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 - ε²Δu+ bu= f posed on bounded domains Ω⊂ R^k for k≥ 1 , with the Dirichlet boundary condition u= 0 on ∂Ω, where 0 < ε≪ 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 ε) almost first-order error bound for a particular FEM comprising piecewise bilinears on a Shishkin mesh is proved in detail for the case where Ω is the unit square in R². Numerical results illustrate the performance of the method.