An a-posteriori error estimate for hp-adaptive DG methods for convection-diffusion problems on anisotropically refined meshes

We prove an a-posteriori error estimate for hp-adaptive discontinuous Galerkin methods for the numerical solution of convection-diffusion equations on anisotropically refined rectangular elements. The estimate yields global upper and lower bounds of the errors measured in terms of a natural norm associated with diffusion and a semi-norm associated with convection. The anisotropy of the underlying meshes is incorporated in the upper bound through an alignment measure. We present a series of numerical experiments to test the feasibility of this approach within a fully automated hp-adaptive refinement algorithm.