Uniform error estimates in the finite element method for a singularly perturbed reaction-diffusion problem

Abstract

Consider the problem −2Δu+u = f with homogeneous Neumann boundary condition in a bounded smooth domain in RN. The whole range 0 < ≤ 1 is treated. The Galerkin finite element method is used on a globally quasi-uniform mesh of size h; the mesh is fixed and independent of . A precise analysis of how the error at each point depends on h and is presented. As an application, first order error estimates in h, which are uniform with respect to , are given.