Numerical Analysis of a 2d Singularly Perturbed Semilinear Reaction-Diffusion Problem

Abstract

A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter ε ² is arbitrarily small, which induces boundary layers. We extend the numerical method and its maximum norm error analysis of the paper [N. Kopteva: Math. Comp. 76 (2007) 631–646], in which a parametrization of the boundary $\partial\Omega$ is assumed to be known, to a more practical case when the domain is defined by an ordered set of boundary points. It is shown that, using layer-adapted meshes, one gets second-order convergence in the discrete maximum norm, uniformly in ε for ε ≤ Ch. Here h > 0 is the maximum side length of mesh elements, while the number of mesh nodes does not exceed Ch ^− 2. Numerical results are presented that support our theoretical error estimates.