Uniform Superconvergence of a Finite Element Method with Edge Stabilization for Convection-Diffusion Problems

Abstract

In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ${||πu−u^h||}_E$ where $πu$ is some interpolant of the solution $u$ and $u^h$ the discrete solution. This supercloseness result implies an optimal error estimate with respect to the $L_2$ norm and opens the door to the application of postprocessing for improving the discrete solution.