Uniform convergence of discontinuous finite element methods for singularly perturbed reaction-diffusion problems

Abstract

In this paper, we study two families of discontinuous finite element methods for singularly perturbed reaction-diffusion problems, which contain multiple scales spanning the whole problem domain because of the small perturbation parameter. To solve such a multiscale problem, an anisotropic mesh is constructed. This mesh uses finer mesh inside the small scale regions (where the boundary layers are located) than elsewhere (large scale regions). Uniform optimal error estimates in the energy norm are obtained for both families on the anisotropical mesh.