A Posteriori Local Discontinuous Galerkin Error Estimation for Two-Dimensional Convection–Diffusion Problems

We present a simple, efficient, and asymptotically correct a posteriori error estimates for a minimal dissipation local discontinuous Galerkin method applied to two-dimensional diffusion and convection–diffusion problems on rectangular meshes. The finite element spaces are obtained by performing a local error analysis and a posteriori error estimates are computed by solving local problems on each element. We present computational results for several problems to show the efficiency and accuracy of our error estimates. It is shown that even in the presence of boundary layers our error estimates converge to the true error under mesh refinement when Shishkin meshes are used.