Maximum-norm a posteriori error bounds for an extrapolated upwind scheme applied to a singularly perturbed convection-diffusion problem

TL;DR

This paper develops robust a posteriori error bounds for an extrapolated upwind scheme applied to singularly perturbed convection-diffusion problems, enabling adaptive mesh refinement to accurately resolve layers and singularities.

Contribution

It introduces a new a posteriori error estimator for an extrapolated upwind scheme, facilitating adaptive mesh generation for singularly perturbed problems.

Findings

Error bounds are robust on arbitrary meshes.

The error estimator effectively guides adaptive mesh refinement.

Numerical results confirm theoretical error estimates.

Abstract

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 stear an adaptive mesh algorithm that generates meshes resolving layers and singularities. Numerical results are presented that illustrate the theoretical findings.