Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003) no. 6, pp. 1013-1043.

We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite 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 different backgrounds. In the course of the analysis, the first 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 different analyses of the estimators is that they allow different 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 efficiency 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 confirmed 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 diffusion problem.

DOI : https://doi.org/10.1051/m2an:2003065
Classification : 65N15,  65N30,  65N50
Mots clés : anisotropic mesh, error estimator, Zienkiewicz-Zhu estimator, recovered gradient
@article{M2AN_2003__37_6_1013_0,
author = {Kunert, Gerd and Nicaise, Serge},
title = {Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {1013--1043},
publisher = {EDP-Sciences},
volume = {37},
number = {6},
year = {2003},
doi = {10.1051/m2an:2003065},
zbl = {1077.65114},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2003065/}
}
Kunert, Gerd; Nicaise, Serge. Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003) no. 6, pp. 1013-1043. doi : 10.1051/m2an:2003065. http://www.numdam.org/articles/10.1051/m2an:2003065/

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