A supercloseness result for the discontinuous Galerkin stabilization of convection–diffusion problems on Shishkin meshes

We consider a convection–diffusion problem with Dirichlet boundary conditions posed on a unit square. The problem is discretized using a combination of the standard Galerkin FEM and an h–version of the nonsymmetric discontinuous Galerkin FEM with interior penalties on a layer–adapted mesh with linear/bilinear elements. With specially chosen penalty parameters for edges from the coarse part of the mesh, we prove uniform convergence (in the perturbation parameter) in an associated norm. In the same norm we also establish a supercloseness result. Numerical tests support our theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007