Maximum Norm A Posteriori Error Estimate for a 2D Singularly Perturbed Semilinear Reaction-Diffusion Problem

Abstract

Abstract. A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green's function of the continuous differential operator in the Sobolev W 1,1 and W 2,1 norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate.

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

“…Interestingly, one gets a similar second-order bound (with a logarithmic factor in the case of the Shishkin mesh) for the error of the standard five-point difference scheme applied to problem (1.1) on the corresponding tensorproduct mesh (see, e.g., [5,9]). Tables 1-3 give the maximum nodal errors (odd rows) and the computational convergence rates r in (N −1 ln p N 2 ) r (even rows) for Triangulations A and B obtained from the three tensor-product meshes.…”

“…Their solutions exhibit sharp boundary and interior layers, so locally anisotropic meshes (fine and anisotropic in layer regions and standard outside) are frequently employed in their numerical solution and, furthermore, have been shown to yield reliable numerical approximations in an efficient way (see, e.g., [5,9,14] and references in [10]). Example.…”

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

“…While the analysis and numerical results in this paper are given for the one-dimensional problem (1.1), much of what is here can be generalized to analogues of (1.1) posed in higher dimensions; compare the one-dimensional nonlinear problem discussed in [10] and the extension of this work to the two-dimensional case in [8], where a theoretical analysis and numerical results are presented. The onedimensional analysis is already so complex that the extra notation required to explain it in two dimensions would only obscure the central ideas that we wish to communicate.…”

Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an $L$-shaped domain

“…Furthermore, problems similar to (1.1), (1.2) were considered in [4,10,13,17]. Schatz and Wahlbin [17] derive pointwise error estimates for the Galerkin finite elements on quasiuniform unrefined meshes in polygonal domains.…”

“…Schatz and Wahlbin [17] derive pointwise error estimates for the Galerkin finite elements on quasiuniform unrefined meshes in polygonal domains. Blatov [4] and more recently Kopteva [10] establish second-order convergence, in the discrete maximum norm, on layer-adapted meshes in a smooth domain. Melenk [13] gives an energy-norm exponential-convergence result for hp-finite element methods applied to a more general reaction-diffusion equation posed in a curvilinear polygon.…”