Untitled - Cdm.unimo.it

Contents ix
7. Derivative Matrices
7.1 Derivative matrices in the frequency space 125
7.2 Derivative matrices in the physical space 129
7.3 Boundary conditions in the frequency space 135
7.4 Boundary conditions in the physical space 138
7.5 Derivatives of scaled functions 146
7.6 Numerical solution 148
8. Eigenvalue analysis
8.1 Eigenvalues of first derivative operators 151
8.2 Eigenvalues of higher-order operators 157
8.3 Condition number 165
8.4 Preconditioners for second-order operators 168
8.5 Preconditioners for first-order operators 172
8.6 Convergence of eigenvalues 174
8.7 Multigrid method 179
9. Ordinary differential equations
9.1 General considerations 181
9.2 Approximation of linear equations 183
9.3 The weak formulation 189
9.4 Approximation of problems in the weak form 197
9.5 Approximation in unbounded domains 207
9.6 Other techniques 210
9.7 Boundary layers 211
9.8 Nonlinear equations 214
9.9 Systems of equations 217
9.10 Integral equations 218
10. Time-dependent problems
10.1 The Gronwall inequality 221
10.2 Approximation of the heat equation 222