Maximum-Norm a Posteriori Error Bounds for an Extrapolated Upwind Scheme Applied to a Singularly Perturbed Convection-Diffusion Problem

Richardson extrapolation is applied to a simple first-order upwind difference scheme for the approximation of solutions of singularly perturbed convection-diffusion problems in one dimension. Robust a posteriori error bounds are derived for the proposed method on arbitrary meshes. It is shown that the resulting error estimator can be used to steer an adaptive mesh algorithm that generates meshes resolving layers and singularities. Numerical results are presented that illustrate the theoretical findings.