Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems

TL;DR

This paper develops robust a posteriori error estimators for SUPG methods applied to singularly perturbed convection-diffusion-reaction equations, ensuring uniform accuracy regardless of the perturbation parameter.

Contribution

The paper introduces a new dual norm and a recovery stabilization procedure to create error estimators that are robust with respect to the perturbation parameter in SUPG methods.

Findings

Error estimators are proven to be robust with respect to the perturbation parameter.

Numerical experiments confirm the theoretical robustness and uniform error dependence.

The approach improves adaptive methods for singularly perturbed problems.

Abstract

In this paper, we investigate adaptive streamline upwind/Petrov Galerkin (SUPG) methods for singularly perturbed convection-diffusion-reaction equations in a new dual norm presented in [Du and Zhang, J. Sci. Comput. (2015)]. The flux is recovered by either local averaging in conforming $H({\rm div})$ spaces or weighted global $L^2$ projection onto conforming $H({\rm div})$ spaces. We further introduce a recovery stabilization procedure, and develop completely robust a posteriori error estimators with respect to the singular perturbation parameter $\varepsilon$. Numerical experiments are reported to support the theoretical results and to show that the estimated errors depend on the degrees of freedom uniformly in $\varepsilon$.