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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 Peclet number in the standard dG-energy norm, as well as with respect to the viscosity parameter.

Ana posteriorierror bound for discontinuous Galerkin approximations of convection–diffusion problems