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