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Mesh Quality and More Detailed Error Estimates of Finite Element Method

Part of:Partial differential equations, boundary value problems

Published online by Cambridge University Press:  09 May 2017

Yunqing Huang,
Liupeng Wang and
Nianyu Yi
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China
Liupeng Wang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China
Nianyu Yi*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China
*
*Corresponding author. Email addresses:huangyq@xtu.edu.cn (Y. Q. Huang), wangliupeng@xtu.edu.cn (L. P. Wang), yinianyu@xtu.edu.cn (N. Y. Yi)
*Corresponding author. Email addresses:huangyq@xtu.edu.cn (Y. Q. Huang), wangliupeng@xtu.edu.cn (L. P. Wang), yinianyu@xtu.edu.cn (N. Y. Yi)
*Corresponding author. Email addresses:huangyq@xtu.edu.cn (Y. Q. Huang), wangliupeng@xtu.edu.cn (L. P. Wang), yinianyu@xtu.edu.cn (N. Y. Yi)

Abstract

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In this paper, we study the role of mesh quality on the accuracy of linear finite element approximation. We derive a more detailed error estimate, which shows explicitly how the shape and size of elements, and symmetry structure of mesh effect on the error of numerical approximation. Two computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the mesh. In compare with the standard a priori error estimates, which only yield information on the asymptotic error behaviour in a global sense, our proposed error estimate considers the effect of local element geometry properties, and is thus more accurate. Under certain conditions, the traditional error estimates and supercovergence results can be derived from the proposed error estimate. Moreover, the estimators Ge and Gv are computable and thus can be used for predicting the variation of errors. Numerical tests are presented to illustrate the performance of the proposed parameters Ge and Gv .

Keywords

Finite element methodserror estimationcell geometrymesh quality

MSC classification

Secondary:65N15: Error bounds65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods65N50: Mesh generation and refinement

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017, pp. 420 - 436
Copyright
Copyright © Global-Science Press 2017 

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