Hybrid A Posteriori Error Estimators for Conforming Finite Element Approximations to Stationary Convection-Diffusion-Reaction equations

TL;DR

This paper introduces a hybrid a posteriori error estimator for stationary convection-diffusion-reaction equations that is reliable, efficient, and robust across different regimes, outperforming traditional residual estimators.

Contribution

The paper proposes a new explicit hybrid error estimator that remains reliable and efficient regardless of the dominant physical regime, with proven independence of problem parameters.

Findings

The hybrid estimator is reliable and efficient in both diffusion- and convection-dominated regimes.

Numerical experiments confirm the robustness and superior accuracy of the hybrid estimator.

The hybrid estimator is less sensitive to reaction size compared to residual estimators.

Abstract

We consider the a posteriori error estimation for convection-diffusion-reaction equations in both diffusion-dominated and convection/reaction-dominated regimes. We present an explicit hybrid estimator, which, in each regime, is proved to be reliable and efficient with constants independent of the parameters in the underlying problem. For convection-dominated problems, the norm introduced by Verf{\"u}rth \cite{verf2005confusion} is used to measure the approximation error. Various numerical experiments are performed to (1) demonstrate the robustness of the hybrid estimator; (2) show that the hybrid estimator is more accurate than the explicit residual estimator and is less sensitive to the size of reaction, even though both of them are robust.