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

Lu :“ ́ε4u` fpx, y;uq “ 0 for px, yq P Ω, u “ 0 on BΩ, (1.1) posed in a, possibly non-Lipschitz, polygonal domain Ω Ă R. Here 0 ă ε ď 1. We also assume that f is continuous on ΩˆR and satisfies fp ̈; sq P L8pΩq for all s P R, and the one-sided Lipschitz condition fpx, y;uq ́ fpx, y; vq ě Cf ru ́ vs whenever u ě v, with some constant Cf ě 0. Then there is a unique solution u PW 2 ` pΩq ĎW 1 q Ă CpΩ̄q for some ` ą 1 and q ą 2 [6, Lemma 1]. We additionally assume that Cf ` ε ě 1 (as a division by Cf ` ε immediately reduces (1.1) to this case). Residual-type a posteriori error estimates in the maximum norm for this equation and its version in R were recently proved in [6] in the case of shape-regular triangulations. In the present paper, we restrict our consideration to Ω in R and linear finite elements, but our focus now shifts to more challenging anisotropic meshes, i.e. we allow mesh elements to have extremely high aspect ratios. (Figure 1.1 below illustrates permitted types of (semi-)anisotropic and isotropic mesh nodes.) Even for the linear Laplace equation (which one gets from (1.1) if ε “ 1, fu “ 0), we are aware of no such error estimates in the maximum norm on reasonably general triangulations under no mesh aspect ratio condition (e.g., [7, 15, 5, 17] assume shape regularity of mesh elements). But still of more interest are anisotropic meshes in the context of singularly perturbed differential equations (such as (1.1) with ε ! 1). For such equations, the maximum norm is sufficiently strong to capture sharp boundary and interior layers in their solutions, while locally anisotropic meshes (fine and anisotropic in layer regions and standard outside) have been shown to yield reliable numerical approximations in an efficient way (see, e.g., [4, 8, 12, 18] and references therein). But such meshes are typically constructed a priori or by heuristic methods. We discretize (1.1) using standard linear finite elements. Let Sh Ă H 0 pΩqXCpΩ̄q be a piecewise-linear finite element space relative to a triangulation T , and let the computed solution uh P Sh satisfy εx∇uh,∇vhy ` xf I h , vhy “ 0 @ vh P Sh, fhp ̈q :“ fp ̈;uhq. (1.2)