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

Gerd Kunert; Serge Nicaise

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 6, page 1013-1043
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Kunert, Gerd, and Nicaise, Serge. "Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes." ESAIM: Mathematical Modelling and Numerical Analysis 37.6 (2010): 1013-1043. <http://eudml.org/doc/194195>.

@article{Kunert2010,
abstract = { 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. },
author = {Kunert, Gerd, Nicaise, Serge},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Anisotropic mesh; error estimator; Zienkiewicz–Zhu estimator; recovered gradient.; finite element methods; anisotropic meshes; error estimators; mesh refinement; Zienkiewicz-Zhu estimator; recovered gradient; elliptic second-order boundary value problems; numerical experiments},
language = {eng},
month = {3},
number = {6},
pages = {1013-1043},
publisher = {EDP Sciences},
title = {Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes},
url = {http://eudml.org/doc/194195},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Kunert, Gerd
AU - Nicaise, Serge
TI - Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 6
SP - 1013
EP - 1043
AB - 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.
LA - eng
KW - Anisotropic mesh; error estimator; Zienkiewicz–Zhu estimator; recovered gradient.; finite element methods; anisotropic meshes; error estimators; mesh refinement; Zienkiewicz-Zhu estimator; recovered gradient; elliptic second-order boundary value problems; numerical experiments
UR - http://eudml.org/doc/194195
ER -

References

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