Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction–diffusion problems

Abstract

The bilinear finite element methods on appropriately graded meshes are considered both for solving singular and semisingular perturbation problems. In each case, the quasi-optimal order error estimates are proved in the ε -weighted H 1 -norm uniformly in singular perturbation parameter ε , up to a logarithmic factor. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε -weighted H 1 -norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.