A robust a-posteriori error estimate for hp-adaptive DG methods for convection-diffusion equations

We derive a robust a-posteriori error estimate for hp-adaptive discontinuous Galerkin (DG) discretizations of stationary convection-diffusion equations. We consider 1-irregular meshes consisting of parallelograms. The estimate yields global upper and lower bounds of the errors measured in terms of the natural energy norm associated with the diffusion and a semi-norm associated with the convection. The ratio of the constants in the upper and lower bounds is independent of the local mesh sizes and weakly depending on the local polynomial degrees. Moreover, it is also independent of the magnitude of the Péclet number of the problem, and hence the estimate is fully robust for convection-dominated problems. We apply our estimator as an energy norm error indicator in an hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples.