig. 283
t k−1
x j−2 x j−1 x j x j+1 x j+2
x
t k
Fig. 284
t k−1
x j−2 x j−1 x j x j+1 x j+2 x
t k
Fig. 285
x j−2 x j−1 x j x j+1 x j+2 x
H h : R Z ↦→ R Z :
fully discrete evolution operator, arising from applying single step timestepping
(8.4.3) to (8.4.1).
2-point numerical flux &
explicit Euler timestepping
upwinding &
explicit trapezoidal rule
2-point numerical flux &
explicit trapezoidal rule
△
Example 8.4.4 (Fully discrete evolutions).
8.4
p. 921
8.4
p. 92
Fully discrete evolution arising from finite volume semi-discretization in conservation form with 2-point
numerical fluxF = F(v,w)
(8.3.12) ➣ (L h ⃗µ) j := − 1 h( F(µj ,µ j+1 )−F(µ j−1 ,µ j ) ) . (8.4.2)
Numerical
Methods
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A consequence of explicit timestepping:
locality of fully discrete evolution operator:
∃m l ,m r ∈ N 0 : (H(⃗µ)) j = H j (µ j−ml ,...,µ j+mr ) . (8.4.8)
If flux functionf does not depend onx,f = f(u) as in (8.2.9), we can expect
Numerica
Methods
for PDEs
in combination with explicit Euler timestepping (ˆ= 1-stage explicit RK-method)
⃗µ (k+1) = ⃗µ (k) +τL h (⃗µ (k) ) .
(H h (⃗µ)) j = µ (k)
j − τ ( (k) F(µ
h j ,µ (k)
j+1 )−F(µ(k) j−1 ,µ(k) j ) ) . (8.4.5)
In the case of explicit trapezoidal rule timestepping [18, Eq. 12.4.6] (method of Heun)
⃗κ = µ (k) + τ 2 L h(⃗µ (k) ) , ⃗µ (k+1) = µ (k) +τL h (⃗κ) .
κ j := (⃗κ) j = µ (k)
j − τ ( (k) F(µ
h j ,µ (k)
j+1 )−F(µ(k) j−1 ,µ(k) j ) ) ,
(H h (⃗µ)) j = µ (k)
j −
h( τ F(κj ,κ j+1 )−F(κ j−1 ,κ j ) ) .
(8.4.6)
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DRAFT
This is the case for (8.4.5) and (8.4.6).
H h is translation-invariant: H j = H ∀j ∈ Z .
By inspection of (8.4.3): ifL h is translation-invariant
(L h (⃗µ)) j = L(µ j−nl ,...µ j+nr ) , j ∈ Z ,
and timestepping relies on an s-stage explicit Runge-Kutta method, then we conclude for m l ,m r in
(8.4.8)
m l ≤ s·n l , m r ≤ s·n r .
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H.
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V.
Gradinar
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DRAFT
✸
8.4
p. 922
8.4
p. 92

Now we revisit a concept from Sect. 6.2.5, see, in particular, Rem. 6.2.41:
✬
Definition 8.4.9 (Numerical domain of dependence).
Consider explicit translation-invariant fully discrete evolution ⃗µ (k+1) := H(⃗µ (k) ) on uniform
spatio-temporal mesh (x j = hj,j ∈ Z,t k = kτ,k ∈ N 0 ) with
✩
Numerical
Methods
for PDEs
Definition 8.4.13 (Courant-Friedrichs-Lewy (CFL-)condition). → Rem. 6.2.41
An explicit translation-invariant local fully discrete evolution ⃗µ (k+1) := H(⃗µ (k) ) on uniform
spatio-temporal mesh (x j = hj, j ∈ Z, t k = kτ, k ∈ N 0 ) as in Def. 8.4.9 satisfies the
Courant-Friedrichs-Lewy (CFL-)condition, if the convex hull of its numerical domain of dependence
contains the maximal analytical domain of dependence:
D − (x j ,t k ) ⊂ convex(Dh − (x j,t k ))
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∃m ∈ N 0 : (H(⃗µ)) j = H(µ j−m ,...,µ j+m ) , j ∈ Z . (8.4.10)
Then the numerical domain of dependence is given by
Dh − (x j,t k ) := {(x m ,t l ) ∈ R×[0,t k ]: j −m(k −l) ≤ m ≤ j +m(k −l)} .
✫
From Thm. 8.2.41 recall the maximal analytical domain of dependence for a solution of (8.2.9)
D − (x,t) := {(x,t) ∈ R×[0,t]: ṡ min (t−t) ≤ x−x ≤ ṡ max (t−t)} .
with maximals speeds of propagation
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✪DRAFT
By definition ofD − (x,t) andD − h (x j,t k ) sufficient for the CFL-condition is
τ
h ≤ ṁ s
←→ timestep constraint! . (8.4.14)
This is a timestep constraint similar to the one encountered in Sect. 6.2.5 in the context of leapfrog
timestepping for the semi-discrete wave equation.
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ṡ min := min{f ′ (ξ): inf 0(x) ≤ ξ ≤ sup u 0 (x)} ,
x∈R x∈R
(8.4.11)
ṡ max := max{f ′ (ξ): inf 0(x) ≤ ξ ≤ sup u 0 (x)} .
x∈R x∈R
(8.4.12)
8.4
p. 925
8.4
p. 92
t
t
(¯x,¯t)
Numerical
Methods
for PDEs
As discussed in Rem. 6.2.41,
Numerica
Methods
for PDEs
(¯x,¯t)
We cannot expect convergence for fixed ratio τ : h, forh → 0 in case the CFL-condition is violated.
ṡ
Refer to Fig. 204 for a “graphical argument”:
D − (¯x,¯t)
D − (¯x,¯t) ⊂ R×[0,T]
1
Fig. 286
x
D − h (¯x,¯t) for 3-point stencil
Fig. 287
x
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
t
(x j ,t k )
h
x
τ
u 0
Fig. 288
(• ˆ= coarse grid, ■ ˆ= fine grid,
ˆ= d.o.d)
✁ Sequence of equidistant space-time grids of
R×[0,T] withτ = γh (τ/h = meshwidth in time/space)
Ifγ > CFL-constraint (8.4.14) then
analytical domain
of dependence
numerical domain
⊄
of dependence
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.4
p. 926
8.4
p. 92

8.4.2 Linear stability
Numerical
Methods
for PDEs
As in Sect. 6.1.4.2 and Sect. 6.2.5:
diagonalization technique (with a new twist)
Numerica
Methods
for PDEs
OPTIONAL SECTION
The new twist is thatL h acts on the sequence spaceR Z !
In Sect. 6.1.4.2 and Sect. 6.2.5 we found that for explicit timestepping
timestep constraints τ ≤ O(h r ) , r ∈ {1,2}, necessary to avoid exponential blow-up
(instability)
Is the timestep constraint (8.4.14) suggested by the CFL-condition also stipulated by stability requirements?
We are going to investigate the question only for the Cauchy problem for scalar linear advection in
1D with constant velocityv > 0:
∂u
∂t +v∂u ∂x = 0 in R×]0,T[ , u(x,0) = u 0(x) ∀x ∈ R . (8.1.5)
Semi-discretization in space on equidistant mesh with meshwidthh > 0
➣ linear, local, and translation-invariant semi-discrete evolution
R. Hiptmair
C. Schwab,
H.
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DRAFT
8.4
p. 929
Numerical
Methods
for PDEs
Idea: trial expression for “eigenvectors”
( ) ⃗ξ ζ := exp(ıξj) , j ∈ Z , −π < ξ ≤ π . (8.4.18)
j
By straightforward computations:
Terminology:
m∑
(L h (⃗µ)) j =
l=−m
c l µ j+l ⇒ L h ζ ξ ( ∑ m
=
)
c l exp(iξl) ζ ξ .
l=−m
} {{ }
“eigenvalue”ĉ h (ξ)
m∑
spectrum ofL h : σ(L h ) = {ĉ h (ξ) := c l exp(ıξl): −π < ξ ≤ π} . (8.4.19)
l=−m
The function ĉ h (ξ) is known as the symbol of the difference operator L h , cf. the
concept of symbol of a differential operator.
Example 8.4.20 (Spectrum of upwind difference operator).
R. Hiptm
C. Schwa
H.
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DRAFT
8.4
p. 93
Numerica
Methods
for PDEs
d⃗µ
m∑
dt (t) = L h(⃗µ(t)) , with (L h (⃗µ)) j = c l µ j+l , j ∈ Z , (8.4.15)
l=−m
for suitable weightsc l ∈ R.
Apply formula (8.4.19) withc 0 = −h v ,c −1 = h v (from (8.4.17)):
{ v
}
For L h from (8.4.17): σ(L h ) =
h (exp(−ıξ)−1): −π < ξ ≤ π
Example 8.4.16 (Upwind difference operator for linear advection).
Finite volume semi-discretization of (8.1.5) in conservation form with Godunov numerical flux (8.3.37)
(= upwind flux (8.3.28))
(L h (⃗µ)) j = − v h (µ j −µ j−1 ) . (8.4.17)
In (8.4.15): c 0 = − v h , c −1 = v h .
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Spectrum of upwind finite difference operator for
linear advection with velocity v > 0 (meshwidth
h > 0)
✄
σ(L h )
− v h
v
h
C
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DRAFT
Note: Lax-Friedrichs numerical flux (8.3.26) yields the sameL h .
Fig. 289
✸
✸
8.4
p. 930
8.4
p. 93

Also here:
diagonalization of semi-discrete evolution leads to decoupled scalar linear ODEs. However,
now we have uncountably many “eigenvectors” ⃗ ζ ξ , so that linear combination becomes integration:
∫π
∫π
⃗µ(t) = ˆµ(t,ξ) ⃗ ζ ξ dξ ⇔ µ j (t) = ˆµ(t,ξ) exp(ıξj)dξ . (8.4.21)
−π
−π
d⃗µ
dt (t) = L h(⃗µ(t)) ⇒ ∂ˆµ
∂t (t,ξ) = ĉ h(ξ)ˆµ(t,ξ) . (8.4.22)
This is a family of scalar, linear ODEs parameterized byξ ∈]−π,π].
Numerical
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DRAFT
Example 8.4.25 (Explicit Euler in Fourier domain).
Explicit Euler timestepping [18, Eq. 12.2.4] for semi-discrete evolution (8.4.15), see also (8.4.5),
⃗µ (k+1) = ⃗µ (k) +τL h ⃗µ (k) .
∫π
∫π ∫π
ˆµ (k+1) (ξ) ⃗ ζ ξ dξ = (Id+τL h ) ˆµ (k) (ξ) ⃗ ζ ξ dξ = ˆµ (k) (ξ)(1+τĉ h (ξ)) dξ .
−π
−π
−π
ˆµ (k+1) (ξ) = ˆµ (k) (ξ)(1+τĉ h (ξ)) .
In Fourier domain a single explicit Euler timestep corresponds to a multiplication of ˆµ :]−π,π] ↦→ C
with the function(1+τĉ h ) :]−π,π] ↦→ C.
Numerica
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R. Hiptm
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DRAFT
Remark 8.4.23 (Fourier series).
Up to normalization the relationship
⃗µ (0) ∈ R Z ↔ ˆµ (0) :]−π,π] ↦→ C
8.4
p. 933
Relate this to an explicit Euler step for the ODE ∂ˆµ
∂t (t,ξ) = ĉ h(ξ)ˆµ(t,ξ) from (8.4.22) with paramter
ξ:
ˆµ (k+1) (ξ) = (1+τĉ h (ξ))ˆµ (k) (ξ) .
✸
8.4
p. 93
from (8.4.21) is the Fourier series transform, which maps a sequence to a2π-periodic function. It has
the important isometry property
➣
∞∑
∫π
|µ j | 2 = 2π |ˆµ(ξ)| 2 dξ .
j=−∞ −π
The symbolĉ h can be viewed as the representation of a difference operator in Fourier domain.
△
Numerical
Methods
for PDEs
Generalize the observation made in the previous example:
∫π
⃗µ (k) = ˆµ (k) (ξ) ⃗ ζ ξ dξ ,
−π
( )
where η (k) (ξ) is the sequence of approximations created by the Runge-Kutta method when
k∈N 0
applied to the scalar linear initial value problem
Numerica
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DRAFT
ẏ = ĉ(ξ)y , y(0) = ˆµ (0) (ξ) .
Clearly, timestepping can only be stable, if blowup|ˆµ (k) (ξ)| → ∞ for k → ∞ can be avoided for all
−π < ξ ≤ π.
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The decoupling manifest in (8.4.22) carries over to Runge-Kutta timestepping in the sense of the
commuting diagram (6.1.54).
We introduce the Fourier transforms of the members of the sequence
(
⃗µ (k)) created by timestep-
k
ping
∫π
⃗µ (k) =
−π
∫π
ˆµ (k) (ξ) ⃗ ζ ξ dξ ⇔ µ (k)
j =
−π
ˆµ (k) (ξ) exp(ıξj)dξ . (8.4.24)
8.4
p. 934
From [18, Thm. 13.1.18] we know:
8.4
p. 93

✬
Theorem 8.4.26 (Stability function of explicit Runge-Kutta methods).
The execution of one step of size τ > 0 of an explicit s-stage Runge-Kutta single step method
(→ Def. 6.1.26) with Butcher scheme c A bT (see (6.1.27)) for the scalar linear ODE ẏ = λy,
λ ∈ C, amounts to a multiplication with the number
✫
Ψ τ λ = 1+zbT (I−zA) −1 1 = det(I−zA+z1b T ) , z := λτ , 1 = (1,...,1) T ∈ R s .
} {{ }
stability function S(z)
Example 8.4.27 (Stability functions of explicit RK-methods).
• Explicit Euler method (8.4.5) :
0 0
1
➣ S(z) = 1+z .
✩
✪
Numerical
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R. Hiptmair
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DRAFT
✤
✣
Stability of RK-timestepping of linear semi-discrete evolution ⇐⇒ max
−π

➣
symbol of difference operator (→ Ex. 8.4.20): ĉ h (ξ) = v h (exp(−ıξ)−1),
stability function:
S(z) = 1+z.
Numerical
Methods
for PDEs
To see this note that the explicit trapezoidal rule is a 2-stage Runge-Kutta method. Hence, the spatial
stencil has width 2 in upwind direction, see Fig. 284.
Numerica
Methods
for PDEs
C
✸
Locus of
Σ := S(τĉ(ξ)) , −π < ξ ≤ π ,
in the complex plane
✄
Σ
1− τv
h
v
h
1
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DRAFT
8.4.3 Convergence
Example 8.4.30 (Convergence of fully discrete finite volume methods for Burgers equation).
R. Hiptm
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= CFL-condition of Def. 8.4.13!
|S(τĉ(ξ))| ≤ 1 ∀−π < ξ ≤ π ⇐⇒ v τ h ≤ 1 .
Fig. 293
Cauchy problem for Burgers equation (8.1.16)
∂u
∂t + ∂
∂x (1 2 u2 ) = 0 in R×]0,T[ , u(x,0) = u 0 (x) , x ∈ R .
Note that the maximal analytic region of dependence for constant velocityv linear advection is merely
a line with slopev in thex−t-plane, see Ex. 8.2.12.
Consider: upwind spatial discretization (8.4.17) & explicit trapezoidal rule: stability function S(z) =
1+z + 1 2 z2
Plots forv = 1,τ = 1
8.4
p. 941
Numerical
Methods
for PDEs
smooth, non-smooth and discontinuous initial data, supported in[0,1]:
u 0 (x) = 1−cos 2 (πx) , 0 ≤ x ≤ 1 , 0 elsewhere ,
u 0 (x) = 1−2∗|x− 2 1| , 0 ≤ x ≤ 1 , 0 elsewhere ,
u 0 (x) = 1 , 0 ≤ x ≤ 1 , 0 elsewhere .
(BUMP)
(WEDGE)
(BOX)
8.4
p. 94
Numerica
Methods
for PDEs
0.8
0.6
Loci Σ for Heun method & upwind finite differencing
vτ/h = 1.111111
vτ/h = 1.000000
vτ/h = 0.833333
5
4.5
Stability for Heun method & upwind finite differencing
➣ maximum speed of propagationṡ = 1.
Bump shaped initial data
1
1
Edge shaped initial data
1
Box shaped initial data
4
0.4
0.8
0.8
0.8
Im
0.2
0
−0.2
max|S(c(ξ))|
3.5
3
2.5
2
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DRAFT
u 0
0.6
0.4
0.2
u 0
0.6
0.4
0.2
u 0
0.6
0.4
0.2
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DRAFT
1.5
0
0
0
−0.4
−0.6
−0.8
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Re
1
0.5
0
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
h/τ v
|S(τĉ(ξ))| ≤ 1 ∀−π < ξ ≤ π ⇐⇒ v τ h ≤ 1 .
= tighter timestep constraint than stipulated by mere CFL-condition (8.4.14).
8.4
p. 942
Fig. 294
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x
(BUMP)
Fig. 295
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x
(WEDGE)
Fig. 296
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x
(BOX)
Spatial discretization on equidistant mesh with meshwidthh > 0 based on finite volume method
in conservation form with
➊ (local) Lax-Friedrichs numerical flux (8.3.26),
➋ Godunov numerical flux (8.3.37).
Initial values ⃗µ (0) obtained from dual cell averages.
8.4
p. 94

Explicit Runge-Kutta (order 4) timestepping with uniform timestepτ > 0.
Fixed ratio: τ : h = 1 (➣ CFL-condition satisfied)
Monitored: error norms (log-log plots)
err 1 (h) := max h ∑ |µ (k)
∥
k>0
j −u(x j ,t k )| ≈ max∥u (k)
k>0
N −u(·,t k) ∥ L
j
1 (R) , (8.4.31)
err ∞ (h) := max max k>0 j∈Z |µ(k) ∥
j −u(x j ,t k )| ≈ max∥u (k)
k>0
N −u(·,t k) ∥ L ∞ (R) . (8.4.32)
Numerical
Methods
for PDEs
for different final timesT = 0.3,4,h ∈ { 1
20 , 1
40 , 1
80 , 1
160 , 1
320 , 1
640 , 1
1280 }. R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Why do we study the particular error norms (8.4.31) and (8.4.32)?
From Thm. 8.2.39 and Thm. 8.2.41 we know that the evolution for a scalar conservation law in 1D
enjoys stability on the norms ‖·‖ L 1 (R) and ‖·‖ L ∞ (R). Hence, these norms are the natural norms
for measuring discretization errors, cf. the use of the energy norm for measuring the finite element
discretization error for 2nd order elliptic BVP.
T = 4, errorerr 1
Burgers 1D: smooth initial data, t=4
Burgers 1D: C 0 initial data, t=4
Burgers 1D: box initial data, t=4
Numerica
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DRAFT
Lax−Friedrichs
Godunov flux
O(h)
err 1
8.4
p. 945
10 −1
10 −2
10 2 10 3
3.5h −1
Fig. 299
10 −1
10 −2
3.5h −1
Lax−Friedrichs
Godunov flux
O(h)
Lax−Friedrichs
Godunov flux
O(h)
err 1
Fig. 300
10 2 10 3
10 −1
10 −2
3.5h −1
err 1
T = 0.3: errorerr ∞
Fig. 301
10 2 10 3
8.4
p. 94
1
0.9
t=0.3
t=4
1
0.9
t=0.3
t=4
Numerical
Methods
for PDEs
Burgers 1D: smooth initial data, t=0.3
Lax−Friedrichs
Godunov flux
O(h)
Burgers 1D: C 0 initial data, t=0.3
Lax−Friedrichs
Godunov flux
O(h)
Burgers 1D: box initial data, t=0.3
Lax−Friedrichs
Godunov flux
O(h)
Numerica
Methods
for PDEs
0.8
0.8
10 −0.4
0.4
0.3
0.4
0.3
10 −1 3.5h −1
Fig. 302
10 2 10 3
10 −1
10 −2
10 2 10 3
3.5h −1
Fig. 303
10 −0.8
Fig. 304
10 2 10 3
0.7
0.7
err ∞
err ∞
err ∞
10 −0.3 3.5h −1
10 −0.5
0.6
0.6
10 −0.6
0.5
0.5
10 −0.7
0.2
0.1
0
−1 −0.5 0 0.5 1 1.5 2 2.5 3
u(x,t)
Fig. 297
“Exact solution”: Initial data (WEDGE)
0.2
0.1
0
−1 −0.5 0 0.5 1 1.5 2 2.5 3
u(x,t)
Fig. 298
“Exact solution”: Initial data (BUMP)
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DRAFT
T = 4: errorerr ∞
10 −0.6
Burgers 1D: smooth initial data, t=4
Lax−Friedrichs
Godunov flux
O(h)
Burgers 1D: C 0 initial data, t=4
Lax−Friedrichs
Godunov flux
O(h)
Burgers 1D: box initial data, t=4
Lax−Friedrichs
Godunov flux
O(h)
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10 −0.3
10 −0.7
These “exact solutions’ were computed with a MUSCL scheme (→ Sect. 8.5.3) on an equidistant
mesh withh = 10 −4
10 −0.8
10 −0.9
10 −0.5 3.5h −1
err ∞
err ∞
10 2 10 3
10 −1 3.5h −1
10
Fig. 305 2 10
Fig. 306
3
10 −0.4
10 −0.5
10 −0.2 3.5h −1
err ∞
Fig. 307
10 2 10 3
Note: for bump initial data (BUMP) we can still expect u(·,0.3) to be smoot, because characteristics
will not intersect before that time, cf. (8.2.14) and Ex. 8.2.15.
8.4
p. 946
8.4
p. 94

Error obtained by comparison with numerical “reference solution” obtained on a very fine spatiotemporal
grid.
Numerical
Methods
for PDEs
Example 8.4.35 (Consistency error of upwind numerical flux).
Numerica
Methods
for PDEs
Oberservations: for either numerical flux function
Assumption: f continuously differentiable u 0 ≥ 0 and f ′ (u) ≥ 0 for u ≥ 0 ➣ no transsonic
rarefactions!
(near) first order algebraic convergence (→ Def. 1.6.20) w.r.t. mesh widthhinerr 1 ,
algebraic convergence w.r.t. mesh width h in err ∞ before the solution develops discontinuities
(shocks),
no covergence in normerr ∞ after shock formation.
✸
R. Hiptmair
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H.
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V.
Gradinaru
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DRAFT
In this case the upwind numerical flux (8.3.28) agrees with the Godunov flux (8.3.37), see Rem. 8.3.38
and
F uw (u(x j ,t),u(x j+1 ,t)) = f(u(x j ),t) , j ∈ Z .
(
⃗τFuw (t) ) j = f(u(x j,t))−f(u(x j+1/2 ,t))
= f ′ (u(x j+1/2 ,t))(u(x j ,t)−u(x j+1/2 ,t))+O(|u(x j ,t)−u(x j+1/2 ,t)| 2 )
= −f ′ (u(x j+1/2 ,t)) ∂u
∂x (x j+1/2 ,t)1 2 h+O(h2 ) forh → 0 ,
by Taylor expansion off andu.
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Best we get:
merely first order algebraic convergenceO(h)
This means that the upwind/Godunov numerical flux is (only) first order consistent.
Heuristic explanation for limited order:
u = u(x,t) ˆ= smooth entropy solution of Cauchy problem
∂u
∂t + ∂
∂x f(u) = 0 in R×]0,T[ , u(x,0) = u 0(x) , x ∈ R . (8.2.9)
We study the so-called consistency error of the numerical fluxF = F(v,w)
(⃗τ F (t)) j = F(u(x j ),u(x j+1 ,t))−f(u(x j+1/2 ,t)) ,j ∈ Z ,
which measures the deviation of the approximate flux and the true flux, when the approximate solution
agreed with the exact solution at the nodes of the mesh.
What we are interested in
where an equidistant spatial mesh is assumed.
Terminology:
behavior of(⃗τ F (t)) j as mesh widthh → 0,
max
j∈Z (⃗τ F(t)) j = O(h q ) forh → 0 ↔ numerical flux consistent of orderq ∈ N . (8.4.34)
Rule of thumb:
Order of consistency of numerical flux function limits (algebraic) order of convergence
of (semi-discrete and fully discrete) finite volume schemes.
8.4
p. 949
Numerical
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R. Hiptmair
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DRAFT
8.4
p. 950
Example 8.4.36 (Consistency error of Lax-Friedrichs numerical flux).
Assumption:
smooth flux function
Recall: The (local) Lax-Friedrichs numerical flux
F LF (v,w) = 2 1(f(v)+f(w))− 2
1 max
min{v,w}≤u≤max{v,w} |f′ (u)|(w−v) , (8.3.26)
is composed of the central flux and a
diffusive flux.
We examine the consistency error for both parts separately, using Taylor expansion
➊
central flux:
1
2 (f(u(x j,t))+f(u(x j+1 ,t)))−f(u(x j+1/2 ,t))
= 1 2 f′ (u(x j+1/2 ,t))(u(x j ,t)−u(x j+1/2 ,t)+u(x j+1 ,t)−u(x j+1/2 ,t))+O(h 2 ) (8.4.37)
= 2 1f′ (u(x j+1/2 ,t)) (∂u
∂x (x j+1/2 ,t)(−1 2 h+ 1 2 h)+O(h2 ) ) +O(h 2 )
= O(h 2 ) forh → 0 .
➣
The central flux is second order consistent.
✸
8.4
p. 95
Numerica
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for PDEs
R. Hiptm
C. Schwa
H.
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Gradinar
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DRAFT
8.4
p. 95

However, due to instability the central flux is useless, see Sect. 8.3.3.1.
➋
diffusive flux part:
u(x j+1 ,t)−u(x j ,t) = ∂u
∂x (x j+1/2 ,t)h+O(h2 ) forh → 0 .
Numerical
Methods
for PDEs
for the same spatial resolution. high-order methods frequently provide more accurate solutions in
the sense of global error norms as first-order methods,
high-order methods often provide better resolution of local features of the solution (shocks, etc.).
Numerica
Methods
for PDEs
F LF (u(x j ,t),u(x j+1 ,t))−f(u(x j+1/2 ,t)) = O(h) forh → 0 ,
because the consistency error is dominated by the diffusive flux.
The observations made in the above examples are linked to a general fact:
✗
✖
Monotone numerical fluxes (→ Def. 8.3.43) are at most first order consistent.
8.5 Higher-order conservative schemes
✸
✔
✕
R. Hiptmair
C. Schwab,
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Gradinaru
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DRAFT
8.5
p. 953
Numerical
Methods
for PDEs
In standard semi-discrete finite volume schemes in conservation form for 2-point numerical flux function,
dµ j
dt (t) = −1 ( F(µj (t),µ
h j+1 (t))−F(µ j−1 (t),µ j (t)) ) , j ∈ Z , (8.3.12)
the numerical flux function is evaluated for the cell averages µ j , which can be read as approximate
values of a projection of the exact solution onto piecewise constant functions (on dual cells)
By Taylor expansion we find foru ∈ C 1
x ∫j+1/2
µ j (t) ≈ 1 u(x,t)dx . (8.3.5)
h
x j−1/2
x ∫j+1/2
u(x j+1/2 ,t)− 1 u(x,t)dx = O(h) forh → 0 ,
h
x j−1/2
h
u
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.5
p. 95
Numerica
Methods
for PDEs
OPTIONAL SECTION
and, unless some lucky cancellation occurs as in
the case of the central flux, see Ex. 8.4.36, this
does not allow more than first order consistency.
O(h)
µ j
O(h)
Formally, high-order conservative finite volume methods are distinguished by numerical flux functions
that are consistent of order≥ 2, see (8.4.34).
However, solutions of (systems of) conservation laws will usually not even be continuous (because of
shocks emerging even in the case of smooth u 0 , see (8.2.15)), let alone smooth, so that the formal
order of consistency may not have any bearing for the (rate of) convergence observed for the method
for a concrete Cauchy problem.
Therefore in the field of numerics of conservation laws “high-order” is desired not so much for the
promise of higher rates of convergence, but for the following advantages:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.5
p. 954
8.5.1 Piecewise linear reconstruction
x j x
Fig. 308x
j+1
Idea: Plug “better” approximations of u(x j±1/2 ,t) into numerical flux function in
(8.3.12)
dµ j
dt (t) = −1 ( F(ν
+
h j (t),νj+1 − (t))−F(ν+ j−1 (t),ν− j (t))) , j ∈ Z , (8.5.1)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
where ν j ± are obtained by piecewise linear reconstruction from the (dual) cell values 8.5
µ j .
p. 95
DRAFT

ν − j (t) := µ j(t)− 1 2 hσ j(t) ,
ν + j (t) := µ j(t)+ 1 2 hσ j(t) ,
with suitable slopesσ j (t) = σ(⃗µ(t)).
j ∈ Z , (8.5.2)
mu0
ν − j
ν + j−1
µ j
ν j
+ µ j+1
ν − j+1
x j−1 x j x j+1
Fig. 309
Analogy: piecewise cubic Hermite interpolation with reconstructed slopes, discussed in the context
of shape preserving interpolation in [18, Sect. 3.6.2]. However, we do not aim for smooth functions
now.
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
13 odefun = @(t,mu) (-1/h*fluxdiff(h,mu,F,slopes));
14 % timestepping by explicity Runge-Kutta method of order 5
15 options = odeset(’abstol’,1E-8,’reltol’,1E-6,’stats’,’on’);
16 [t,MU] = ode45(odefun,[0 T],mu0,options);
17 % Graphical output
18 [X,T] = meshgrid(x,t);
19 f i g u r e ; s u r f(X,T,MU/h); colormap(copper);
20 x l a b e l (’{\bf x}’,’fontsize’,14);
21 y l a b e l (’{\bf t}’,’fontsize’,14);
22 z l a b e l (’{\bf u}’,’fontsize’,14);
23 ufinal = MU(end,:)’;
24 end
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Definition 8.5.5 (Linear reconstruction).
Given an (infinite) mesh M := {]x j−1 ,x j [} j∈Z (x j−1 < x j ), a linear reconstruction operator
R M is a mapping
R M : R Z ↦→ {v ∈ L ∞ (R): v linear on]x j−1/2 ,x j+1/2 [ ∀j ∈ Z} ,
taking a sequence ⃗µ ∈ R Z of cell averages to a possibly discontinuous function R M ⃗µ that is
piecewise linear on dual cells.
Linear reconstruction & (8.5.1) ➣ semi-discrete evolution in conservation form, cf. (8.3.11)
For 2-point numerical flux F = F(u,w)
dµ j
(F(ν
dt (t) = −1 + )
h j (t),ν− j+1 (t))−F(ν+ j−1 (t),ν− j (t)) , j ∈ Z . (8.5.6)
Code 8.5.8: Conservative FV with linear reconstruction: ode45 timestepping
1 f u n c t i o n ufinal = highresevl(a,b,N,u0,T,F,slopes)
2 % finite volume discrete evolution in conservation form with linear
3 % reconstruction, see (8.5.6)
4 % Cauchy problem over time [0,T] restricted to finite interval [a,b],
5 % equidistant mesh with meshwidth N cells, meshwidth h := b−a /N.
6 % 2-point numerical flux function F = F(v,w) passed in handle F
7 % 3-point slope recostruction rule passed as handle slopes = @(v,u,w) ...
8 % (Note: no division by h must be done in slope computation)
9 % returns cell averages for approximate solution at final time in a row vector
10 h = (b-a)/N; x = a+0.5*h:h:b-0.5*h; % cell centers
11 mu0 = h*u0(x)’; % vector of initial cell averages (column vector)
12 % right hand side function for MATLAB ode solvers
8.5
p. 957
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.5
p. 958
Code 8.5.10: Operator L h for spatial semidiscretization with conservative FV with linear reconstruction
and 2-point numerical flux
1 f u n c t i o n fd = fluxdiff(h,mu,F,slopes)
2 % MATLAB function that realizes the right hand side operator L h for the ODE
3 % (8.4.1) arising from conservative finite volume semidiscretization of the
4 % Cauchy problem for a 1D scalar conservation law (8.2.9).
5 % h: meshwidth of equidistant spatial grid
6 % mu: (finite) vector ⃗µ of cell averages
7 % F: handle to 2-point numerical flux function F = F(v,w)
8 % slope: handle to slope function σ j = slopes(µ j−1 ,µ j ,µ j+1 )
9 n = l e n g t h(mu); sigma = zeros(n,1); fd = zeros(n,1);
10 % Computation of slopes σ j , uses µ 0 = µ 1 ,
11 % m N+1 = µ N , which amounts to constant extension of state beyond domain of
12 % influence [a,b] of non-constant intial data.
13 sigma(1) = slopes(mu(1),mu(1),mu(2));
14 f o r j=2:n-1, sigma(j) = slopes(mu(j-1),mu(j),mu(j+1)); end
15 sigma(n) = slopes(mu(n-1),mu(n),mu(n));
16 % Compute linear reconstruction at endpoints of dual cells
17 nup = mu+0.5*sigma; % ν j + at right endpoint
18 num = mu-0.5*sigma; % nu − j at left endpoint
19 % Also here: constant continuation of data outside [a,b] !
20 fd(1) = F(nup(1),num(2)) - F(mu(1),num(1));
21 f o r j=2:n-1
22 fd(j) = F(nup(j),num(j+1)) - F(nup(j-1),num(j)); % see (8.5.6)
23 end
24 fd(n) = F(nup(n),mu(n)) - F(nup(n-1),num(n));
25 end
“Natural” choice: central slope (averaged slope)
σ j (t) = 1 ( µj+1 (t)−µ j (t)
+ µ )
j(t)−µ j−1 (t)
= 1 µ j+1 (t)−µ j−1 (t)
. (8.5.11)
2 h h 2 h
8.5
p. 95
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.5
p. 96

t
By Taylor expansion: for u ∈ C 2 (that is, u sufficiently smooth), central slope (8.5.11), ν ± j according
to (8.5.2)
|ν − j (t)−u(x j−1/2 ,t)|,|ν+ j (t)−u(x j+1/2 ,t)| = O(h2 ) .
Numerical
Methods
for PDEs
Example 8.5.13 (Linear reconstruction with central slope).
Cauchy problem of Ex. 8.3.19:
Numerica
Methods
for PDEs
Example 8.5.12 (Convergence of FV with linear reconstruction).
Cauchy problem for Burgers equation (8.1.16) (flux function f(u) = 1 2 u2 ) from Ex. 8.2.38 with
C 1 bump initial data (BUMP)
Equidistant spatial mesh with meshwidthh =
Linear reconstruction with central slope (8.5.11)
Godunov numerical flux (8.3.37):
F = F GD
2n-order Runge-Kutta timestepping (method of Heun), timestepτ = 0.5h (“CFL = 0.5”)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1
0.8
Cauchy problem for Burgers equation (8.1.16) (flux functionf(u) = 1 2 u2 ) from Ex. 8.2.38 (“box”
intial data)
Equidistant spatial mesh with meshwidthh =
Linear reconstruction with central slope (8.5.11)
Godunov numerical flux (8.3.37):
F = F GD
timestepping based on adaptive Runge-Kutta method ode45 of MATLAB
(opts = odeset(’abstol’,1E-7,’reltol’,1E-6);).
h = 0.066667, t=1.002437
finite volume solution
exact solution
1
0.8
h = 0.066667, t=2.002624
finite volume solution
exact solution
1
0.8
h = 0.066667, t=3.001343
finite volume solution
exact solution
1
0.8
h = 0.066667, t=4.000000
finite volume solution
exact solution
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
0.6
0.6
0.6
0.6
Monitored: ApproximateL 1 - andL ∞ -norms of error at final timeT = 0.3 (exact solution still smooth
at this time, see Ex. 8.4.30)
8.5
p. 961
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.2
Fig. 311
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
8.5
p. 96
1
0.8
Burgers equation, BUMP initial data
t = 0
t=0.1
t=0.2
t=0.3
✁ “exact solution”
Numerical
Methods
for PDEs
u(t,x)
1
0.8
0.6
0.4
4
3.5
3
Numerica
Methods
for PDEs
0.6
0.2
0
2.5
u
0.4
0.2
0
−0.5 0 0.5
x
1 1.5
Fig. 310
computed by means of a high-order finite volume
method (WENO) on a equidistant mesh with 2 14
points., U. Fjordholm (SAM)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
4
3.5
3
2.5
t
2
1.5
1
0.5
0
−0.5
0
0.5
1
x
1.5
2
2.5
3
2
1.5
1
3.5 0.5
0
0 0.5 1 1.5 2 2.5 3
x
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Error plots
Emergence of spurious oscillations in the vicinity of shock (in violation of structural properties of the
exact solution, see (8.2.40).)
Observation: 2nd-order convergence in both norms
Compare: Oscillations occurring in FV schemes relying on central flux, see Ex. 8.3.19.
✸
8.5
p. 962
✸
8.5
p. 96

In Ex. 8.3.19, 7.2.17, the spurious oscillations can be blamed on the unstable central flux/central
finite differences. Maybe, this time the central slope formula is the culprit. Thus, we investigate slope
reconstruction connected with backward and forward difference quotients.
Numerical
Methods
for PDEs
It seems to be the very process of linear reconstruction that triggers oscillations near shocks. Theses
oscillations can be traced back to “overshooting” of linear reconstruction at jumps.
Numerica
Methods
for PDEs
Example 8.5.14 (Linear reconstruction with one-sided slopes).
1.4
Centered slope selection
One-sided slopes for use in (8.5.2)
Right slope:
Left slope:
σ j (t) = µ j+1(t)−µ j (t)
h
σ j (t) = µ j(t)−µ j−1 (t)
h
, (8.5.15)
. (8.5.16)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Slope from central differencing:
σ j = 1
2h (µ j+1 −µ j−1 ) . (8.5.11)
u
1.2
1
0.8
0.6
0.4
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Same setting as in Ex. 8.5.13, with central slope replaced with one-sided slopes.
Left slope:
0.2
0
−0.2
function
averages
reconstruction
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x
Fig. 314
8.5
p. 965
8.5
p. 96
1
0.8
h = 0.066667, t=1.001099
finite volume solution
exact solution
1
0.8
h = 0.066667, t=2.001189
finite volume solution
exact solution
1
0.8
h = 0.066667, t=3.000629
finite volume solution
exact solution
1
0.8
h = 0.066667, t=4.000000
finite volume solution
exact solution
Numerical
Methods
for PDEs
1.6
Right slope selection
Numerica
Methods
for PDEs
0.6
0.6
0.6
0.6
1.4
0.4
0.4
0.4
0.4
1.2
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
Right slope:
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.2
Fig. 312
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
Slope from forward differencing:
σ j = 1 h (µ j+1 −µ j ) . (8.5.15)
u
1
0.8
0.6
1
0.8
0.6
0.4
h = 0.066667, t=1.002046
finite volume solution
exact solution
1
0.8
0.6
0.4
h = 0.066667, t=2.000722
finite volume solution
exact solution
1
0.8
0.6
0.4
h = 0.066667, t=3.000618
finite volume solution
exact solution
1
0.8
0.6
0.4
h = 0.066667, t=4.000000
finite volume solution
exact solution
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
0.4
0.2
0
−0.2
function
averages
reconstruction
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.2
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.2
Fig. 313
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x
Fig. 315
Observation: spurious oscillations/overshoots, massive and global for (8.5.15), moderate close to
shock for (8.5.16).
✸
8.5
p. 966
8.5
p. 96

1.2
Left slope selection
Numerical
Methods
for PDEs
Related:
shape preserving Hermite interpolation, see [18, Sect. 3.6.2], achieved by using
Numerica
Methods
for PDEs
1
Slope from backward differencing:
0.8
0.6
zero slope, in case of local slopes with opposite sign, see [18, (3.6.7)],
harmonic averaging of local slopes, see [18, (3.6.9)].
σ j = 1 h (µ j −µ j−1 ) . (8.5.16)
u
0.4
0.2
0
−0.2
−0.4
function
averages
reconstruction
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Remark 8.5.22 (Consequence of monotonicity preservation).
A monotonicity preserving linear reconstruction operatorR M (→ Def. 8.5.18)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x
Fig. 316
• respects the range of cell averages
8.5.2 Slope limiting
min{µ k µ k+1 ,...,µ m } ≤ (R⃗µ)(x) ≤ max{µ k ,µ k+1 ,...,µ m } , x k < x < x m . (8.5.23)
↔ “range preservation” by entropy solutions, see Thm. 8.2.39.
• does not allow the creation of new extrema
8.5
p. 969
♯{extrema of R M ⃗µ} ≤ ♯{extrema of ⃗µ} . (8.5.24)
↔ preservation of number of extrema in entropy solution, Sect. 8.2.7.
8.5
p. 97
Guarantee for suppression of “overshoots” (→ Figs. 314, 315, 316)
Numerical
Methods
for PDEs
△
Numerica
Methods
for PDEs
local monotonicity preservation of linear reconstruction
Definition 8.5.18 (Monotonicity preserving linear interpolation).
An linear reconstruction operatorR M (→ Def. 8.5.5) is monotonicity preserving, if
(R M ⃗µ)(x j ) = µ j ∧ µ j ≤ µ j+1 ⇒ R M ⃗µ non-decreasing in ]x j ,x j+1 [ ,
µ j ≥ µ j+1 ⇒ R M ⃗µ non-increasing in ]x j ,x j+1 [ .
Monotonicity preserving linear
reconstruction:
constant at plateaus
constant at (local) extrema
x j−1 x j x j+1 x j+2 Fig. 317
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.5
p. 970
Remark 8.5.28 (Linearity and monotonicity preservation).
The linear reconstruction operators (→ Def. 8.5.5) based on the slope formulas (8.5.11) (central
slope), (8.5.15) (forward slope), (8.5.16) (backward slope) are linear in the sense that
✬
✫
R M (α⃗µ+β⃗ν) = αR M (⃗µ)+βR M ⃗ν) ∀⃗µ,⃗ν ∈ R Z , α,β ∈ R . (8.5.29)
Lemma 8.5.30 (Linear monotonicity preserving reconstruction trivial).
Every linear, monotonicity preserving (→ Def. 8.5.18) linear reconstruction yields piecewise
constant functions.
Proof. Define⃗ǫ k ∈ R Z ,k ∈ Z, by
ǫ k j = {
1 fork = j ,
0 else.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
✩
DRAFT
✪
8.5
p. 97

The⃗ǫ k form a basis of R Z . Thus, { due to linearity, R M is fixed by its action on the basis vectors⃗ǫ k
and its image is spanned by R M ⃗ǫ k} k∈Z .
Numerical
Methods
for PDEs
✬
Lemma 8.5.34 (Monotonicity preservation of minmod reconstruction).
Minmod reconstruction (→ Def. 8.5.33) is monotonicity preserving (→ Def. 8.5.18)
✩
Numerica
Methods
for PDEs
However, monotonicity preservation entails thatR M ⃗ǫ k is piecewise constant, see Fig. 317.
✷
✫
✪
Necessary (for monotonicity preservation):
Non-linear linear reconstruction
!?
△
Proof. w.l.o.g. assumeµ j+1 ≥ µ j ⇒ σ j ≥ 0 ∧σ j+1 ≥ 0
⇒ µ j + 1 2 hσ j ≤ 1 2 (µ j +µ j+1 ) ≤ µ j+1 − 1 2 hσ j+1
✷
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Terminology: effect ofminmod-function inR mm : slope limiting: minmod = slope limiter
Example 8.5.35 (Linear reconstruction with minmod limiter).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
What is a non-linear linear reconstruction in the context of high-order finite volume methods for 1D
conservation laws?
Same setting as in Ex. 8.5.13, Cauchy problem as in Ex. 8.3.19:
A simple consideration, see Fig. 317
µ j−1 ≤ µ j and µ j ≥ mu j+1 ⇒ R M ⃗µ ≡ const on ]x j−1/2 ,x j+1/2 [ , (8.5.32)
8.5
p. 973
Cauchy problem for Burgers equation (8.1.16) (flux functionf(u) = 1 2 u2 ) from Ex. 8.2.38 (“box”
intial data)
Equidistant spatial mesh with meshwidthh = 1
15
8.5
p. 97
for any monotonicity preserving (→ Def. 8.5.18) linear reconstruction operatorR M (→ Def. 8.5.5).
➣ monotonicity preserving linear reconstructionR M ⃗µ must be constant at local extrema of ⃗µ!
Numerical
Methods
for PDEs
Linear reconstruction with minmod limited slope (→ Def. 8.5.33)
( µj −µ j−1
σ j := minmod , µ )
j+1 −µ j
.
h h
Numerica
Methods
for PDEs
Definition 8.5.33 (Minmod reconstruction).
The minmod reconstruction R mm is a piecewise
linear reconstruction (→ Def. 8.5.5) defined
by
(R mm ⃗µ)(x) = µ j +σ j (x−x j )
for x j−1/2 < x < x j+1/2 ,j ∈ Z ,
(
µj+1 −µ j
σ j := minmod , µ )
j −µ j−1
,
x j+1 −x j x j −x j−1
⎧
⎪⎨ v ,vw > 0, |v| < |w| ,
minmod(v,w) := w ,vw > 0, |w| < |v| ,
⎪⎩
0 ,vw ≤ 0 .
u
Minmod monotonicity preserving linear interpolation
1
0.8
0.6
0.4
0.2
µ j
0 const. reconst.
reconstruction
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x
Fig. 318
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
1
0.8
0.6
0.4
0.2
Godunov numerical flux (8.3.37):
F = F GD
timestepping based on adaptive Runge-Kutta method ode45 of MATLAB
(opts = odeset(’abstol’,1E-7,’reltol’,1E-6);).
h = 0.066667, t=1.003384
finite volume solution
exact solution
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
1
0.8
0.6
0.4
0.2
h = 0.066667, t=2.001321
finite volume solution
exact solution
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
1
0.8
0.6
0.4
0.2
h = 0.066667, t=3.000543
finite volume solution
exact solution
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
1
0.8
0.6
0.4
0.2
h = 0.066667, t=4.000000
finite volume solution
exact solution
Fig. 319
0
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.5
p. 974
8.5
p. 97

t
1
0.8
4
3.5
Numerical
Methods
for PDEs
1
0.8
Burgers equation (transsonic rarefaction), N = 60
u
low order: Godunov
high order: minmod
1
0.8
Burgers equation (transsonic rarefaction), N = 60
u
low order: Godunov
high order: minmod
Numerica
Methods
for PDEs
u(t,x)
0.6
0.4
0.2
0
4
3.5
3
3
2.5
2
1.5
µ j
(0.50)
0.6
0.4
0.2
0
−0.2
µ j
(2.00)
0.6
0.4
0.2
0
−0.2
2.5
−0.4
−0.4
2
1.5
1
0.5
t
0
−0.5
0
0.5
1
x
1.5
2
2.5
3
3.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3
x
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
−0.6
−0.8
−1
−2.5 −2 −1.5 −1 −0.5
x
0 0.5 1 1.5
Fig. 320
−0.6
−0.8
−1
−2.5 −2 −1.5 −1 −0.5
x
0 0.5 1 1.5
Fig. 321
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Observation: spurious oscillations successfully suppressed!
Observation: Better resolution of rarefaction fan compared with the conservative finite volume method
based on of Godunov numerical flux without linear reconstruction. Good resolution of shock.
✸
This improved resolution is the main rationale for the use of piecewise linear reconstruction.
Example 8.5.36 (Improved resolution by limited linear reconstruction).
Same setting as in Ex. 8.3.30: Cauchy problem for Burgers equation (8.1.16) (flux function
f(u) = 1 2 u2 ) from Ex. 8.2.38 (shifted “box” intial data, u 0 (x) = −1 for x ∉ [0,1], u 0 (x) = 1 for
x ∈ [0,1])
Equidistant spatial mesh with meshwidthh = 1
15
“High-order” method based on linear reconstruction with minmod limited slope (→ Def. 8.5.33)
( µj −µ j−1
σ j := minmod , µ )
j+1 −µ j
.
h h
Godunov numerical flux (8.3.37):
F = F GD
timestepping based on adaptive Runge-Kutta method ode45 of MATLAB
(opts = odeset(’abstol’,1E-10,’reltol’,1E-8);).
8.5
p. 977
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
8.5.3 MUSCL scheme
= Monotone Upwind Scheme for Conservation Laws
Case of equidistant spatial mesh with meshwidthh > 0:
Conservative finite volume spatial discretization (8.5.1) with monotone consistent 2-point flux, e.g.,
Godunov numerical flux (8.3.37)
Piecewise linear reconstruction (→ Def. 8.5.5) withminmod slope limiting (→ Def. 8.5.33):
ν ± j := µ j ± 1 2 minmod( µ j+1 −µ j ,µ j −µ j−1
) . (8.5.39)
✸
8.5
p. 97
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
2nd-order Runge-Kutta timestepping for (8.5.1): method of Heun, cf. (8.4.6):
If the right hand side of (8.5.1) is abbreviated by
8.5
p. 978
L h (⃗µ) := − 1 h( F(ν
+
j (t),ν − j+1 (t))−F(ν+ j−1 (t),ν− j (t))) ,
8.5
p. 98

then the fully discrete scheme (uniform timestepτ > 0) reads (8.5.1)
⃗κ := ⃗µ (k) + 1 2 τL h(⃗µ (k) ) ,
⃗µ (k+1) := ⃗µ (k) +τhL h (⃗κ) .
Example 8.5.41 (Adequacy of 2nd-order timestepping).
(8.5.40)
Numerical
Methods
for PDEs
Observation: 2nd-order Runge-Kutta method (8.5.40) provides same accuracy as “overkill integration”
by means of ode45 with tigth tolerances.
➣
For the sake of efficiency balance order of spatial and temporal discretizations and use Heun
timestepping.
Numerica
Methods
for PDEs
R. Hiptmair Example 8.5.42 (Convergence of MUSCL scheme).
C. Schwab,
H.
Harbrecht
V.
Gradinaru
Same setting as in Ex. 8.3.30: Cauchy problem for Burgers equation (8.1.16) (flux function A. Chernov
f(u) = 1 2 u2 DRAFT
) from Ex. 8.2.38 (shifted “box” intial data, u 0 (x) = −1 for x ∉ [0,1], u 0 (x) = 1 for
x ∈ [0,1])
Numerical experiments of Ex. 8.4.30 repeated for
Equidistant spatial mesh with meshwidthh = 15
1
conservative finite volume discretization with Godunov numerical flux andminmod-limited linear
Linear reconstruction with minmod limited slope (→ Def. 8.5.33)
( µj −µ j−1
σ j := minmod , µ )
reconstruction, see Ex. 8.5.35 (ode45 timestepping),
j+1 −µ j
.
MUSCL scheme as introduced above with fixed timestepτ = 0.5h.
h h
8.5
Godunov numerical flux (8.3.37): F = F Monitored: “discrete” error norms (8.4.31), (8.4.32)
GD
p. 981
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
8.5
p. 98
Two options for timestepping
1. timestepping based on adaptive Runge-Kutta method ode45 of MATLAB
(opts = odeset(’abstol’,1E-10,’reltol’,1E-8);).
2. Heun timestepping (8.5.40) with uniform timestepτ = h
Burgers equation (transsonic rarefaction), N = 60
Burgers equation (transsonic rarefaction), N = 60
Numerical
Methods
for PDEs
L 1 −norm of error at t=0.3
10 −2
L ∞ −norm of error at t=0.3
10 −1
Numerica
Methods
for PDEs
µ j
(0.50)
1
0.8
0.6
0.4
0.2
0
u
ode45, 157 timesteps
Heun timestepping, cfl = 1.0
µ j
(2.00)
1
0.8
0.6
0.4
0.2
0
u
ode45, 725 timesteps
Heun timestepping, cfl = 1.0
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
10 −3
10 −1 meshwidth h
BUMP initial data
WEDGE initial data
BOX initial data
O(h 3/2 )
10 −1
10 −2
Fig. 324
10 0 meshwidth h
10 −2
10 −1
BUMP initial data
WEDGE initial data
BOX initial data
O(h)
10 −2
Fig. 325
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
x
Fig. 322
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
x
Fig. 323
8.5
p. 982
8.5
p. 98

Numerical
Methods
for PDEs
Numerica
Methods
for PDEs
L 1 −norm of error at t=4
10 −1 meshwidth h
10 −2
L ∞ −norm of error at t=4
10 0
10 −1
meshwidth h
9
Finite Elements for the Stokes
Equations
10 −3
BUMP initial data
WEDGE initial data
BOX initial data
O(h)
10 −1
10 −2
Fig. 326
10 −2
10 −1
BUMP initial data
WEDGE initial data
BOX initial data
10 −2
Fig. 327
Observation: 2nd-order Heun method produces solutions whose convergence and accuracy matches
those of solutions obtained by highly accurate high-order Runge-Kutta timestepping.
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
OPTIONAL CHAPTER
Dependencies:
• Modelling of fluid flow Sect. 7.1.1
• Variational calculus for equilibrium problems, entire Ch. 2
• Basic concepts and algorithms for finite elements, Sects. 3.1, 3.3, 3.4
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• Finite differences Sect. 4.1
➣
• MATLAB FEM library [6], Sect. 3.5
8.6 Outlook: systems of conservation laws
✸
8.6
p. 985
Numerical
Methods
for PDEs
This chapter has a strong modelling component, similar to Ch. 1.
9.0
p. 98
Numerica
Methods
for PDEs
Supplementary and further reading:
Books (chapters) dealing with the (mathematical foundations of) discretization of the Stokes boundary
value problem:
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
[15, Ch. 12]: Concise introduction to the Stokes PDEs and Galerkin discretization. Modelling is not
discussed.
[5, Ch. 12]: Numerical analysis of variational saddle point problems
[3, III.§5]: Concise presentation of principles of finite element discretization of the Stokes problem
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.1 Viscous fluid flow
8.6
p. 986
9.1
p. 98

Task:
simulation of stationary fluid flow
computation of the velocityv = v(x) of a fluid moving in a containerΩ ⊂ R d ,d = 2,3, under
the influence of an external force fieldf : Ω ↦→ R d .
(d = 2? ↔ translational symmetry ➥ dimensionally reduced model)
Numerical
Methods
for PDEs
In this case acceptably accurate modelling can neglect inertia forces ➣ creeping flow
Viscous fluids “stick to the walls of the container”
no-slip boundary conditions: v = 0 on ∂Ω . (9.1.5)
Numerica
Methods
for PDEs
✎ notation: as before, bold typeface for vector valued functions
Recall: description of fluid motion through a velocity field → Sect. 7.1.1
We restrict ourselves to incompressible fluids → Def. 7.1.7
Thm. 7.1.12 Constraint divv = 0 . (9.1.2)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
configuration space for viscous incompressible fluid
{
v : Ω ↦→ R
V :=
d }
continuous,
. (9.1.6)
divv = 0 , v |∂Ω = 0
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
configuration space for incompressible fluid
{
V := v : Ω ↦→ R d }
continuous , divv = 0 . (9.1.3)
9.1
p. 989
We appeal to an extremal principle to derive governing equations for incompressible creeping flow:
the state of the system renders a physical quantity minimal.
For the elastic string (→ Sect. 1.2), taut membrane (→ Sect. 2.1.1), electrostatic field (→ Sect. 2.1.2)
this quantity was the total potential energy. For stationary viscous fluid flow, this role is played by the
energy dissipation:
9.1
p. 99
Numerical
Methods
for PDEs
energy dissipation = conversion of kinetic energy into internal energy (heat)
(↔ entropy production)
Numerica
Methods
for PDEs
Flow regimes of an incompressible Newtonian fluid (a fluid, for which stress is linearly proportional to
strain) are distinguished by the size of a fundamental non-dimensional quantity, the
Reynolds number
Re := ρVL
µ ,
where (ford = 3) ρ ˆ= density ([ρ] = kgm −3 )
V ˆ= mean velocity ([V] = ms −1 )
L ˆ= characteristic length of region of interest ([L] = m)
µ ˆ= dynamic viscosity ([µ] = kgm −1 s −1 )
Reynolds number = ratio of inertia forces:viscous (friction) forces
The Reynolds number becomes small, if
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
AXIOM: energy dissipation functional for viscous fluid ([P diss ] = W)
∫
P diss (v) = µ‖curlv(x)‖ 2 dx (9.1.7)
Ω
rotation/curl ˆ= first-order differential operator
⎛
∂v 2
− ∂v ⎞
3
∂x 3 ∂x 2
curlv :=
∂v 3
− ∂v 1
⎜∂x 1 ∂x 3
⎝∂v 1
− ∂v ⎟
2⎠
∂x 2 ∂x 1
for d = 3 , curlv := ∂v 1
− ∂v 2
for
∂x 2 ∂x 1
d = 2 . (9.1.8)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
• the speed of the flow is very small (slowly flowing fluids), or
• the flow is studied at tiny length scales (micro flows), or
• the fluid is highly viscous (“sticky”).
9.1
p. 990
9.1
p. 99

0
0.1
0.2
0.3
45
40
35
Numerical
Methods
for PDEs
To that end we study a related problem in finite dimensional context R n :
x ∗ = argmax x T Ax , (9.1.15)
x T Ax=b T x
Numerica
Methods
for PDEs
x 2
0.4
0.5
30
25
with s.p.d. A ∈ R n,n , b ∈ R n . With the transformationy = A −1/2 x (→ [18, Rem. 5.3.2]) we arrive
at the equivalent maximization problem
0.6
0.7
20
15
y ∗ = argmax ‖y‖ 2 .
‖y‖ 2 =(A −1/2 b) T y
0.8
10
Fig. 328
0.9
5
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 1
Fig. 329
“eddy field”,divv = 0
Plot of‖curlv‖ for eddy field
curlv = “density of eddies/vortices in flow field”
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
A −1 b
The set {y : ‖y‖
2 = (A −1/2 b) T y} is a sphere through
0 around 1 2 A−1/2 b and we are looking for its point farthest
away from 0. By “geometric considerations” this will be the
pointy ∗ = A −1/2 b ➣ x ∗ = A −1 b.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Thus, in viscous fluid flow the conversion of kinetic energy into heat due to friction presumably happens
in vortical flow patterns (eddies).
9.1
p. 993
Recall: relationship between linear systems of equations and quadratic minimization problems, see
[18, Sect. 5.1.1] and Sect. 2.1.3.
9.1
p. 99
Second law of thermodynamics for creeping flow:
Maximization of energy dissipation in flow (9.1.9)
entropy production
Numerical
Methods
for PDEs
x ∗ = A −1 b can be obtained as solution of
x ∗ = argmin
x∈R n 1
2 x T Ax−b T x . (9.1.16)
To have faith that this reasoning applies to (9.1.14) as well, the bilinear form (u,v) ↦→ ∫ Ω curlu ·
curlvdx should be positive definite (→ Def. 2.1.25) ➤ see Lemma 9.2.1 below.
Numerica
Methods
for PDEs
First law of thermodynamics: conservation of energy/power balance
∫
∫
µ‖curlv(x)‖ 2 dx = f ·vdx . (9.1.10)
Ω
Ω
dissipated energy
energy injected through forces
First equilibrium condition for viscous stationary flow:
{∫
}
v ∗ = argmax µ‖curlv(x)‖ 2 dx: v ∈ V , v satisfies (9.1.10)
Ω
ˆ= constrained optimization problem with constraint (9.1.10).
Goal:
Convert (9.1.14) into a “more standard” optimization problem.
(9.1.14)
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
9.1
p. 994
Another issue, of course, is, whether the above arguments remain true for (infinite dimensional) function
spaces ➤ theory of variational calculus [30, Ch. 49], not elaborated here.
Second equilibrium condition for viscous stationary flow, cf. (2.1.4), (2.1.14):
∫
∫
v ∗ = argmin 1
2 µ‖curlv(x)‖ 2 dx− f ·vdx . (9.1.17)
v∈V Ω
Ω
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.2
p. 99

9.2 The Stokes equations
Numerical
Methods
for PDEs
ˆ= quadratic minimization problem (→ Def. 2.1.21) on function spaceV.
Numerica
Methods
for PDEs
9.2.1 Constrained variational formulation
✬
Lemma 9.2.1 (−∆ = curlcurl−graddiv).
Forv ∈ C 2 (Ω),v |∂Ω = 0, holds
∫ ∫ ∫
‖curlv‖ 2 dx+ |divv| 2 dx = ‖Dv‖ 2 F dx .
Ω Ω Ω
✫
✩
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✪
Rewrite quadratic form (µ ≡ const)
v = (v 1 ,...,v d ) T :
∫
d∑
µ‖Dv‖ 2 F dx = µ ‖gradv i ‖ 2 dx .
Ω
i=1
By the first Poincaré-Friedrichs inequality of Thm. 2.2.16
∫
‖v‖ 2 L 2 (Ω) ≤ diam(Ω)2 ‖Dv‖ 2 F dx ∀v ∈ V ⊂ (H1 0 (Ω))3 .
Ω
Bilinear formafrom (9.2.4) is positive definite (→ Def. 2.1.25).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
( ) d ∂vi
✎ notations: Dv := : Ω ↦→ R d,d Jacobian,
∂x j
i,j=1
‖M‖ F ˆ= Frobenius matrix norm (→ [18, Def. 6.5.24])
9.2
p. 997
Remark 9.2.7 (Decoupling of velocity components ?).
Rewrite (9.2.4) in terms of componentsv i of velocity (with force fieldf = (f 1 ,f 2 ,f 3 ) T ):
(9.2.4) ⇔ argmin
v∈V
3∑ ( ∫
12
i=1
Ω
∫
µ‖gradv i ‖ 2 )
dx− f i v i dx . (9.2.8)
Ω
9.2
p. 99
Proof (of Lemma 9.2.1)
Use the variant of Green’s first formula Thm. 2.4.7
∫ ∫
∂u ∂v
vdx = − udx ∀u,v ∈ C 1 (Ω) , u,v = 0 on∂Ω , (9.2.3)
∂x j ∂x j
Ω Ω
and the fact that different partial derivatives can be interchanged, which implies
∫ ∫
∂u ∂v ∂u ∂v
dx = dx , k,j = 1,...,d .
∂x j ∂x k ∂x k ∂x j
Ω Ω
Then use the definitions ofcurl anddiv.
✷
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Well, three copies of (2.1.14) ?!
NO!
divv = 0 constraint (9.1.2) links components of velocity fieldv.
This constraint in the space V represents the crucial difference compared to minimization problems
(2.1.4), (2.1.14) underlying scalar 2nd-order elliptic variational equations.
As in Sect. 2.2: put (9.2.4) into Hilbert space (more precisely, Sobolev space) framework, where we
have existence and uniqueness of solutions.
△
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
In light of the propertiesdivv = 0,v = 0 on∂Ω, for eligible fluid velocity fields, see (9.1.6), we have
the equivalence:
(9.1.17)
Lemma 9.2.1
⇐⇒
v ∗ = argmin
v∈V
∫
1
2 µ‖Dv‖ 2 F dx
Ω
} {{ }
=:a(v,v)
∫
−
f ·vdx . (9.2.4)
Ω
} {{ }
=:l(v)
9.2
p. 998
(9.2.8) offers hint on how to choose suitable Sobolev spaces.
Remember: function spaces for a (linear) variational problem are chosen as the largest (Hilbert)
spaces on which the involved bilinear forms and linear forms are still continuous, cf.
(2.2.1), (3.1.2).
appropriate Sobolev space for (9.2.4):
(9.2.8), (9.1.5)
H 1 0
{v (div0,Ω) := ∈ (H0 1 }
(Ω))3 : divv = 0
9.2
p. 100

((H 1 0 (Ω))3 ˆ= space of vector fields with components inH 1 0 (Ω), alternative notationH1 0 (Ω)).
Numerical
Methods
for PDEs
This remark motivates an approach that removes the constraint from trial and test space (and incorporates
it into the variational formulation).
Numerica
Methods
for PDEs
As in Sect. 2.3.1 derive the linear variational problem
v ∈ H 1 0 (div0,Ω): a(v,w) = l(w) ∀w ∈ H1 0 (div0,Ω) ,
from (9.2.8), which reads in concrete terms:
9.2.2 Saddle point problem
}
Seekv ∈ H 1 0
{v (div0,Ω) := ∈ (H0 1(Ω))3 : divv = 0 such that
∫
Ω
∫
gradv i·gradw i dx =
∫
Ω
Ω
∫
Dv : Dwdx =
f i w i dx ∀w ∈ H 1 0 (div0,Ω) , i = 1,2,3 ,
Ω
⇕
f ·wdx ∀w ∈ H 1 0 (div0,Ω) . (9.2.9)
✎ notation: A : B := ∑ a ij b ij for matricesA,B ∈ R m,n (“componentwise dot product”).
i,j
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Idea: weak enforcement of divergence constraint (9.1.2)
Remark 9.2.20 (Heuristics behind Lagrangian multipliers).
Setting:
through Lagrange multiplier
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.2
p. 1001
U,Q ˆ= real Hilbert spaces with inner products(·,·) U ,(·,·) Q ,
J : U ↦→ R convex and differentiable functional,
B : U ↦→ Q linear operator (defining constraint)
9.2
p. 100
For this linear variational problem we verify
• Assumption 5.1.1 from Poincaré-Friedrichs inequality, see above,
• Assumption 5.1.2 forf ∈ (L 2 (Ω)) d by Cauchy-Schwarz inequality, see (2.2.15), (2.3.13),
• Assumption 5.1.3, sinceH 1 0 (div0,Ω) is a closed subspace ofH1 (Ω).
Thm. 5.1.4 ➡ existence & uniqueness of solutions of (9.2.9)
Remark 9.2.11 (H 1 0 (div0,Ω)-conforming finite elements).
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Methods
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Linearly constrained minimization problem
Introduce Lagrangian functional:
L(v,p) := J(v)+(p,Bv) Q
v ∗ = argmin J(v) . (9.2.21)
v∈U,Bv=0
v ∗ = argmin
v∈U
supL(v,p) , (9.2.22)
p∈Q
because, ifBv ≠ 0, the value of the inner supremum will be+∞, and, thus, such av can never be a
candidate for a minimizer.
Terminology:
Terminology:
p is called a Lagrange multiplier,Qthe multiplier space.
a min-max problem like (9.2.22) = saddle point problem
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
In principle, the linear variational problem could be tackled by means of a finite element Galerkin
discretization.
However, finding finite element spaces ⊂H 1 0 (div0,Ω) is complicated [27]: Continuous, piecewise
polynomial, locally supported, and divergence free basis fields exist only for polynomial degree≥ 4!
△
9.2
p. 1002
✬
Lemma 9.2.23 (Necessary conditions for solution of saddle point problem). → [30, Ch. 50]
Any solution v ∗ of (9.2.22) will be the first component of a zero (v ∗ ,p ∗ ) of the derivative (“gradient”)
of the Lagrangian functionalL.
✫
✩
✪
9.2
p. 100

(v ∗ ,p ∗ ) will satisfy
L(v ∗ +tw,p ∗ )−L(v ∗ ,p ∗ )
lim
= 0 ∀w ∈ U ,
t→0 t
L(v ∗ ,p ∗ +tq)−L(v ∗ ,p ∗ )
lim
= 0 ∀q ∈ Q .
t→0 t
because by the very structure of the saddle point problem, see Fig. 330 for illustration,
(9.2.24)
L(v ∗ ,p) ≤ L(v ∗ ,p ∗ ) ≤ L(v,p ∗ ) ∀v ∈ U, p ∈ Q . (9.2.25)
Computing these “directional derivatives” as in Sect. 1.3.1 (for the elastic string energy functional
there), we obtain
〈DJ(v ∗ ),w〉 + (p ∗ ,Bw) Q = 0 ∀w ∈ U ,
(q,Bv ∗ ) Q = 0 ∀q ∈ Q .
(9.2.26)
This is a variational saddle point problem .
Special case: quadratic functionalJ : U ↦→ R → Def. 2.1.17
J(v) := 1 2 a(v,v)−l(v) ,
with a positive definite, symmetric bilinear forma : U ×U ↦→ R (→ Defs. 1.3.13, 2.1.25), continuous
linear forml : U ↦→ R.
(2.3.6)
In this special case (9.2.26) becomes a
Seekv ∗ ∈ U,p ∗ ∈ Q
〈DJ(v ∗ ),w〉 = a(v ∗ ,w)−l(w) , w ∈ U .
linear variational saddle point problem:
a(v ∗ ,w) + (p ∗ ,Bw) Q = l(w) ∀w ∈ U ,
(q,Bv ∗ ) Q = 0 ∀q ∈ Q .
(9.2.27)
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9.2
p. 1005
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Solution of min-max problem:
saddle point
(non-extremal critical point)
The saddle point is a minimum when approached
from the “U-direction”, and a maximum, when approached
from the “Q-direction”.
Adapt abstract approach outline in Rem. 9.2.20 to (9.2.9):
Hilbert spaces:
U = H 1 0 (Ω),Q = L2 (Ω),
3
2.5
2
1.5
1
0.5
0
−0.5
−1
1
p
0
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
v
Fig. 330
Constraintdivv = 0 ➣ B := div : U ↦→ Q continuous,
J ↔v ↦→ 1 ∫
2 µ‖Dv‖ 2 dx− ∫ f ·vdx, a strictly convex quadratic functional (→ Def. 2.1.17)
Ω Ω
Lagrangian functional for (9.2.9)
∫ ∫ ∫
L(v,p) = 2
1 µ‖Dv‖ 2 F dx− f ·vdx+ divvpdx , v ∈ H 1 0 (Ω), p ∈ L2 (Ω) . (9.2.28)
Ω Ω Ω
Next use formula for derivative of quadratic functionals, see Sect. 2.3.1, (2.3.6), which yields a concrete
specimen of (9.2.27).
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9.2
p. 100
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For rigorous mathematical treatment of constrained optimization in Banach spaces refer to [30, Ch. 49
& Ch. 50]. A discussion in finite-dimensional setting is given in [18, Sect. 7.4.1].
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Stokes problem:
Linear variational saddle point problem for viscous flow (preliminary version)
seek velocityv ∈ H 1 0 (Ω), Lagrange multiplierp ∈ L2 (Ω)
∫
µDv : Dwdx + ∫ divwpdx = ∫ f ·wdx ∀w ∈ H 1 0 (Ω) ,
Ω ∫ Ω Ω
divvqdx = 0 ∀q ∈ L 2 (Ω) .
Ω
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Lagrange multiplierp = pressure ([p] = Nm −2 )
9.2
p. 1006
No differential constraints in test/trial spaces for (9.2.33)!
9.2
p. 100

Exercise 9.2.1. Also for stationary heat conduction derive the variational model from the two laws of
thermodynamics:
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Proof. (crude outline; this sketch of the proof is included, because its ideas carry over to the discrete
setting.)
Preparatory considerations: a(v,w) := ∫ Ω µDv : Dwdx is an inner product onH1 0 (Ω).
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• maximization of dissipation with dissipation functional given byJ D (j) = ∫ σ‖j‖ 2 dx,
Ω
• power balancedivj = f.
Then temperature turns out to be a Lagrangian multiplier to ensure the conservation of energy.
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a-orthogonal decomposition
H 1 0 (Ω) = H1 0 (div0,Ω)⊕V⊥
➊ Unique solutionv ∈ H 1 0 (div0,Ω) of (9.2.9) ➣ uniquev-solution for (9.2.33)
(first test withw ∈ H 1 0 (div0,Ω), then withw ∈ V ⊥ .)
➋ Use the following profound result from functional analysis [3, Thm. 5.3]:
✬
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✩
Remark 9.2.31 (Ensuring uniqueness of pressure).
∫ ∫
Notice: divvdx = v·ndS = 0, sincev |∂Ω = 0.
Ω ∂Ω
Pressure solutionpin (9.2.33) can be unique only up to an constant!
Compare: Non-uniqueness of solution of 2nd-order elliptic Neumann problem, Rem. 2.8.13.
Theorem 9.2.36 (Existence of stable velocity potentials).
∃C = C(Ω) > 0: ∀q ∈ L 2 ∗(Ω): ∃v ∈ H 1 0 (Ω): q = divv ∧ ‖v‖ H 1 (Ω) ≤ C‖q‖ L 2 (Ω) .
✫
✪
Idea: Assume f = 0, test first equation with w ∈ V ⊥ satisfying divw = p ➤ ‖p‖ L 2 (Ω) = 0 ⇔
p = 0, for any pressure solutionp ∈ L 2 ∗(Ω).
9.2
9.2
p. 1009
uniqueness of pressure solution
p. 101
Remedy, cf. (2.8.14)
←→
Choose
∫
p ∈ L 2 ∗(Ω) := {q ∈ L 2 (Ω): qdx = 0} . (9.2.32)
Ω
constraint on trial/test spaceL 2 (Ω)
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➌ Existence of pressure solution from Riesz representation theorem (→ functional analysis) and
Thm. 9.2.36, not elaborated here.
✷
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Remaining issue: (9.2.32) introduces another constraint into (9.2.33)!
Stokes problem: Variational saddle point problem for viscous flow
seek velocityv ∈ H 1 0 (Ω), Lagrange multiplierp ∈ L2 ∗(Ω)
∫
µDv : Dwdx + ∫ divwpdx = ∫ f ·wdx ∀w ∈ H 1 0 (Ω) ,
Ω ∫ Ω Ω
divvqdx = 0 ∀q ∈ L 2 ∗(Ω) .
Ω
(9.2.33)
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Relax, Lagrangian multipliers can deal with this, too. Now we study their use to enforce a zero mean
constraint in the simpler setting of 2nd-order elliptic Neumann BVPs.
Remark 9.2.39 (Enforcing zero mean). → [2]
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✬
Theorem 9.2.34 (Existence and uniqueness of weak solutions of Stokes problem).
The linear variational saddle point problem (9.2.33) (“Stokes problem”) has a unique solution.
✫
✩
✪
9.2
p. 1010
As in Sect. 2.4, Rem. 2.8.13, we consider a 2nd-order linear Neumann BVP (with zero
Neumann boundary conditions,h = 0), cf. (2.8.15),
∫
∫
u ∈ H∗(Ω):
1 κ(x)gradu·gradvdx = fvdx ∀v ∈ H∗(Ω) 1 .
Ω
Ω
9.2
p. 101

with the constrained trial/test space
∫
H∗(Ω) 1 := {v ∈ H 1 (Ω): v(x)dx = 0} . (2.8.14)
Ω
The related quadratic minimization problem reads (→ Sect. 2.1.3)
∫
∫
u = argmin J(v) , J(v) := 1
v∈H∗(Ω)
1 2 κ(x)‖gradv‖ 2 dx− fvdx .
Ω
Ω
Idea:
enforce linear constraint ∫ Ωv(x)dx = 0 by means of Lagrangian multiplier,
see Rem. 9.2.20
Here: scalar constraint (Q = R) ➤ scalar multiplierp ∈ R
Lagrangian functional:
∫
L(v,p) = J(v)+p v(x)dx , v ∈ H 1 (Ω) , p ∈ R .
Ω
related (augmented) linear variational saddle point problem, specialization of (9.2.27):
seeku ∈ H 1 (Ω),p ∈ R
∫
∫ ∫
κ(x)gradu·gradvdx + p vdx = fvdx ∀v ∈ H 1 (Ω) ,
Ω ∫
Ω Ω
vdx = 0 .
Ω
(9.2.40)
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DRAFT
9.2
p. 1013
As in Sect. 2.4: derivation of the BVP in PDE form corresponding to (9.2.41).
Approach: Remove spatial derivatives from test functions by integration by parts (1.3.22)
Assuming sufficient smoothness of solution (v,p), constant µ and (9.2.41) and taking into account
boundary conditions, apply Green’s formula of Thm 2.4.7:
∫
d∑
∫
d∑
∫
µ∇v : ∇wdx = µ gradv i·gradw i dx = −µ ∆v i w i dx ,
Ω
i=1 Ω
i=1
∫ ∫
Ω
divwpdx = − gradp·wdx .
Ω
Ω
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9.2
p. 101
The same technique can be applied to (9.2.33).
Stokes variational saddle point problem with pressure normalization:
△
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(9.2.41) ⇒
−µ∆v−gradp = f
∫ divv = 0 in Ω ,
pdx = 0
Ω
v = 0 on ∂Ω .
(9.2.42)
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seek velocityv ∈ H 1 0 (Ω), pressurep ∈ L2 (Ω), multiplierλ ∈ R
∫ ∫
∫
µ∇v : ∇wdx + divwpdx = f ·wdx ∀w ∈ H 1 0 (Ω) ,
Ω ∫ Ω ∫ Ω
divvqdx + λ qdx = 0 ∀q ∈ L 2 (Ω) ,
Ω ∫ Ω
pdx = 0 .
Ω
(9.2.41)
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✎ notation: ∆ ˆ= componentwise Laplacian, see (2.4.13) (“vector Laplacian”)
Remark 9.2.44 (Pressure Poisson equation).
Manipulating the PDEs in (9.2.42):
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9.2.3 Stokes system
div· (9.2.42) −µdiv∆v+divgradp = divf inΩ ,
−µ∆(divv)+∆p = divf inΩ ,
divv=0
∆p = divf .
9.2
p. 1014
9.2
p. 101

Appearance:
Problems
(9.2.42) can be solved by solvingd+1Poisson equations,
☛
☛
first solve pressure Poisson equation∆p = divf
then solve Dirichlet boundary value problems for velocity components
−∆v i = f i + ∂p
∂x i
in Ω , v i = 0 on ∂Ω .
above manipulations only valid for sufficiently smooth u (not guaranteed).
we cannot solve a “Poisson equation”, we also need boundary conditions forp:
not available!
9.3 Saddle point problems: Galerkin discretization
Example 9.3.1 (Naive finite difference discretization of Stokes system).
BVP (9.2.42) onΩ =]0,1[ 2 ,µ ≡ 1,f = cos(πx 1 ) ( 0
1
)
,
Finite difference discretization on (→ Sect. 4.1) equidistant tensor product grid ✄
Unknowns: v 1,ij ,v 2,ij ,p ij ˆ= approximations ofv 1 (ih,jh),v 2 (ih,jh),p(ih,jh),0 < i,j < N.
Zero boundary values forv 1 ,v 2 , andp
5-point stencil discretization of−∆, see (4.1.1)
Central finite difference approximation ofgradp, e.g.,
∂p
≈ 1 ( )
pi+1,j −p
∂x 1|(ih,jh) 2h i−1,j , 1 ≤ i,j < N .
x 2
1
0.8
0.6
0.4
0.2
0
Stirring force field
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15
16 % Build matrix representation of gradp. Note the efficient assembly based on
the
17 % special structure of the matrices.
18 % Auxiliary 1D central finite difference matrix
19 e = ones(unk, 1); CD = spdiags ([-h/2*e h/2*e], [-1 1], unk,unk);
20 % Central difference matrix for ∂ : ∂x This matrix is a block
21
1
% diagonal matrix with N −1 diagonal blocks corresponding to the grid rows.
Its diagonal
22 % blocks are skew-symmetric and bidiagonal with non-zero first off-diagonals
9.3 only.
23 P1 = kron(speye(unk),CD);
p. 1017
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Code 9.3.3: Central finite difference discretization of Stokes system
1 f u n c t i o n [u1,u2,p] = StokesFD(N,f)
2 % Naive finite difference discretization of the Stokes system (9.2.42)
3 % N: number of grid cells in each direction.
4 % f: handle to a (vector valued!) function implementing the force field f
5 % Return values u1, u2 give the velocity components v = (v 1 ,v 2 ) T
6 % in a matrix whose entries correspond to the vertices of the mesh,
7 % p returns the preassure.
8 h = 1/N; % mesh width
9 x = h:h:1-h; % coordinates of interior grid points
10 unk = N-1; % number of interior points in each direction
11 n = 3 * unk^2; % total number of unknowns for v and p
12 % A line-by-line numbering (lexikographic numbering) of the grid points is
assumed,
13 % see Sect. 4.1, Fig. 335.
14 A = g a l l e r y (’poisson’, unk); % Matrix for 5-point stencil discretization of −∆
24 % Central difference matrix for ∂
∂x : This matrix is
25
2
% a block matrix with non-zero first off-diagonal blocks only. Each non-zero
block is
26 % a multiple of the identity.
27 P2 = kron(CD,speye(unk));
28 % Build the complete n×n system matrix and make sure that it is a sparse
matrix.
29 Z = sparse (unk^2, unk^2); % Major mistake would be z = zeros(unk^2,unk^2);
30 H = [A Z P1; Z A P2; P1’ P2’ Z];
31
32 % Assemble the right hand side (sampling of f at interior grid points)
33 F = zeros(n,1);
34 pidx1 = 1; pidx2 = n/3+1;
35 f o r j = 1: s i z e (x), f o r i = 1: s i z e (x)
36 frc = h^2*f(x(i),x(j));
37 F(pidx1) = frc(1); F(pidx2) = frc(2);
38 pidx1 = pidx1+1; pidx2 = pidx2 + 1;
39 end, end
40 % Direct solution of sparse indefinite symmetric system
41 X = H\F;
42
43 % Convert vectors of nodal values into matrix representations of grid functions
44 u1 = r o t 9 0 (reshape(X(1:unk^2),unk,unk));
45 u2 = r o t 9 0 (reshape(X(unk^2+1:2*unk^2),unk,unk));
46 p = r o t 9 0 (reshape(X(2*unk^2+1:end),unk,unk));
47
48 end
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9.3
p. 101
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finite difference grid
Fig. 331
0 0.2 0.4 0.6 0.8 1
x 1
force fieldf
Fig. 332
9.3
p. 1018
9.3
p. 102

0 0.2 0.4 0.6 0.8 1
1
0.8
0.02
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Discrete linear variational saddle point problem:
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x 2
0.6
0.015
0.01
0.005
0
v N ∈ U N
p N ∈ Q N
:
a(v N ,w N ) + b(w N ,p N ) = l(w N ) ∀w N ∈ U N ,
b(v N ,q N ) = 0 ∀q N ∈ Q N .
(9.3.15)
0.4
0.2
0
x 1
Fig. 333
−0.005
−0.01
−0.015
1
0.8
0.6
0.4
0.6
0.4
0.2
0.2
0 0
pressure
velocity solution,N = 50
solution,N = 50
Physically meaningless solution marred by massive spurious oscillations of the pressure.
0.8
1
Fig. 334
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Second step of Galerkin discretization:
Introduce ordered bases
B U := {b 1 N ,...,bN N },
B Q := {β 1 N ,...,βM N } of
U N , N := dimU N ,
Q N , M := dimQ N .
(N +M)×(N +M) linear system of equations symmetric indefinite matrix!
(
A B T
B 0
)( ( ⃗ν ⃗ϕ
=
⃗π)
0)
, (9.3.16)
(9.3.17)
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9.3.1 Pressure instability
Lesson learned:
discretizing saddle point problems can be tricky!
Now, we examine the Galerkin discretization (→ Sect. 3.1) of the linear variational problem (9.2.33)
(Practical schemes will rely on (9.2.41), but here, for the sake of simplicity, we skirt the treatment of
zero mean constraint.)
Shorthand notation for (9.2.33) (↔ abstract linear variational saddle point problem, see (9.2.27))
v ∈ U := H 1 0 (Ω) ,
p ∈ Q := L 2 ∗(Ω)
:
a(v,w) + b(w,p) = l(w) ∀w ∈ U ,
b(v,q) = 0 ∀q ∈ Q .
✸
(9.3.13)
with concrete bilinear forms
∫
∫
a(v,w) := µDv : Dwdx , b(v,q) := divvqdx . (9.3.14)
Ω
Ω
First step of Galerkin discretization:
Replace
H 1 0 (Ω)
L 2 ∗(Ω)
with finite dimensional subspaces
U N ⊂ H 1 0 (Ω)
Q N ⊂ L 2 ∗(Ω)
with Galerkin matrices, right hand side vectors
9.3 A :=
(a(b j ) (∫
) N N
N ,bi N ) p. 1021
i,j=1 = ΩµDb j N (x) : Dbi N (x)dx ∈ R N,N , (9.3.18) 9.3
i,j=1
p. 102
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9.3
p. 1022
B :=
(b(b j ) (∫
N ,βi N ) 1≤i≤M
=
1≤j≤N Ω
⃗ϕ :=
(l(b j ) (∫ N
N ) j=1 = Ω
and basis expansions
divb j N (x)βi N (x)dx )
1≤i≤M
1≤j≤N
∈ R M,N , (9.3.19)
f(x)·b j N (x)dx ) N
j=1
∈ R N , (9.3.20)
N∑
v N = ν j b j M∑
N , p N = π j β j N . (9.3.21)
j=1 j=1
Issue: existence, uniqueness and stability of solutions of (9.3.15)
!
Existence, uniqueness and stability of solutions of discrete variational saddle point problems
cannot be inferred from these properties for the continuous saddle point problem
(→ Thm. 9.2.34).
(Unlike in the case of linear variational problems with s.p.d.
Thm. 3.1.5)
bilinear forms, cf.
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9.3
p. 102

A simple consideration:
M > N ⇒ Ker(B) ≠ {0} ⇒ non-uniquenesss of p N .
➣ dimU N ≥ dimQ N is a necessary condition for uniqueness of solutionp N of (9.3.15)
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But dimU N ≥ dimQ N is not enough: even if this condition is satisfied, the pressure may not be
unique:
Example 9.3.23 (Checkerboard instability for quadrilateral P1-P0 pair). (→ [3, § 6])
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Some “natural” finite element Galerkin schemes for (9.2.33)↔(9.3.13) fail to meet this condition:
Example 9.3.22 (Unstable P1-P0 finite element pair on triangular mesh).
Notation: (cf. S 0 p(M)):
S −1 p (M)
discontinuous functions
locally polynomials of degreep,cf. P p (R d )
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M = uniform tensor product mesh of]0,1[ 2 ✄
velocity spaceU N = (S 0 1,0 (M))2
pressure spaceQ N = S −1
0 (M)∩L2 ∗(Ω)
IfK ∈ N mesh cells in one coordinate direction,
dimU N = 2(K −1) 2 , dimQ N = K 2 −1 .
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The spaces S −1 p (M) are the natural finite element spaces for test/trial functions ∈ L 2 (Ω), because
this function space does not enforce any continuity conditions on piecewise smooth functions. Conversely,H
1 (Ω) does, see Thm. 2.2.17.
Regular triangular mesh of]0,1[ 2
Finite element spaces for (9.2.33)
9.3
p. 1025
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dimQ N < dimU N forK ≥ 4 .
C 1 C 2
9.3
p. 102
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U N := (S1,0 0 (M))2 ,
e 1
Q N := S0 −1 (M)∩L2 ∗(Ω) (M-piecewise constants) .
Consider interior grid pointp = (ih,jh),1 ≤ i,j ≤ K, with
e 4 p e 2 adjacent quadratic cellsC
K ∈ N ˆ= 1 ,C 2 ,C 3 ,C 4 , see figure.
no. of mesh cells in one coordinate
Denote by p i the piecewise constant values of p
direction,
N on C i ,
i = 1,2,3,4.
dimU N = 2(K −1) 2 , dimQ N = 2K 2 −1 .
e
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3
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Fig. 336
A. Chern
DRAFT
DRAFT
dimQ N > dimU N (
Fig. 335
b p N,1 ˆ= nodal basis function for x 1 velocity component at vertex p: b p 1
N,1 = bp N · , where b
0)
p N
is the 2D “tent function” (→ Fig. 87) associated withp.
In this case we end up with a singular linear system (9.3.16), which will make the linear solver bail
out or produce a pressure solution, which is polluted by “noise” fromKer(B).
supp(b p N,1 ) = C 1∪C 2 ∪C 3 ∪C 4
9.3
Apply Gauss’ theorem Thm. 2.4.5 on C
✸ p. 1026
i taking into account that b p N ⊥ normals at e 2, e 4 , and b p N ‖ 9.3
p. 102

normals ate 1 ,e 3 ,
∫
∫ ∫ ∫
divb p N,1 p N dx = p 1 b
∫e p N dx−p 2 b p N dx+p 3 b p N dx−p 4 b p N dx
1 e 1 e 3 e 3
Ω
= 1 2 (p 1−p 2 +p 3 −p 4 ) .
Similarly, ifb p N,2 is the nodal basis function atpfor thex 2-component of the velocityv N , then
∫
divb p N,2 p N dx = 1 2 (p 1+p 2 −p 3 −p 4 ) .
p 1 = 1,p 2 = −1,p 3 = 1,p 4 = −1 ⇒
Now, realize that the setting is translation invariant!
Ω
∫
Ω
∫
divb p N,1 p N dx =
Ω
divb p N,2 p N dx = 0 . (9.3.25)
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
BVP (9.2.42) onΩ =]0,1[ 2 ,µ ≡ 1,f = cos(πx 1 ) ( 0
1
)
, see Ex. 9.3.1
P1-P0 finite element Galerkin discretization on equidistant tensor product quadrilateral mesh,
as in Ex. 9.3.23
Code 9.3.28: P1-P0 finite difference discretization of augmented Stokes problem
1 f u n c t i o n [v1,v2,p] = stokesP1P0FD(N,frc)
2 % P1-P0 finite element discretization of Stokes problem (9.2.41) on a
3 % quadrilateral tensor product mesh, see Ex. 9.3.23.
DRAFT
4 % N: number of mesh cells in one coordinate direction.
5 % f: function handle of type symbol64(x1,x2) to right hand side
6 % force field f
7 h = 1/N; nv = (N-1)^2; nc = N^2; % meshwidth, ♯V(M), ♯M
8 % Assemble system matrix from Kronecker products of 1D Galerkin matrices
9 % Tridiagonal 1D mass matrix for linear finite elements
10 M = h*spdiags (ones(N-1,3)*diag([1/6 2/3 1/6]),[-1 0 1],N-1,N-1);
11 % Tridiagonal 1D Galerkin matrix for d2
, see (1.5.60)
dx 2
12 D = spdiags (ones(N-1,3)*diag([-1 2 -1]),[-1 0 1],N-1,N-1)/h;
13 % 1D Galerkin matrix for p.w. linear/p.w. constant finite elements and the
bilinear
9.3
∫ 1
du
14 % form dx qdx
p. 1029
0
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.3
p. 103
+1
−1 +1
+1
−1 +1
+1
−1 +1
+1
−1 +1
+1
−1
−1
−1
−1
−1
−1 +1
+1
−1 +1
+1
−1 +1
+1
−1 +1
+1
−1 +1
+1
−1
−1
−1
−1
−1
−1 +1
+1
−1 +1
+1
−1 +1
+1
−1 +1
+1
−1 +1
+1
−1
−1
−1
−1
−1
−1 +1
+1 −1
−1 +1
+1 −1
−1 +1
+1 −1
−1 +1
+1 −1
−1 +1
+1 −1
−1 +1
+1 −1
−1 +1
+1 −1
−1 +1
+1 −1
−1 +1
+1 −1
−1 +1
+1 −1
−1 +1
Fig. 337
By (9.3.25) the discrete pressure with alternating
values ±1 in checkerboard fashion will belong to
Ker(B) for this finite element Galerkin method (for
oddK).
Observation:
{p N ∈ Q N : b(v N ,p N ) = 0 ∀v N ∈ U N } ≠ ∅ .
= 1-dimensional space of checkerboard modes
✁ p.w. constant checkerboard mode
Example 9.3.26 (P1-P0 quadrilateral finite elements for Stokes problem).
✸
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
15 G = spdiags (ones(N,2)*diag([-1 1]),[-1 0],N,N-1);
16 % 1D mass matrix for p.w. linear and p.w. constant finite elements
17 C = 0.5*h*spdiags (ones(N,2),[-1 0],N,N-1);
18 % constraint on pressure, see Rem. 9.2.39
19 Delta = kron(M,D)+kron(D,M); % 9-point stencil matrix for discrete Laplacian
20 divx = kron(C,G); divy = kron(G,C); % discrete divergence
21 % Complete saddle point system matrix including Lagrangian multiplier for
enforcing mean zero
22 A = [ Delta , sparse (nv,nv) , divx’ , sparse (nv,1) ;...
23 sparse (nv,nv) , Delta , divy’ , sparse (nv,1) ;...
24 divx , divy , sparse (nc,nc) , ones(nc,1); ...
25 sparse (1,nv) , sparse (1,nv) , ones(1,nc) , 0 ];
26 % Assembly of right hand side
27 phi = zeros(2*nv+nc+1,1); x = h:h:1-h; idx = 1;
28 f o r j=1:N-1, f o r i=1:N-1,
29 phi([idx idx+nv]) = h*h*frc(x(i),x(j)); idx = idx+1;
30 end, end;
31 % Direct solve of (singular for even N) linear saddle point system
32 u = A\phi;
33 % Recover velocity and pressure values on the grid
34 v1 = r o t 9 0 (reshape(u(1:nv),N-1,N-1));
35 v2 = r o t 9 0 (reshape(u(nv+1:2*nv),N-1,N-1));
36 p = r o t 9 0 (reshape(u(2*nv+1:end-1),N,N));
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.3
p. 1030
9.3
p. 103

0 0.2 0.4 0.6 0.8 1
1
P1−P0 velocity vector field, N=50
P1−P0 pressure field, N=50
0.3
Numerical
Methods
for PDEs
Example 9.3.29 (P2-P0 finite element scheme for the Stokes problem).
Numerica
Methods
for PDEs
0.8
0.4
0.3
0.2
x 2
0.6
p N
(x 1
,x 2
)
0.2
0.1
0
−0.1
0.1
0
Ω =]0,1[ 2 , u(x) = (cos(π/2(x 1 +x 2 )),−cos(π/2(x 1 +x 2 ))) T , p(x) = sin(π/2(x 1 −x 2 )), f and
inhomogeneous Dirichlet boundary values foruaccordingly
0.4
0.2
0
x 1
Fig. 338
−0.2
−0.3
−0.4
1
0.8
0.6
0.4
x 2
0.2
0
0
0.2
0.4
x 1
0.6
0.8
−0.1
−0.2
1
−0.3
Fig. 339
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
Sequence of (a) uniform triangular meshes, created by regular refinement,
(b) randomly perturbed meshes from (a) (still uniformly shape-regular & quasiuniform).
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
Observation: ☞
☞
pressure solution marred by checkerboard mode
computed velocity field ok!
DRAFT
“P2-P0-scheme” velocity finite element space U N = (S 0 2,0 (M))2 (continuous, piecewise
quadratic→Sect. 3.4.1, Ex. 3.4.2),
pressure finite element spaceQ N = S −1
0 (M)∩L2 ∗(Ω) (piecewise constant)
DRAFT
9.3.2 Stable Galerkin discretization
In the previous examples we found a subspace ofQ N , which dodgesdivv N in the bilinear formb.
We arrive at the important heuristic insight:
✗
✖
Idea:
divv N must be “large enough to fix the pressure”p N ∈ Q N
How to get a larger trial space for the velocity?
Use larger velocity trial/test spacesv N
➣ larger spacedivv N
➣ more control ofp N
Raise polynomial degree!
✔
✕
✸
9.3
p. 1033
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab, 10
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Monitored: Error norms‖u−u N ‖ 1 ,‖u−u N ‖ 0 ,‖p−p N ‖ 0
Code 9.3.31: LehrFEM driver script P2-P0 finite element method for Stokes problem
1 % LehrFEm driver script for computing solutions of the steady Stokes problem on
the unit square
2 % using piecewise quadratic finite elements for the velocity and piecewise
constants for the
3 % pressure.
4 NREFS = 4; % Number of red refinements
5 NU = 1; % Viscosity
6 % Dirichlet boundary data
7 GD_HANDLE = @(x,varargin)[cos( p i /2*(x(:,1)+x(:,2)))
-cos( p i /2*(x(:,1)+x(:,2)))];
8 % Right hand side source (force field)
9 F_HANDLE = @(x,varargin)[ sin ( p i *x(:,1)) zeros( s i z e (x(:,1))) ];
11 % Initialize mesh
12 Mesh = load_Mesh(’Coord_Sqr.dat’,’Elem_Sqr.dat’);
13 Mesh.ElemFlag = ones( s i z e (Mesh.Elements,1),1);
14 Mesh = add_Edges(Mesh);
15 Loc = get_BdEdges(Mesh);
16 Mesh.BdFlags = zeros( s i z e (Mesh.Edges,1),1);
17 Mesh.BdFlags(Loc) = -1;
18 f o r i = 1:NREFS, Mesh = refine_REG(Mesh); end
19
20 % Assemble Galerkin matrix and load vector
21 A = assemMat_Stokes_P2P0(Mesh,@STIMA_Stokes_P2P0,NU,P7O6());
22 L = assemLoad_Stokes_P2P0(Mesh,P7O6(),F_HANDLE);
23
24 % Incorporate Dirichlet boundary data
9.3 25 [U,FreeDofs] = assemDir_Stokes_P2P0(Mesh,-1,GD_HANDLE); L = L - A*U;
26
p. 1034
9.3
p. 103
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Methods
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C. Schwa
H.
Harbrech
V.
Gradinar
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DRAFT
9.3
p. 103

27 % Solve the linear system (direct solver)
28 U(FreeDofs) = A(FreeDofs,FreeDofs)\L(FreeDofs);
29
30 % Plot and print solution
31 plot_Stokes(U,Mesh,’P2P0’);
32 t i t l e (’{\bf Steady Stokes equation (P2 elements)}’);
33 x l a b e l ([’{\bf # Dofs : ’ i n t 2 s t r ( s i z e (U,1)) ’}’]);
34 c o l o r b a r ;
35 p r i n t (’-depsc’,’func_P2P0.eps’)
Code 9.3.35: Assembly of global Galerkin matrix for P2-P0 finite element method for Stokes problem
1 f u n c t i o n varargout = assemMat_Stokes_P2P0(Mesh,EHandle,varargin)
2 % Assemble Galerking matrix for P2-P0 finite element discretization of Stokes
problem
3 % (9.2.41): piecewise quadratic continuous velocity components and piecewise
4 % constant pressure approximation.
5 %mesh LehrFEM mesh data structure, complete with edge information,
6 %Sect. 3.5.2 The struct MESH must at least contain the following fields:
7 % COORDINATES M-by-2 matrix specifying the vertices of the mesh.
8 % ELEMENTS N-by-3 or matrix specifying the elements of the mesh.
9 % EDGES P-by-2 matrix specifying the edges of the mesh.
10 % ELEMFLAG N-by-1 matrix specifying additional elementinformation.
11 % EHandle passes function for computation of element matrix.
12 % See Sect. 3.5.3 for a discussion of the generic assembly algorithm
13 nCoordinates = s i z e (Mesh.Coordinates,1);
14 nElements = s i z e (Mesh.Elements,1);
15 nEdges = s i z e (Mesh.Edges,1);
16 % Preallocate memory for the efficient initialization of sparse matrix,
Ex. 3.5.18
17 I = zeros(196*nElements,1); J = zeros(196*nElements,1); A =
zeros(196*nElements,1);
18 % Local assembly: loop over all cells of the mesh
19 loc = 1:196;
20 f o r i = 1:nElements
21 % Extract vertex coordinates
22 vidx = Mesh.Elements(i,:);
23 Vertices = Mesh.Coordinates(vidx,:);
24 % Compute 14×14 element matrix: there are 6 local shape functions for the
finite
25 % element space S2 0 (M), and 1 (constant) local shape function for
26 % S0 −1 (M): 6+6+1 = 13 local shape functions for the P2-P0 scheme
27 Aloc = EHandle(Vertices,Mesh.ElemFlag(i),varargin{:});
28 % Add contributions to global Galerkin matrix: the numbering convention is a
follows:
29 % d.o.f. for x 1 -components of the velocity are numbered first, then
30 % x 2 -components of the velocity, then the pressure d.o.f.
31 eidx = [Mesh.Vert2Edge(vidx(1),vidx(2)) ...
32 Mesh.Vert2Edge(vidx(2),vidx(3)) ...
33 Mesh.Vert2Edge(vidx(3),vidx(1))];
34 % Note: entries of an extra last row/column of the Galerkin matrix
corresponding to
35 % pressure d.o.f. are filled with one to enfore zero mean pressure, see
36 % Ex. 9.2.39
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C. Schwab,
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DRAFT
9.3
p. 1037
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37 idx = [vidx,eidx+nCoordinates,... % ↔ v 1
38 vidx+nCoordinates+nEdges,eidx+2*nCoordinates+nEdges, % ↔ v 2
39
40
i+2*(nEdges+nCoordinates), % ↔ p
2*(nEdges+nCoordinates)+nElements+1]; % ↔ zero mean multiplier
41 I(loc) = set_Rows(idx,14); J(loc) = set_Cols(idx,14); A(loc) = Aloc(:);
42 loc = loc+196;
43 end
44
45 % Assign output arguments for creation of sparse matrix
46 i f (nargout > 1), varargout{1} = I; varargout{2} = J; varargout{3} = A;
47 else , varargout{1} = sparse (I,J,A); end
48 r e t u r n
Code 9.3.37: Computation of element matrix for P2-P0 finite element method for Stokes problem
1 f u n c t i o n Aloc = STIMA_Stokes_P2P0(Vertices,ElemInfo,nu,QuadRule,varargin)
2 % Computation of element matrix for P2-P0 finite element discretization of 2D
Stokes problem
3 % Vertices passes the location of the vertices of the triangle
4 % nu is the viscosity parameter
5 % QuadRule specifies local quadrature rule, see Rem. 3.5.39
6 % The function returns a 14×14 dense matrix
7 Aloc = zeros(14,14); % Preallocate memory
8 % Compute element mapping
9 bK = Vertices(1,:); BK = [Vertices(2,:)-bK; Vertices(3,:)-bK];
10 inv_BK_t = transpose(inv (BK)); det_BK = abs(det(BK));
11 % Compute gradients element shape functions and their values at quadrature
points
12 grad_N = grad_shap_QFE(QuadRule.x);
13 grad_N(:,1:2) = grad_N(:,1:2)*inv_BK_t;
14 grad_N(:,3:4) = grad_N(:,3:4)*inv_BK_t;
15 grad_N(:,5:6) = grad_N(:,5:6)*inv_BK_t;
16 grad_N(:,7:8) = grad_N(:,7:8)*inv_BK_t;
17 grad_N(:,9:10) = grad_N(:,9:10)*inv_BK_t;
18
19
grad_N(:,11:12) = grad_N(:,11:12)*inv_BK_t;
% The first 6 rows/columns of the element matrix correspond to the x 1 -component
of the
20 % velocity. the corresponding block of the element matrix agrees with that for
−∆
21 % discretized by means of quadratic Lagrangian finite elements. The local
shape functions are
22 % described in Ex. 3.4.2.
DRAFT
23
24
25
Aloc(1,1) = nu*sum(QuadRule.w.*sum(grad_N(:,1:2).*grad_N(:,1:2),2))*det_BK;
Aloc(1,2) = nu*sum(QuadRule.w.*sum(grad_N(:,1:2).*grad_N(:,3:4),2))*det_BK;
Aloc(1,3) = nu*sum(QuadRule.w.*sum(grad_N(:,1:2).*grad_N(:,5:6),2))*det_BK;
26 Aloc(1,4) = nu*sum(QuadRule.w.*sum(grad_N(:,1:2).*grad_N(:,7:8),2))*det_BK;
27 Aloc(1,5) = nu*sum(QuadRule.w.*sum(grad_N(:,1:2).*grad_N(:,9:10),2))*det_BK;
28 Aloc(1,6) = nu*sum(QuadRule.w.*sum(grad_N(:,1:2).*grad_N(:,11:12),2))*det_BK;
29 Aloc(2,2) = nu*sum(QuadRule.w.*sum(grad_N(:,3:4).*grad_N(:,3:4),2))*det_BK;
30 Aloc(2,3) = nu*sum(QuadRule.w.*sum(grad_N(:,3:4).*grad_N(:,5:6),2))*det_BK;
31 Aloc(2,4) = nu*sum(QuadRule.w.*sum(grad_N(:,3:4).*grad_N(:,7:8),2))*det_BK;
32 Aloc(2,5) = nu*sum(QuadRule.w.*sum(grad_N(:,3:4).*grad_N(:,9:10),2))*det_BK;
33 Aloc(2,6) = nu*sum(QuadRule.w.*sum(grad_N(:,3:4).*grad_N(:,11:12),2))*det_BK;
9.3
34
35
Aloc(3,3) = nu*sum(QuadRule.w.*sum(grad_N(:,5:6).*grad_N(:,5:6),2))*det_BK;
Aloc(3,4) = nu*sum(QuadRule.w.*sum(grad_N(:,5:6).*grad_N(:,7:8),2))*det_BK;
36
p. 1038
Aloc(3,5) = nu*sum(QuadRule.w.*sum(grad_N(:,5:6).*grad_N(:,9:10),2))*det_BK;
Numerica
Methods
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R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.3
p. 103
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.3
p. 104

37 Aloc(3,6) = nu*sum(QuadRule.w.*sum(grad_N(:,5:6).*grad_N(:,11:12),2))*det_BK;
38 Aloc(4,4) = nu*sum(QuadRule.w.*sum(grad_N(:,7:8).*grad_N(:,7:8),2))*det_BK;
39 Aloc(4,5) = nu*sum(QuadRule.w.*sum(grad_N(:,7:8).*grad_N(:,9:10),2))*det_BK;
40 Aloc(4,6) = nu*sum(QuadRule.w.*sum(grad_N(:,7:8).*grad_N(:,11:12),2))*det_BK;
41 Aloc(5,5) = nu*sum(QuadRule.w.*sum(grad_N(:,9:10).*grad_N(:,9:10),2))*det_BK;
42 Aloc(5,6) = nu*sum(QuadRule.w.*sum(grad_N(:,9:10).*grad_N(:,11:12),2))*det_BK;
43
44
Aloc(6,6) = nu*sum(QuadRule.w.*sum(grad_N(:,11:12).*grad_N(:,11:12),2))*det_BK;
% the same for the x 2 -component of the velocity
45 Aloc(7,7) = Aloc(1,1); Aloc(7,8) = Aloc(1,2); Aloc(7,9) = Aloc(1,3);
46 Aloc(7,10) = Aloc(1,4); Aloc(7,11) = Aloc(1,5); Aloc(7,12) = Aloc(1,6);
47 Aloc(8,8) = Aloc(2,2); Aloc(8,9) = Aloc(2,3); Aloc(8,10) = Aloc(2,4);
48 Aloc(8,11) = Aloc(2,5); Aloc(8,12) = Aloc(2,6); Aloc(9,9) = Aloc(3,3);
49 Aloc(9,10) = Aloc(3,4); Aloc(9,11) = Aloc(3,5); Aloc(9,12) = Aloc(3,6);
50 Aloc(10,10) = Aloc(4,4); Aloc(10,11) = Aloc(4,5); Aloc(10,12) = Aloc(4,6);
51 Aloc(11,11) = Aloc(5,5); Aloc(11,12) = Aloc(5,6); Aloc(12,12) = Aloc(6,6);
52 % Interaction of pressure shape functon (constant ≡ 1) with velocity:
evaluation of
53 % local bilinear form b K .
54 % First for x 1 -components, then for
55 Aloc(1,13) = sum(QuadRule.w.*grad_N(:,1))*det_BK;
56 Aloc(2,13) = sum(QuadRule.w.*grad_N(:,3))*det_BK;
57 Aloc(3,13) = sum(QuadRule.w.*grad_N(:,5))*det_BK;
58 Aloc(4,13) = sum(QuadRule.w.*grad_N(:,7))*det_BK;
59 Aloc(5,13) = sum(QuadRule.w.*grad_N(:,9))*det_BK;
60
61
Aloc(6,13) = sum(QuadRule.w.*grad_N(:,11))*det_BK;
% Next for x 2 -components of velocity
62 Aloc(7,13) = sum(QuadRule.w.*grad_N(:,2))*det_BK;
63 Aloc(8,13) = sum(QuadRule.w.*grad_N(:,4))*det_BK;
64 Aloc(9,13) = sum(QuadRule.w.*grad_N(:,6))*det_BK;
65 Aloc(10,13) = sum(QuadRule.w.*grad_N(:,8))*det_BK;
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
9.3
p. 1041
Raising the polynomial degree has cured the instability!
Oberservation: algebraic convergence ‖u−u N ‖ 1 = O(h M ),
‖u−u N ‖ 0 = O(h 2 M ),
‖p−p N ‖ 0 = O(h M ).
The pair U N = S2,0 0 (M), Q N = S0 −1 (M) is the first combination of finite element spaces that we
find to provide a stable Galerkin discretization of the variational Stokes problem (9.2.33)↔(9.2.26).
Recall the concept of stability/well-posedness for linear problems, see Sect. 2.3.2, “stability estimate”
of Thm. 3.1.5,
∥ solution
∥ ∥ ≤ C ∥ ∥ right hand side
∥ ∥ for all data,
✸
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.3
p. 104
66 Aloc(11,13) = sum(QuadRule.w.*grad_N(:,10))*det_BK;
67 Aloc(12,13) = sum(QuadRule.w.*grad_N(:,12))*det_BK;
68 % Entry corresponding to zero mean multiplier
69 Aloc(13,14) = det_BK;
70 % Fill in lower triangular part
71 tri = t r i u (Aloc); Aloc = tri+ t r i l (tri’,-1);
72 r e t u r n
Numerical
Methods
for PDEs
where relevant norms have to be considered.
For the Stokes problem: relevant norms = norms of Sobolev spaces fitting (9.2.33)
Numerica
Methods
for PDEs
For velocityv : use “energy norm” ‖·‖ a := a(·,·) 1/2 ∼ ‖·‖ H 1 (Ω)
, cf. Def. 2.1.27
For pressurep : use‖·‖ L 2 (Ω) .
Convergence rate of discretization error for P2P0 elements
Convergence rate of discretization error for P2P0 elements
10 −1 p = 1.97
Discretization error [log]
10 −2
10 −3
10 −4
10 −5
10 0
10 −1
Mesh width [log]
V, L2 Norm
V, H1 Semi−norm
P, L2 Norm
p = 1.00
10 −2
Fig. 340
Discretization error [log]
10 −1 p = 2.06
10 −2
10 −3
10 −4
10 −5
10 0
10 −1
Mesh width [log]
V, L2 Norm
V, H1 Semi−norm
P, L2 Norm
p = 1.05
10 −2
Fig. 341
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Definition 9.3.38 (Stable finite element pair).
A pair of finite element spacesU N ⊂ H 1 0 (Ω),Q N ⊂ L 2 ∗(Ω) is a stable finite element pair, if the
solution(v N ,p N ) of the discrete saddle point problem (9.3.15) satisfies
|l(w N )| ≤ C l ‖w N ‖ a ∀w N =⇒ ∃C > 0: ‖v N ‖ a +‖p N ‖ L 2 (Ω) ≤ CC l ,
where C > 0 may depend only on Ω, the coefficient µ, and the shape regularity measure (→
Def. 5.3.26) ofM.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Structured meshes
Randomly perturbed meshes
9.3
p. 1042
9.3
p. 104

We have already encountered an estimate like
|l(w N )| ≤ C l ‖w N ‖ a ∀w N ∈ U N , (9.3.40)
when finding that the existence of solutions of quadratic minimization problems (→ Def. 2.1.21)
hinges on the continuity of the involved linear form, see (2.2.1).
Numerical
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for PDEs
✬
Theorem 9.3.53 (inf-sup condition).
The finite element spacesU N ⊂ H 1 0 (Ω),Q N ⊂ L 2 ∗(Ω) provide a stable finite element pair (→
Def. 9.3.38) for the Stokes problem (9.2.33)/ (9.2.26) if there is a constantβ > 0 depending only
onΩand the shape regularity measure (→ Def. 5.3.26) ofMsuch that
|b(w
sup N ,q N )|
≥ β‖q
w N ∈U N
‖w N ‖ N ‖ L 2 (Ω) ∀q N ∈ Q N . (9.3.54)
a
✩
Numerica
Methods
for PDEs
Let us embark on a mathematical analysis of the stability issue, which turns out to be much simpler
than expected.
Remark 9.3.43 (Stable velocity solution).
Consider (9.2.33)↔(9.2.26), and Galerkin discretization (9.3.15), define the subspace
N(b N ) := {w N ∈ U N : b(w N ,q N ) = 0 ∀q N ∈ Q N } ⊂ U N . (9.3.44)
From 2nd equaton ➣ for any solution(v N ,p N ) of (9.3.15): v N ∈ N(b N )
Test the first equation of (9.3.15) withw N ∈ N(b N )
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
✫
The estimate (9.3.54) is kown as
inf-sup condition
LBB (Ladyzhenskaya-Babuska-Brezzi) condition
It is the linchpin of the numerical analysis of finite element methods for the Stokes problem, see [13].
△
✪
R. Hiptm
C. Schwa
H.
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V.
Gradinar
A. Chern
DRAFT
a(v N ,w N ) = l(w N )
w N :=v N
=⇒ ‖v N ‖ 2 a ≤ l(v N) (9.3.40)
≤ C l ‖v N ‖ a .
perfect stability of any velocity Galerkin solution
9.3
p. 1045
Numerical
Methods
for PDEs
9.3.3 Convergence
9.3
p. 104
Numerica
Methods
for PDEs
This explains the observation made in Ex. 9.3.26: reasonable approximation for velocity v despite
pressure instability.
△
Abstract considerations (easier this way!):
Remark 9.3.50 (Stability of pressure solution: inf-sup condition).
Goal: stability of pressure solutionp N ∈ Q N of (9.3.15)
From the first equation of (9.3.15)
‖p N ‖ L 2 (Ω) ≤ C sup
w N ∈U N
l(w N )
‖w N ‖ a
(9.3.51)
best constant in (9.3.40)
a(v N ,w N )+b(w N ,p N ) = l(w N ) ∀w N ∈ U N ,
and the stability of the velocity solution (→ Rem. 9.3.43) we conclude (9.3.51), once we know
b(w N ,p N ) = g(w N ) ∀w N ∈ U N ⇒ ‖p N ‖ L 2 (Ω) ≤ C sup
w N ∈U N
|g(w N )|
‖w N ‖ a
. (9.3.52)
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
A. Chernov
DRAFT
9.3
p. 1046
H ˆ= normed vector space, norm‖·‖ (think of a function space),
c : H ×H ↦→ R bilinear form onH, not necessarily s.p.d. (→ Def. 2.1.25),
l : H ↦→ R linear form onH,
Assumption: c is continuous, cf. Rem. 7.2.2, (3.1.2)
∃C c > 0: |c(u,v)| ≤ C c ‖u‖‖v‖ ∀u,v ∈ H . (9.3.61)
We consider the linear variational problem (→ Rem. 1.4.5)
u ∈ H: c(u,v) = l(v) ∀v ∈ H , (9.3.62)
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.3
p. 104

and its Galerkin discretization, based on finite-dimensional subspaceH N ⊂ H, cf. (3.1.4),
Assumption:
Trick!
”Trick”
stability
u N ∈ H N : c(u N ,v N ) = l(v N ) ∀v N ∈ H N . (9.3.63)
u N solves (9.3.63)
=⇒ ∃C s > 0: ‖u N ‖ ≤ sup
w N ∈H N
|l(w N )|
‖w N ‖ . (9.3.64)
For any v N ∈ H N the differenceu N −v N (u N solution of (9.3.63)) solves
Triangle inequality
c(u N −v N ,w N ) = l(w N )−c(v N ,w N ) ∀w N ∈ H N .
(9.3.64)
|l(w
=⇒ ‖u N −v N ‖ ≤ C s sup N )−c(v N ,w N )|
w N ∈H N
‖w N ‖
(9.3.62) |c(u−v
= C s sup N ,w N )|
w N ∈H N
‖w N ‖
(9.3.61)
≤ C c C s ‖u−v N ‖ .
‖u−u N ‖ ≤ ‖u−v N ‖+‖u N −v N ‖ (9.3.65)
≤ (1+C c C s )‖u−v N ‖ ∀v N ∈ H N .
(9.3.65)
‖u−u N ‖ ≤ (1+C c C s ) inf
v N ∈H N
‖u−v N ‖ . (9.3.66)
(9.3.66) is a fundamental insight into the properties of Galerkin discretizations, cf. Thm. 5.1.10 that
was confined to s.p.d. bilinear forms:
For the Galerkin discretization of linear variational problems:
Stability ⇒ Quasi-optimality(∗)
Terminology: Quasi-optimality of Galerkin solutions: withC > 0 independent of data and discretization
parameters
‖u−u N ‖
} {{ }
↑
(norm of) discretization error
≤ C inf ‖u−v N ‖ , (9.3.67)
v N ∈H N
} {{ }
↑
best approximation error
Numerical
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R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
9.3
p. 1049
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
H := H 1 0 (Ω)×L2 (Ω) (combination of two function spaces!)
Role ofcplayed by
c
(( (
v w
p)
q))
Right hand side functional ”l (( w
q
)) = l(w)”
Galerkin trial/test space H N := U N ×Q N .
,
:= a(v,w)+b(w,p)+b(v,q) . (9.3.69)
Then, along the lines of the above abstract considerations, one can show the following a priori error
estimate:
✬
Theorem 9.3.70 (Convergence of stable FE for Stokes problem).
If U N , Q N is a stable finite element pair (→ Def. 9.3.38) for the Stokes problem (9.2.33), then
the corresponding finite element Galerkin solution(v N ,p N ) satisfies
(
)
‖v−v N ‖ H 1 (Ω) +‖p−p N‖ L 2 (Ω) ≤ C inf ‖v−w N ‖
w N ∈U H 1 (Ω) + inf ‖p−q N ‖
N q N ∈Q L 2 (Ω) ,
N
with a constantC > 0 that depends only onΩ,µ, and the shape regularity of the finite element
mesh.
✫
Note: the a priori error bound of Thm. 9.3.70 involves the sum of the best approximation errors for
both the velocity and pressure trial/test spaces.
Example 9.3.72 (Convergence of P2-P0 scheme for Stokes equation).
✩
Numerica
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for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
9.3
p. 105
Numerica
Methods
for PDEs
✪
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Interpretation of error curves observed in Ex. 9.3.29:
Application of abstract theory to finite element discretization of Stokes problem (9.2.33):
9.3
p. 1050
Smooth solutions for bothvandp:
9.3
p. 105

Sect. 5.3.5
➣
inf
w N ∈S 0 2,0 (M) ‖v−w N ‖ H 1 (Ω) ≤ Ch2 M ‖v‖ H 3 (Ω) (Thm. 5.3.42),
inf
q N ∈S −1
0 (M) ‖p−q N ‖ L 2 (Ω) ≤ Ch M‖p‖ H 1 (Ω) ,
with constants depending only on the shape regularity measure (→ Def. 5.3.26) of triangulationM.
Numerical
Methods
for PDEs
➣ Excellent approximation of eithervorpalone may not lead to an accurate solution.
Recall similar situation for method of lines, where errors of spatial discretization and timestepping add
up, see “Meta-Thms.” 6.1.63, 6.2.42.
Numerica
Methods
for PDEs
The observed O(h) algebraic convergence in the H 1 (Ω)-norm (for v N ) and L 2 (Ω)-norm (for p N )
results, because
the larger best aproximation error ofS −1
0 (M) dominates. ✸
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
For the sake of efficiency
balance
inf ‖v−w N ‖
w N ∈U H 1 (Ω) and+ inf ‖p−q N ‖
N q N ∈Q L 2 (Ω) N
Too ambitious: we have no chance of guessing the best approximation errors a priori.
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Thus we settle for a more modest asymptotic balance condition, cf. the considerations in Sect. 6.1.5.
△
9.4 The Taylor-Hood element
A: The ultimate cure for instability
chose trial/test space for velocity large enough→very large (to play safe).
9.4
p. 1053
Numerical
Methods
for PDEs
Guideline for viable and efficient choice of Galerkin finite element spaces for Stokes problem:
➊ The pair(U N ,Q N ) of finite element spaces must be stable (→ Def. 9.3.38)
➋ The velocity finite element spaceU N should provide the same rate of algebraic convergence
of theH 1 (Ω)-best approximation error w.r.t. h M → 0 as the pressure space inL 2 (Ω).
➌ The velocity finite element spaceU N should guarantee ➊ and ➋ with as few degrees of
freedom as possible.
9.4
p. 105
Numerica
Methods
for PDEs
B: Well, but a large finite element space leads to a large system of linear equations, that is, high
computational cost.
A: Never mind, a large space buys good accuracy, which is what we also want!
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
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DRAFT
Note that the stable finite element pair (S2,0 0 (M),S−1 0 (M)) does not meet the efficiency criterion,
because the velocity space offers a better asymptoic rate of convergence than the pressure space,
see Ex. 9.3.72.
R. Hiptm
C. Schwa
H.
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V.
Gradinar
A. Chern
DRAFT
Remark 9.4.2 (Efficient finite element discretization of Stokes problem).
There is a stable, perfectly balanced pair of spaces:
Thm. 9.3.70, cf. discussion in Ex. 9.3.72: the finite element discretization error for a stable finite
element pair(U N ,Q N ) (→ Def. 9.3.38) for the Stokes problem (9.2.33) is the sum of approximation
errors for the velocityvinU N and the pressurepinQ N .
9.4
p. 1054
9.4
p. 105

Taylor-Hood finite element method for Stokes problem:
M: triangular/tetrahedral or rectangular/hexahedral mesh of Ω, may be hybrid, see
Sect. 3.3.1
Velocity space:
Pressure space:
U N := S 0 2,0 (M) ⊂ H1 0 (Ω)
Q N := S1 0 (M) (continuous pressure)
Numerical
Methods
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Monitored: Error norms‖u−u N ‖ H 1 (Ω) ,
‖u−u N ‖ L 2 (Ω) ,‖p−p N‖ L 2 (Ω)
Observation: algebraic convergence
‖u−u N ‖ H 1 (Ω) = O(h2 M ) ,
‖u−u N ‖ L 2 (Ω) = O(h3 M ) ,
Discretization error [log]
10 −2
10 −3
10 −4
10 −5
Convergence rate of discretization error for Taylor−Hood elements
V, L2 Norm
V, H1 Semi−norm
P, L2 Norm
p = 2.15
Numerica
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10 −1 p = 3.11
R. Hiptmair
C. Schwab,
H.
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V.
Gradinaru
A. Chernov
DRAFT
‖p−p N ‖ L 2 (Ω) = O(h2 M ) .
10 −6
10 −7
10 0
10 −1
Mesh width [log]
10 −2
Fig. 344
✸
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Fig. 342
sites of velocity local shape functions
Fig. 343
sites of pressure local shape functions
Balanced approximation properties of finite element spaces (for sufficiently smooth velocity and pressure
solution):
9.4
p. 1057
Numerical
Methods
for PDEs
9.4
p. 105
Numerica
Methods
for PDEs
velocity:
pressure:
inf ‖v−w N ‖
w N ∈U H 1 (Ω) ≤ Ch2 M ‖v‖ H 3 (Ω) by Thm. 5.3.42,
N
inf ‖p−q N ‖
q N ∈S0 −1 L 2 (M) (Ω) ≤ Ch2 M ‖p‖ H 2 (Ω) by Thm. 5.3.27.
✬
Theorem 9.4.3 (Stability and convergence of Taylor-Hood finite element). → [28]
The Taylor-Hood element provides a stable finite element pair for the Stokes problem (→
Def. 9.3.38) and for sufficiently smooth velocity and pressure solution
( )
‖v−v N ‖ H 1 (Ω) +‖p−p N‖ L 2 (Ω) ≤ Ch2 M ‖v‖ H 3 (Ω) +‖p‖ H 2 (Ω) ,
with a constantC > 0 that depends only onΩ,µ, and the shape regularity of the finite element
mesh.
✫
Example 9.4.4 (Convergence of Taylor-Hood method for Stokes problem).
✩
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
✪
DRAFT
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Stokes problem (9.2.41) as in Ex. 9.3.29
perturbed triangular meshes as in Ex. 9.3.29
Taylor-Hood finite element Galerkin discretization
9.4
p. 1058
10.4
p. 106

10
Adaptive Finite Element
Discretization
Numerical
Methods
for PDEs
11 Multilevel iterative solvers
Numerica
Methods
for PDEs
10.1 Concept of adaptivity
11.1 Solving finite element linear systems
10.2 A priori hp-adaptivity
10.2.1 Graded meshes in 1D
10.2.2 Triangular graded meshes
10.2.3 hp-approximation in 1D
R. Hiptmair
C. Schwab,
H.
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V.
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DRAFT
10.4
p. 1061
Numerical
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11.2 Subspace correction
11.2.1 Successive subspace correction algorithm (SSC)
11.2.2 Gauss-Seidel iteration
11.2.3 Hierarchical basis multigrid
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
11.5
p. 106
Numerica
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
11.5
p. 1062
12.5
p. 106

12 Sparse Grids Galerkin Methods
12.1 The curse of dimension
12.2 Hierarchical basis
12.3 Sparse grids
12.4 Approximation on sparse grids
12.5 Sparse grids algorithms
Index
H 1 -semi-norm, 147
L 2 -norm, 144
2-regularity
of Dirichlet problem, 580
convergence
exponential, 156
a priori estimates, 483
a-orthogonal, 462
affine linear function
in 2D, 287
affine transformation, 378
algebraic convergence, 156
algorithm
numerical, 73
analytic solution, 72
angle condition
for Delaunay mesh, 448
anisotropic diffusion, 750
artificial diffusion, 745
artificial viscosity, 745
assembly, 299
cell oriented, 365
in FEM, 358
linear finite elements, 306
barycentric coordinate representation
of local shape functions, 371
barycentric coordinates, 300
basis
change of, 280
best approximation error, 464
beta function, 375
bilinear form, 51, 66
continuous, 717
positive definite, 186
bilinear transformation, 406
boundary conditions, 26, 69, 234
Dirichlet, 234, 245
homogeneous, 247
Neumann, 237, 246
no slip, 981
radiation, 246
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5
p. 1065
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5
p. 1066
boundary fitting, 417
parabolic, 417
boundary flux
compuation of, 564
boundary layer, 720
boundary value, 167
boundary value problem
elliptic, 248
boundary value problem (BVP), 167
bounding box, 351
Burger’s equation, 849
Burgers equation, 823
calculus of variations, 45
Cauchy problem, 815, 826, 840
for one-dimensional conservation law, 831
for wave equation, 665
cell, 315
cell contributions, 359
central flux, 876
central slope, 952
CFL-condition, 691, 919
characteristic curve, 832
characteristic method, 779
Chebychev nodes, 130
checkerboard mode, 1020
circumcenter, 447
classical solution, 61, 257
coefficient vector, 278
collocation, 124, 125
spline, 133
compatibility condition, 258
convective terms, 708
convergence, 140, 468
algebraic , 156
asymptotic, 150
convex function, 189
corner singular function, 539
Courant-Friedrichs-Levy condition (CFL), 691
Crank-Nicolson method, 621
creeping flow, 981
curve, 26, 27
length, 28
parametrization, 27
cut-off function, 571
d’Alembert solution, 666
Delaunay mesh
angle condition, 448
delta distribution, 222
dielectric tensor, 178
difference quotient, 137
differential operator, 129, 167
diffusion tensor, 750
diffusive flux, 829
diffusive terms, 708
Dirac delta function, 222
Dirichlet boundary conditions, 234, 245
for linear FE, 390
Dirichlet data, 215, 262
Dirichlet problem, 234
variational formulation, 214
discrete maximum principle, 590
discrete model, 33
compatibility conditions
forH 1 (Ω), 208
complete space, 459
composite midpoint rule, 114
composite trapezoidal rule, 114
computational domain, 239
computational effort, 527
condition number, 99
conditionally stable, 654
configuration space, 26
for incompressible fluid, 979
for taut membrane, 174
congruent matrices, 281
conservation
of energy, 675
conservation form, 869
conservation law, 827
differential form, 830
integral form, 829
one-dimensional, 830
scalar, 830
conservation of energy, 240
consistency
of a variational problem, 751
consistency error, 942
continuity
of a linear form, 205
of linear functional, 557
control volume, 443, 827
convection-diffusion equation, 708
convective cooling, 246
discrete variational problem, 274, 278
discretization, 73, 272
discretization error, 140, 464
discretization parameter, 468
displacement, 69
displacement function, 69
dissipation, 614
in fluid, 984
DistMesh, 351
divergence
of a vectorfield, 228
domain, 167
computational, 239
spatial, 71, 172
domain of dependence, 668
domain of influence, 668
dual mesh, 446
dual problem, 561
duality estimate, 561
dynamic viscosity, 980
eddy
in a fluid, 983
edge, 315
elastic energy, 32
mass-spring model, 33
elastic string, 24
electric field, 177
electric scalar potential, 178
electromagnetic field energy, 178
electrostatic field energy, 178
electrostatics, 177
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
12.5
p. 106
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
12.5
p. 106

element, 315
element load vector, 361
element stiffness matrix, 361
elliptic
linear scalar second order PDE, 244
elliptic boundary value problem, 248
energy
conservation, 240
of electrostatic field, 178
energy conservation
for wave equation, 671
energy norm, 187
entropy, 982
equidistant mesh, 107, 137, 323
equilibrium length
of spring, 32
equilibrium principle, 34
equivalence
of norms, 504
Euler equations, 823
Euler method, 619
evolution operator
fully discrete, 913
semi-discrete, 911
evolution problem, 600
semi-discrete, 617
spatial variational formulation, 608
expansion shock, 849
explicit Euler method, 619
exponential convergence, 156
face, 315
General entropy solution for 1D scalar Riemann problem,
858
generic constants, 525
global shape functions, 322
Godunov numerical flux, 896
gradient, 174
of a function, 175
gravitational force, 30
Green’s first formula, 229
grid
1D, 107, 137
grid function, 141
h-refinement, 533
hanging node, 318
hat function, 211, 290
heat capacity, 603
heat conductivity, 242
heat equation, 603
heat flux, 239, 242
computation of, 564
convective, 707
diffusive, 707
heat source, 241
Hessian, 495
Heun method, 914
Hilbert space, 459
homogeneous boundary conditions, 247
Hooke’s law, 31
hyperbolic evolution problem, 662
discrete case, 672
field energy
electromagnetic, 178
finite difference methods, 430
finite differences
1D, 136
in 2D, 430
finite elements
parametric, 410
finite volume methods, 442
flow field, 701
flow map, 704
flux function, 827, 832
force density, 175
Fourier’s law, 241
if fluid, 707
Frobenius norm, 499
functional, 555
linear, 557
fundamental lemma of calculus of variations, 232
Galerkin discretization, 76, 273
Galerkin matrix, 359
Galerkin orthogonality, 462
Galerkin solution, 78
quasi-optimality, 464
Galerkin test space, 78
Galerkin trial space, 78
Gamma function, 375
Gauss’ theorem, 229, 243, 706, 838
Gauss-Legendre quadrature, 385
Gauss-Lobatto quadrature, 385
implicit Euler method, 619
implicit midpoint rule, 621
increments
Runge-Kutta, 622
index mapping matrix, 363
inf-sup condition, 1037
inflow, 827
inflow boundary, 722
initial conditions, 600
initial value problem
stiff, 631
initial-boundary value problems (IBVP), 601
parabolic, 605
integrated Legendre polynomials, 88
integration by parts
in 1D, 58
multidimensional, 229
intermediate state, 892
interpolant
piecewise linear, 451
interpolation error, 494
interpolation error estimates
anisotropic, 510
in 1D, 485
interpolation nodes, 330
inviscid, 821
kinetic energy, 671
L(π)-stability, 645
L-shaped domain, 475
L-stability, 645
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5
p. 1069
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5
p. 1070
Lagrange functional
for zero mean constraint, 1003
Lagrange multiplier, 993
Lagrangian finite elements, 329, 367
on quadrilaterals, 404
Lagrangian functional, 994
Lagrangian method, 779
for advecti0n, 771
Laplace equation, 233
Lax entropy condition, 857
Lax-Friedrichs flux, 882
LBB condition, 1037
leapfrog, 681
Legendre polynomials, 89
integrated, 88
LehrFEM, 345
lexikographic ordering, 431
lifting theorem, 537, 544
linear boundary fitting, 551
linear form, 51
continuity, 205
linear function
in 2D, 287
linear functional, 51
linear interpolation
in 1D, 485
in 2D, 493, 494
linear variational problem, 65
Linearity, 248
linearization
of variational problems, 418
simplicial, 318
triangular, 316
mesh data structure, 352
mesh file format, 346
triangular mesh, 347
mesh generation, 351
mesh generator, 346
mesh width, 471
method of characteristics, 721
method of lines, 617
midpoint rule
composite, 114
minmod, 966
mixed boundary conditions, 247
Mixed Neumann–Dirichlet problem, 238
model
continuous, 73
discrete, 33, 73
monomial basis, 87
monotonicity preserving linear interpolation, 962
multi-index notation, 319
multiplicative trace inequality, 266
MUSCL scheme, 971
NETGEN, 351
Neumann boundary conditions, 237, 246
Neumann data
admissibilty conditions, 264
Neumann problem, 257
compatibility condition, 258
variational form, 257
Newton update, 426
load vector, 278, 359
local linearization, 423
local operations, 366
local shape function, 325
barycentric representation, 402
local shape functions
quadratic, 331
mass lumping, 683
mass-spring model, 31
elastic energy, 33
material coordinate, 29
material derivative, 801
material tensor, 215
mathematical modelling, 22
maximu principle
discrete, 590
maximum principle, 586, 713
Maxwell’s equations
static case, 178
mean value formula, 495
membrane, 170
potential energy, 174, 175
membrane problem
variational formulation, 214
mesh, 315
1D, 107, 137
data structures, 352
equidistant, 107, 137
node, 284
non-conforming, 318
quadrilateral, 316
Newton’s method, 422
in function space, 422
termination, 427
Newton’s second law of motion, 659
Newton-Cotes formula, 385
Newton-Galerkin iteration, 426
nodal basis, 290
nodal interpolation operators, 521
nodal value, 290
node, 284
1D, 107, 137
quadrature, 94
norm, 142
on function space, 143
numerical domain of dependence, 917
numerical flux, 444, 869
numerical flux function, 444
numerical quadrature, 93
nodex, 376
weights, 376
offset function, 52, 387
for linear FE, 388
order of quadrature rule, 377
outflow, 827
outflow boundary, 722
output functional, 555
p-refinement, 533
parametric finite elements, 410, 411
parametric quadrature rule, 376
parametrization
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
12.5
p. 107
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
12.5
p. 107

of curve, 27
particle method, 779
PDE
lLinear scalar second order elliptic, 244
perpendicular bisector, 447
phase space, 826
Phythagoras’ theorem, 463
piecewise linear interpolant, 451
piecewise linear reconstruction, 948
piecewise quadratic interpolation, 522
Poincaré-Friedrichs inequality, 203, 264
point force, 57
Poisson equation, 233, 469
Poisson matrix, 434
polar coordinates, 223
polynomials
degree, 319
multivariate, 319
univariate, 85
positive definite
bilinear form, 186
uniformly, 180
postprocessing, 141
potential energy, 671
of taut membrane, 174
pressure, 998
pressure Poissson equation, 1006
problem parameters
for elastic string, 30
problem size, 527
procedural form
reaction term
in 2nd-order BVP, 282
reference elements, 411
regular refinement, 466
reversibility, 675
Reynolds number, 980
Riemann problem, 846
Riesz representation theorem, 459
right hand side vector, 359
Robin boundary conditions, 246
rubber band, 24
Runge-Kutta
increments, 622
Runge-Kutta method, 622
Runge-Kutta methods
stability function, 929
saddle point problem, 994
linear, 996, 1012
variational, 995
SDIRK timestepping, 645
semi-discrete evolution problem, 617
semi-norm, 147
sensitivity
of a problem, 218
shape functions
global, 322
shape regularity measure, 502
shock, 847
physical, 857
subsonic, 895
supersonic, 895
of functions, 271
product rule, 612
in higher dimensions, 228
production term, 827
pullback, 396
quadratic functional, 183
quadratic local shape functions, 331
quadratic minimization problem, 64
quadratic minimization problems, 182
quadrature formula, 94
quadrature node, 94
quadrature nodes, 376
quadrature rule, 376
on triangle, 381
order, 377
parametric, 376
quadrature rules
Gauss-Legendre, 385
Gauss-Lobatto, 385
quadrature weight, 94
quadrature weights, 376
quadrilateral mesh, 316
quasi-optimality, 464, 1040
Radau timestepping, 645
radiation boundary conditions, 246, 255
rarefaction
subsonic, 895
supersonic, 895
transonic, 895
rarefaction wave/fan, 855
shock speed, 847
similarity solution, 853
simplicial mesh, 318
singular perturbation, 724
slope limiter, 967
slope limiting, 961
Sobolev norms, 506
Sobolev semi-norms, 508
Sobolev space H 1 (Ω), 201
Sobolev space H0 1 (Ω), 199
Sobolev spaces, 193, 506
solution
analytic, 72
approximate, 73
source term, 167
space-time-cylinder, 600
sparsity pattern, 297
spatial domain, 172
spectrum, 923
spline
cubic, 134
spline collocation, 133
spring constant, 32
Störmer scheme, 680
stability, 1033
of linear variational problem, 218
stability domain, 931
stability function
of explicit Runge-Kutta methods, 929
of RK-SSM, 645
Stable finite element pair, 1034
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5
p. 1073
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5
p. 1074
stiff IVP, 631
stiffness
of spring, 32
stiffness matrix, 278, 359
sparsity, 325
Stokes problem
variational form, 999
Strang splitting, 771
streamline, 702
streamline diffusion, 745
strong form, 61
subsonic rarefaction, 895
subsonic shock, 895
supersonic rarefaction, 895
supersonic shock, 895
supremum norm, 143
symbol
of a difference operator, 923
T-matrix, 363
Taylor expansion, 47
Taylor-Hood finite element, 1046
tensor product polynomials, 321
tensor-product grid, 431
tent function, 108, 290
test function, 52
test space, 52
TETGEN, 351
thermodynamics
2nd law, 984
trace theorem, 266
trajectory, 702
vertex, 315
virtual work principle, 50
von Neumann stability analysis, 931
Voronoi cell, 446
Voronoi dual mesh, 446
vortex, 983
wave equation, 662
weak form, 61
weak solution, 840
weight
quadrature, 94
well-posedness, 1033
width
of a mesh, 471
transformation of functions, 396
transformation techniques, 410
translation-invariant, 916
transonic rarefaction, 895
transport equation, 767
transsonic rarefaction fan, 888
trapezoidal rule
composite, 113, 114
trial space, 52, 125
triangle inequality, 142
triangular mesh, 316
triangular mesh: file format, 347
triangulation, 315
two-point boundary value problem, 61
two-step method, 680
uniformly positive, 243
upwind quadrature, 738, 741
upwinding, 734
variational crime, 545
variational equation
linear, 65
non-linear, 49
variational formulation
spatial, 608
variational problem
discrete, 274, 278
linear, 65
non-linear, 419
perturbed, 546
vector Laplacian, 1005
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
12.5
p. 107
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
12.5
p. 107

Examples and Remarks
L 2 interpolation error, 583
L 2 -convergence of FE solutions, 580
L 2 -estimates on non-convex domain, 587
L ∞ interpolation error estimate in 1D, 525
H 1 0(div0,Ω)-conforming finite elements, 1002
|·| H1 (Ω)-semi-norm, 205
(Bi)-linear Lagrangian finite elements on hybrid meshes,
346
[Membrane with free boundary values, 238
ode45 for discrete parabolic evolution, 634
“PDEs” for univariate functions, 25
“Physics based” discretization, 76
1D convection-diffusion boundary value problem, 726
Adequacy of 2nd-order timestepping, 981
Admissible Dirichlet data, 265
Admissible Neumann data, 267
Affine transformation of triangles, 380
Approximate computation of norms, 152
Approximate Dirichlet boundary conditions, 392
Approximate sub-steps for Strang splitting time, 783
Boundary conditions inH 1 0(Ω), 203
Boundary value problems, 170
Boundary values for conservation laws, 839
Breakdown of characteristic solution formula, 843
Causes for non-smoothness of solutions of elliptic BVPs,
549
Central flux for Burgers equation, 884
Central flux for linear advection, 887
Characteristics for advection, 841
Checkerboard instability for quadrilateral P1-P0 pair, 1027
Choice of basis for polynomial spectral Galerkin methods,
89
Choice of timestepping for m.o.l. for transient convectiondiffusion,
773
Coefficients/data in procedural form, 77
Collocation approach on “complicated” domains, 433
Collocation points for polynomial spectral collocation, 133
Collocation: smoothness requirements for coefficients, 131
Compatible boundary and initial data, 611
Computation of heat flux, 570, 577
Conditioning of linear variational problems, 221
Conditioning of spectral Galerkin system matrices, 100
Conrner singular functions, 543
Consequence of monotonicity preservation, 971
Consistency error of Lax-Friedrichs numerical flux, 952
Consistency error of upwind numerical flux, 951
Continuity of interpolation operators, 513
Convective cooling, 249
Convergence of Euler timestepping, 629
Convergence of fully discrete finite volume methods for
Burgers equation, 943
Approximation of mean temperature, 563, 566
Arrays storing 2D triangular mesh, 357
Assembly algorithm for linear Lagrangian finite elements,
309
Assembly for linear Lagrangian finite elements on triangular
mes, 367
Assembly for quadratic Lagrangian FEM, 370
Assembly of right hand side vector for linear finite elements,
315
Asymptotic nature of a priori estimates, 530
Barycentric representation of local shape functions, 405
Bases for polynomial spectral collocation, 133
Behavior of generalized eigenvalues ofA⃗µ = λM⃗µ, 640
Benefit of variational formulation of BVPs, 111
Bilinear Lagrangian finite elements, 339
Blow-up for leapfrog timestepping, 694
Boundary conditions andL 2 (Ω), 200
Boundary conditions for 2nd-order parabolic IBVPs, 612
Boundary conditions for linear advection, 827
Boundary conditions for wave equation, 669
Convergence of fully discrete timestepping in one spatial
dimension, 653
Convergence of FV with linear reconstruction, 961
Convergence of Lagrangian FEM forp-refinement, 484
Convergence of linear and quadratic Lagrangian finite elements
inL 2 -norm, 477
Convergence of linear and quadratic Lagrangian finite elements
in energy norm, 473
Convergence of MUSCL scheme, 983
Convergence of P2-P0 scheme for Stokes equation, 1052
Convergence of SUPG and upwind quadrature FEM, 765
Convergence of Taylor-Hood method for Stokes problem,
1058
Crank-Nicolson timestepping, 627
Decoupling of velocity components ?, 999
Delaunay-remeshing in 2D, 796
Derivative of non-linear u ↦→ a(u;·), 428
Difference stencils, 923
Differentiating a functional on a space of curves, 51
Differentiating bilinear forms with time-dependent arguments,
618
Diffusive flux, 837
Dimensionless equations, 31
Dimensions of Lagrangian finite element spaces on triangular
meshes, 534
Discontinuity connecting constant states, 852
Domain of dependence/influence for 1D wave equation,
constant coefficient case, 673
Effect of added diffusion, 753
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5
p. 1077
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5
p. 1078
Efficient finite element discretization of Stokes problem,
1054
Efficient implementation of assembly, 373
Elastic string shape by finite element discretization, 125
Elliptic lifting result in 1D, 541
Energy conservation for leapfrog, 690
Energy norm, 148
Energy norm and H 1 (Ω)-norm, 508
Enforcing zero mean, 1012
Ensuring uniqueness of pressure, 1009
Entropy solution of Burgers equation, 866
Euler equations, 831
Euler timestepping, 625
Euler timestepping for 1st-order form of semi-discrete wave
equation, 681
Evaluation of local shape functions at quadrature points,
404
Explicit Euler in Fourier domain, 935
Exploring convergence experimentally, 163
Extended MATLAB mesh data structure, 358
Extra regularity requirements, 64
Extra smoothness requirement for PDE formulation, 237
Finding continuous replacement functionals, 579
Finite differences for convection-diffusion equation in 1D,
732
First-order semidiscrete hyperbolic evolution problem, 680
Fourier series, 933
Fully discrete evolutions, 921
Gap between interpolation error and best approximation
error, 523
Interpolation nodes for cubic and quartic Lagrangian FE in
2D, 337
L(π)-stable Runge-Kutta single step methods, 651
Lagrangian finite elements on hybrid meshes, 348
Lagrangian method for convection-diffusion in 1D, 799
Lagrangian method for convection-diffusion in 2D, 803
Laplace operator, 236
Lax-Friedrichs flux for Burgers equation, 890
Leapfrog timestepping, 687
Linear FE discretization of 1D convection-diffusion problem,
733
Linear finite element Galerkin discretization for elastic string
model, 119
Linear finite element space for homogeneous Dirichlet problem,
294
Linear reconstruction with central slope, 963
Linear reconstruction with minmod limiter, 975
Linear reconstruction with one-sided slopes, 965
Linear variational problems, 67
Linearity and monotonicity preservation, 972
Local interpolation onto higher degree Lagrangian finite
element spaces, 526
Local quadrature rules on quadrilaterals, 387
Local quadrature rules on triangles, 384
Mass lumping, 689
Material coordinate, 31
Mathematical modelling, 24
Maximum principle for finite difference discretization, 595
Maximum principle for higher order Lagrangian FEM, 603
General asymptotic estimates, 537
Generic constants, 529
Geometric interpretation of CFL condition in 1D, 697
Geometric obstruction to Voronoi dual meshes, 451
Godunov flux for Burgers equation, 906
Good accuracy on “bad” meshes, 520
Graph description of string shape, 71
Grid functions, 144
heat conduction, 251
Heat conduction with radiation boundary conditions, 424
Heuristics behind Lagrangian multipliers, 1003
Higher order timestepping for 1D heat equation, 656
Impact of choice of basis, 282
Impact of linear boundary fitting on FE convergence, 556
Impact of numerical quadrature on finite element discretization
error, 553
Implementation of non-homogeneous Dirichlet b.c. for linear
FE, 393
Implementation of spectral Galerkin discretization for elastic
string problem, 103
Implementation of spectral Galerkin discretization for linear
2nd-order two-point BVP, 97
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Implicit Euler method of lines for transient convection-diffusion,
770
Imposing homogeneous Dirichlet boundary conditions, 345
Improved resolution by limited linear reconstruction, 977
Inefficiency of conditionally stable single step methods,
660
12.5
Initial time, 607
Internal layers, 754
p. 107
Maximum principle for linear FE for 2nd-order elliptic BVPs,
602
Mesh file format for MATLAB code, 352
Minimal regularity of membrane displacement, 179
Mixed boundary conditions, 250
Naive finite difference discretization of Stokes system, 1017
Naive finite difference scheme, 872
Non-continuity of boundary flux functional, 574
Non-differentiable function inH 1 0(]0,1[), 213
Non-existence of solutions of positive definite quadratic
minimization problem, 194
Non-homogeneous Dirichlet boundary conditions in LehrFEM,
395
Non-linear variational equation, 52
Non-polynomial “bilinear” local shape functions, 413
Non-smooth external forcing, 58
Norms on grid function spaces, 151
Numerical quadrature in LehrFEM, 386
Offset function for finite element Galerkin discretization,
118
Offset functions and Galerkin discretizatio, 82
offset functions for linear Lagrangian FE, 391
One-sided difference approximation of convective terms,
737
Ordered basis of test space, 84
Output functionals, 561
P1-P0 quadrilateral finite elements for Stokes problem,
1030
P2-P0 finite element scheme for the Stokes problem, 1035
Parametrization of a curve, 29
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
12.5
p. 108

Piecewise gradient, 291
Piecewise linear functions (not) inH 1 , 211
Piecewise quadratic interpolation, 527
Point charge, 225
Point particle method for pure advection, 787
Pressure Poisson equation, 1016
Properties of weak solutions, 848
Pullback of functions, 399
Quadratic functionals with positive definite bilinear form in
2D, 191
Quadratic minimization problem, 66
Quadratic minimization problems on Hilbert spaces, 201
Quadratic tensor product Lagrangian finite elements, 343
Quasi-locality of solution of scalar elliptic boundary value
problem, 254
Radiative cooling, 250
Relationship between discrete minimization problem and
discrete variational problem, 80
Scalar elliptic boundary value problem in one space dimension,
253
Scaling of entries of element matrix for−∆, 306
Semi-Lagrangian method for convection-diffusion in 1D,
814
Semi-Lagrangian method for convection-diffusion in 2D,
818
Smoothness of solution of scalar elliptic boundary value
problem, 253
Smoothness requirements for collocation trial space, 130
Solution formula for sourceless transport, 775
Space of square integrable functions, 198
Unstable P1-P0 finite element pair on triangular mesh,
1025
Upwind flux and expansion shocks, 905
Upwind flux and transsonic rarefaction, 895
Upwind flux for Burgers equation, 893
Upwind quadrature discretization, 749
Vanishing viscosity for Burgers equation, 857
Variational formulation for convection-diffusion BVP, 723
Variational formulation for heat conduction with Dirichlet
boundary conditions, 257
Variational formulation for pure Neumann problem, 260
Variational formulation: heat conduction with general radiation
boundary conditions, 258
Virtual work principle, 52
Wave equation as first order system in time, 670
Well-posed 2nd-order linear elliptic variational problems,
464
Sparse sitffness matrices, 299
Numerical
Spatial difference operators for linear advection, 930 Methods
Spatial discretization options, 623
for PDEs
Spatial domains, 175
Specification of local quadrature rules, 383
Spectral Galerkin discretization of non-linear variational
problem, 107
Spectral Galerkin discretization with quadrature, 95
Spectrum of elliptic operators, 645
Spectrum of upwind difference operator, 932
Spurious Galerking solution for 2D convection-diffusion BVP,
740
Stability and CFL condition, 940
Stability domains, 939
Stability functions of explicit RK-methods, 937
Stability of pressure solution: inf-sup condition, 1046
Stable velocity solution, 1045
Streamline-diffusion discretization, 760
Streamlines, 728
Supports of global shape functions in 1D, 326
Supports of global shape functions on triangular mesh,
326
Tense string without external forcing, 44
Timestepping for ODEs, 77
Transformation of basis functions, 94
Transformation techniques for bilinear transformations, 419
Triangular mesh: file format, 350
Triangular quadratic Lagrangian finite elements, 332
Uniqueness of solutions of Neumann problem, 261
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5 Imcompressible flow field, 715
p. 1081 L(π)-stability, 651
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
Definitions
H 1 -semi-norm, 150
Affine transformation, 381
Characteristic curve for one-dimensional scalar conservation
law, 840
congruent matrices, 284
Consistent modifications of variational problems, 758
Consistent numerical flux function, 881
Courant-Friedrichs=Levy (CFL-)condition, 927
Cubic spline, 138
element load vector, 364
element stiffness matrix, 364
Energy norm, 190
Higher order Lagrangian finite element spaces, 332
Higher order Sobolev norms, 510
Higher order Sobolev semi-norms, 512
Higher order Sobolev spaces, 510
Quadratic minimization problem, 187
Riemann problem, 854
Runge-Kutta method, 628
Shape regularity measure for simplex, 506
shock, 855
Singularly perturbed problem, 730
Sobolev space H 1 (Ω), 204
Sobolev space H 1 0(Ω), 202
Space L 2 (Ω), 199
sparse matrix, 299
Stable finite element pair, 1044
Support of a function, 113
Supremum norm, 146
Tensor product Langrangian finite element spaces, 343
tensor product polynomials, 324
Uniformly positive definite tensor field, 183
Lax entropy condition, 865
Legendre polynomials, 91
Linear interpolation in 2D, 498
Linear reconstruction, 957
Local shape functions, 329
Material derivative, 808
Mean square norm/L 2 -norm, 147
mesh, 318
mesh width, 475
Minmod reconstruction, 974
Monotone numerical flux function, 912
Monotonicity preserving linear interpolation, 970
multivariate polynomials, 322
norm, 145
Numerical domain of dependence, 925
parametric finite elements, 415
Positive definite bilinear form, 189
pullback, 399
Quadratic functional, 186
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
12.5
p. 108
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
Weak solution of Cauchy problem for conservation law,
848
12.5
p. 1082
12.5
p. 108

MATLAB codes
assemMat_QFE, 371
sparse (MATLAB-function), 371
add_Edge2Elem, 358
add_Edges, 358
init_Mesh, 358
Loading a mesh from file, 353
Numerical
Methods
for PDEs
R. Hiptmair
C. Schwab,
H.
Harbrecht
V.
Gradinaru
A. Chernov
DRAFT
12.5
p. 1085
‖·‖ 0 ˆ= norm onL 2 (Ω), 199
‖·‖ ∞ ˆ= supremum norm of a function/maximum norm of
a vector, 146
‖u‖ Hm (Ω) ˆ=m-th order Sobolev norm, 510
‖u‖ L∞ (Ω) ˆ= supremum norm ofu : Ω ↦→ Rn , 146
‖·‖ L2 (Ω) ˆ=L2 -norm of a function, 147
‖·‖ L2 (Ω) ˆ= norm on L2 (Ω), 199
‖·‖ 0 ˆ=L 2 -norm of a function, 147
V(M), 318
Ω, 170
Ω ˆ= spatial domain or parameter domain, 29
Φ ∗ , 399
|u| Hm (Ω) m-th order Sobolev semi-norm, 512
|·| H1 (Ω) ˆ=H1 -semi-norm of a function, 150
A,B,C,... (matrices), 281
A : B ˆ= componentwise dot product of matrices, 1001
M −T hat= inverse transposed of matrix, 418
a K ˆ= restriction of bilinear form a to cell K, 302
· ˆ= inner product of vectors in R n , 36
χ I ˆ= characteristic function of an interval I ⊂ R, 876
curl ˆ= rotation/curl of a vector field, 992
ü := ∂u
∂t , 666
˙u(t) ˆ= 2 (partial) derivative w.r.t. time, 615
l K restriction of linear form l to cell K, 312
Df
(t) ˆ= material derivative w.r.t. velocity field v, 808
Dv
grad ˆ= gradient of a scalar valued function, 177
ĉ(ξ) ˆ= symbol of a finite difference operator, 931
1 = (1,...,1) T , 937
dS ˆ= integration over a surface, 232
M, 318
∇F(x) := gradF(x) ˆ= nabla nontation for gradient,
178
diam(Ω) ˆ= diameter ofΩ ⊂ R d , 176
nnz, 299
∂Ω ˆ= boundary of domain Ω, 176
ρ K ˆ= shape regularity measure of cellK, 506
ρ M ˆ= shape regularity measure of a meshM, 506
⃗µ,⃗ϕ, ⃗ ξ,... (coefficient vectors), 281
S 0 p,0(M) ˆ= DegreepLagrangian finite element space with
zero Dirichlet boundary conditions., 345
S 0 1,0 (M) ˆ= space of p.w. linear C0 -finite elements, 110
H 1 0(Ω) Sobolev space, 202
H 1 0(Ω) ˆ= componentwise H 1 0(Ω)-vectorfields, 1001
h M ˆ= mesh width of meshM, 475
h M ˆ= meshwidth of a grid, 109
x j−1 /2
:= 1 2 (x j +x j−1 ) ˆ= midpoint of cell in 1D, 876
‖·‖ ˆ= Euclidean norm of a vector ∈ R n , 30
Numerica
Methods
for PDEs
R. Hiptm
C. Schwa
H.
Harbrech
V.
Gradinar
A. Chern
DRAFT
12.5
p. 108
Numerical
Methods
for PDEs
Numerica
Methods
for PDEs
List of Symbols
C0([0,1]) 2 := {v ∈ C 2 ([0,1]):v(0) = v(1) = 0}, 48
C0 ∞ (Ω) ˆ= smooth functions with support inside Ω, 207
C k ([a,b]) ˆ= k-times continuously differentiable functions
on[a,b] ⊂ R, 29
Cpw([a,b]), k 58
D − (x,t) ˆ= maximal analytical domain of dependence of
(x,t), 926
D α u ˆ= multiple partial derivatives, 510
L 2 ∗ (Ω) := {q ∈ L2 (Ω): ∫ Ωqdx = 0}, 1010
M i ˆ=i-th integrated Legendre polynomial, 90
O(f(N)) ˆ= Landau-O forN → ∞, 159
S(z) ˆ= stability function of Runge-Kutta method, 937
n, 248
n ˆ= exterior unit normal vectorfield, 232
H h ˆ= fully discrete evolution operator, 921
L h ˆ= semi-discrete evolution operator doe 1D conservation
law, 919
P p (R) ˆ= space of univariate polynomials of degree ≤ p,
87
P p (R d ), 322
P p (R d ) ˆ= space ofd-variate polynomials, 322
Q p (R d ), 324
V(M) ˆ= set of vertices of a mesh, 287
R. Hiptmair
∆ ˆ= Laplace operator, 236
C. Schwab,
H.
∆ ˆ= vector Laplacian, 1016
Harbrecht
V.
divj ˆ= divergence of a vectorfield, 231
Gradinaru
A. Chernov
E(M), 359
DRAFT
I 1 , 498
Γ in ˆ= inflow boundary for advection BVP, 728
H m (Ω) ˆ=m-th order Sobolev space, 510
H 1 0(div0,Ω) ˆ= componentwise H0(Ω)-vectorfields 1 with
vanishing divergence., 1001
S 0 1
(M), 289
I 1 ˆ= piecewise linear interpolation on finite element mesh,
455
P n ˆ=n-th Legendre polynomial, 91
Sp 0(M) ˆ=H1 (Ω)-conforming Lagrangian FE space, 332
L ∞ (Ω) ˆ= space of (essentially) bounded functions on Ω,
146
12.5
L 2 (Ω) ˆ= space of square-integrable functions onΩ, 199 p. 1086
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