Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements

TL;DR

This paper extends improved Zienkiewicz-Zhu (ZZ) a posteriori error estimators for conforming linear finite element methods to more general diffusion problems with full tensors and different flux recovery spaces.

Contribution

It generalizes previous ZZ estimators to handle full diffusion tensors and both piecewise constant and linear flux recovery in $H(div)$ spaces.

Findings

Enhanced error estimation accuracy for diffusion problems with full tensors.

Applicability to both piecewise constant and linear flux recovery methods.

Theoretical validation of the extended estimators.

Abstract

In \cite{CaZh:09}, we introduced and analyzed an improved Zienkiewicz-Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise "constant" flux recovery in the $H(div;\Omega)$ conforming finite element space. This paper extends the results of \cite{CaZh:09} to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear $H(div)$ space.