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

new finite element method is presented for a general class of singularly perturbed reaction-diffusion problems -ε ^2 u +bu=f posed on bounded domains ⊂ℝ^k for k≥ 1 , with the Dirichlet boundary condition u=0 on ∂ , 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 is the unit square in ℝ^2 . Numerical results illustrate the performance of the method.