An a posteriori error bound for discontinuous Galerkin approximations of convection-diffusion problems

An a posteriori bound for the error measured in the discontinuous energy norm for a discontinuous Galerkin (dG) discretization of a linear one-dimensional stationary convection-diffusion-reaction problem with essential boundary conditions is presented. The proof is based on a conforming recovery operator inspired from a posteriori error bounds for the dG method for first-order hyperbolic problems. As such, the bound remains valid in the singular limit of vanishing diffusion. Detailed numerical experiments demonstrate the independence of the quality of the a posteriori bound with respect to the Péclet number in the standard dG-energy norm, as well as with respect to the viscosity parameter.