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

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{lunate}-weighted H¹-norm uniformly in singular perturbation parameter ε{lunate}, up to a logarithmic factor. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε{lunate}-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.