Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

Abstract

We consider a posteriori error estimators that can be applied to anisotropic tetrahedral nite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz{Zhu (ZZ) type error estimators are derived which originate from dierent backgrounds. In the course of the analysis, the rst estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient. The advantage of keeping two dierent analyses of the estimators is that they allow dierent and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and eciency of the ZZ error estimation is obtained. The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and conrmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diusion problem.