Flux Reconstruction for Goal-Oriented A Posteriori Error Estimation

TL;DR

This paper introduces a new goal-oriented a posteriori error estimator that combines dual weighted residuals with equilibrated error estimation, demonstrating reliable and optimally convergent adaptivity in finite element methods.

Contribution

It presents a novel localized flux reconstruction algorithm for goal-oriented error estimation, including the first rigorous analysis over quadrilateral meshes with hanging nodes.

Findings

Demonstrates practical reliability through numerical experiments

Confirms theoretical predictions of the estimator's performance

Achieves optimally convergent adaptive finite element solutions

Abstract

We propose a new heuristic goal-oriented a posteriori error estimator that connects the dual weighted residual method with equilibrated a posteriori error estimation. Our numerical experiments demonstrate the practical reliability of the error estimator, confirming theoretical predictions, as well as optimally convergent adaptivity even over singular domains and coarse meshes. The central algorithm is a localized flux reconstruction, which has been implemented in the finite element library deal.II. For a solid preparation we assess the performance of the equilibrated a posteriori error estimator of the energy norm in numerical experiments. Moreover, we give what seems to be first rigorous discussion in the numerical literature of localized flux reconstruction over quadrilateral meshes with hanging nodes.