A weighted and balanced FEM for singularly perturbed reaction-diffusion problems

A new finite element method is presented for a general class of singularly perturbed reaction-diffusion problems -epsilon(2) Delta u + bu = f posed on bounded domains Omega subset of R-k for k >= 1, with the Dirichlet boundary condition u = 0 on partial derivative Omega, where 0 < epsilon << 1. The method is shown to be quasioptimal (on arbitrary meshes and for arbitrary conforming finite element spaces) with respect to a weighted norm that is known to be balanced when one has a typical decomposition of the unknown solution into smooth and layer components. A robust (i.e., independent of epsilon) almost first-order error bound for a particular FEM comprising piecewise bilinears on a Shishkin mesh is proved in detail for the case where Omega is the unit square in R-2. Numerical results illustrate the performance of the method.