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

Abstract

A singularly perturbed reaction-diffusion equation is posed in a two-dimensional $L$-shaped domain $\Omega$ subject to a continuous Dirchlet boundary condition. Its solutions are in the Hölder space $C^{2/3}(\bar \Omega )$ and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is further refined in a neighboorhood of the vertex of angle $3\pi /2$. We establish almost second-order convergence of our numerical method in the discrete maximum norm, uniformly in the small diffusion parameter. Numerical results are presented that support our theoretical error estimate.