Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations

Abstract

Abstract. We give a counterexample of an anisotropic triangulation on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. Our example is given in the context of a singularly perturbed reaction-diffusion equation, whose exact solution exhibits a sharp boundary layer. Furthermore, we give a theoretical justification of the observed numerical phenomena using a finite-difference representation of the considered finite element methods. Both standard and lumped-mass cases are addressed.

Maximum Norm A Posteriori Error Estimation for Parabolic Problems Using Elliptic Reconstructions

“…Table 6.1 gives the maximum nodal errors, the computational convergence rates, and the two estimators E : maxtE p6.4q , E p6.7q u and E p6.6q with their effectivity indices (computed as the ratio of the estimator to the error). Here E p6.¤q denotes the right-hand side of (6.¤), in which we set Θ Θ I 1 (while, by Lemma 8.1 below, Θ Θ I À h ), and also replace quantities of type mint1, ε ¡1 au by their smoother analogues a ε a , e.g., E p6.6q max zN The mesh is chosen so that the linear interpolation error }u ¡ u I } V ;V À N ¡2 ; however, as ε Ñ 0, the convergence rates deteriorate from 2 to 1 (this phenomenon is noted and explained in [10]). For the considered ranges of ε and N , the aspect ratios of the mesh elements take values between 1 and 3.6e+8.…”

Maximum-Norm A Posteriori Error Estimates for Singularly Perturbed Reaction-Diffusion Problems on Anisotropic Meshes

“…While the maximum mesh aspect ratios vary between 2 and 3.35e+7, the effectivity indices do not exceed 85 in all our experiments. Considering these variations (and the observation made by [10]), the estimator appears to perform reasonably well.…”

“…The mesh is chosen so that the linear interpolation error }u´u I } 8 ;8 À N´2; however, as ε Ñ 0, the convergence rates deteriorate from 2 to 1 (this phenomenon is noted and explained in [10]). For the considered ranges of ε and N , the aspect ratios of the mesh elements take values between 1 and 3.6e+8.…”

How accurate are finite elements on anisotropic triangulations in the maximum norm?

“…We start with the singularly perturbed reaction-diffusion equation considered in [5] − ε 2 u + u = 0, (2.1) Table 2.3: Laplace equation with the exact solution u = e −x/ε in the domain (0, 2ε)×(0, 1), lumped-mass linear elements on the uniform mesh, M = 1 4 N : maximum nodal errors and computational rates p in N −p posed in a rectangular domain Ω, subject to Dirichlet boundary conditions, with the exact solution u = e −x/ε . The maximum nodal errors of the linear finite element method applied to this equation are presented in Tables 2.1 and 2.2.…”