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A singularly perturbed reaction-diffusion equation is posed in a two-dimensional L-shaped domain Ω subject to a continuous Dirchlet boundary condition. Its solutions are in the Holder space C 2/3 (Ω) 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 3π/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.

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