A new strategy for finite element method adaptation

CMM-2011 – Computer Methods in Mechanics9–12 May 2011, Warsaw, PolandA new strategy for finite element method adaptationPiotr OpiołaInstitute of Computer ScienceFaculty of Mathematics and Computer ScienceJagiellonian Universityul. prof. Stanisława Łojasiewicza 6, 30-348 Kraków, PolandAbstractThere are several adaptive approaches to finite element method, including the r-method, the p-method and the h-method. The adaptivemethods are used to optimize the mesh and the approximation function by minimizing a monitor function. The monitor function usuallyconsists of an approximation error estimate and some chosen measures of the mesh quality, such as smoothness or orthogonality. Inthe paper, we present a new approach, in which the monitor function consists of an approximation error estimate, the number of meshpoints and a numerical error estimate. The optimization is performed by adding and removing the mesh points, and by changing theapproximation order. The method has been tested on several boundary and initial problems, including the Laplace’s equation, the heatequation and the wave equation.Keywords: adaptive finite element method, partial differential equationsWe define a mesh as a finite set M = {x ⊂ Ω and1. Introductionmax x∈Ω (5) i} i=1...Na neighborhood function N : Ω → P(M). The neighborhoodThe finite element method is a powerful tool for solving partialdifferential equations, mainly appreciated for being general The domain is divided into elements. Two points x, y belong tofunction maps a point x ∈ Ω into a nonempty set of its neighbors.and flexible. However, the process of adjusting the method to the same element iff N (x) = N (y). Having a mesh (M, N ),a particular problem is usually difficult and exacting. To make we define the approximation ũ : Ω → R asthis process more convenient, an automatic mechanism is necessary.Such a mechanism is provided by adaptive finite elementmethods, which have been developed for years [1, 2], still being‖x − x i‖w i(x) = 1 −, ∑x j ∈N (x)‖x − xj‖(6)improved.The novelty of our method is in the intuitive formulation of∑the monitor function, flexibility of the approximation function,xand effective mesh generation algorithm for higher dimensions. ũ(x) = i ∈N (x)wi(x)ui + wi(x)ri(ui + qi · (x − xi))∑xSimilarly to e.g. [3], the monitor function consists of an approximationerror estimate and measures of the mesh quality. The(7)i ∈N (x) wi(x) + ,wi(x)ridifference is that orthogonality and smoothness as the mesh qualitymeasures are replaced by the number of mesh points and ai ∈ R, r i ≥ 0, u i ∈ R, q i ∈ R n ,where ror rnumerical error estimate. Such a formulation helps in controllingthe method efficiency with two meaningful parameters: thei ∈ R, r i ≥ 0, u i ∈ R n , q i ∈ R n×n are the approximationparameters. The parameters, as well as the mesh points M,are sought to minimize the monitor function E, defined asexpected approximation error and the expected number of points.E = 1 2. Problem descriptionα Eε + 1 β EN + E δ, (8)We consider three first-order BVPs∇u(x) = f(x), for x ∈ Ω ⊂ R n , (1)whereE ε = max |ũ(x) − u(x)|,x∈Ω(9)divu(x) = f(x), for x ∈ Ω ⊂ R n , (2) E N = N, (10)∇ vu(x) = f(x), for x ∈ Ω ⊂ R n , (3)1Ewhere ∇ v is the directional derivative in the direction v. We seek δ = maxi≠j ‖x i − x . j‖ (11)a solution u : Ω → R for Eqn. (1) and (3), or u : Ω → R n forEqn. (2), in the C 1 class. The boundary conditions are given byThe parameters α and β have the meanings of the expected errorand the expected number of mesh points respectively. Theu(x) = g(x), for x ∈ Γ ⊂ ∂Ω. (4) parameters provide a convenient way to control the balance betweenIf there is more than one solution for a given problem (1), (2) orprecision and efficiency of the method.(3), we want to obtain any of the solutions. If there is no solution,we solve the optimization problemE ε ≈ max |ũ(xi; M) − ũ(xi; M\{xi})|x i ∈M(12)

CMM-2011 – Computer Methods in MechanicsThe numerical results, including the error estimate, the realerror, the number of iterations and the number of mesh points,are presented in Tables 1, 2 and 3. The method has been testedfor various values of the expected approximation error α and theexpected number of mesh points β.References9–12 May 2011, Warsaw, Poland[1] E. Kita, “Error estimation and adaptive mesh refinement inboundary element method, an overview,” Engineering Analysiswith Boundary Elements, vol. 25, no. 7, 2001.[2] K. Segeth, “A review of some a posteriori error estimates foradaptive finite element methods,” Mathematics and Computersin Simulation, vol. 80, no. 8, pp. 1589–1600, 2010.[3] G. Bono and A. Awruch, “An adaptive mesh strategy for highcompressible flows based on nodal re-allocation,” Journal ofthe Brazilian Society of Mechanical Sciences and Engineering,vol. 30, no. 3, pp. 189–196, 2008.