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A construction of higher-order finite volume methods


Authors: Zhongying Chen, Yuesheng Xu and Yuanyuan Zhang
Journal: Math. Comp. 84 (2015), 599-628
MSC (2010): Primary 65N30, 65N12
Published electronically: July 28, 2014
MathSciNet review: 3290957
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Abstract: We provide a method for the construction of higher-order finite volume methods (FVMs) for solving boundary value problems of the two dimensional elliptic equations. Specifically, when the trial space of the FVM is chosen to be a conforming triangle mesh finite element space, we describe a construction of the associated test space that guarantees the uniform local-ellipticity of the family of the resulting discrete bilinear forms. We show that the uniform local-ellipticity ensures that the resulting FVM has a unique solution which enjoys an optimal error estimate. We characterize the uniform local-ellipticity in terms of the uniform boundedness (below by a positive constant) of the smallest eigenvalues of the matrices associated with the FVMs. We then translate the characterization to equivalent requirements on the shapes of the triangle meshes for the trial spaces. Four convenient sufficient conditions for the family of the discrete bilinear forms to be uniformly local-elliptic are derived from the characterization. Following the general procedure, we construct four specific FVMs which satisfy the uniform local-ellipticity. Numerical results are presented to verify the theoretical results on the convergence order of the FVMs.


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Additional Information

Zhongying Chen
Affiliation: Guangdong Province Key Laboratory of Computational Science, School of Mathematics and Computational Sciences, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China
Email: lnsczy@mail.sysu.edu.cn

Yuesheng Xu
Affiliation: Guangdong Province Key Laboratory of Computational Science, School of Mathematics and Computational Sciences, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China – and – Department of Mathematics, Syracuse University, Syracuse, New York 13244
Email: yxu06@syr.edu

Yuanyuan Zhang
Affiliation: Guangdong Province Key Laboratory of Computational Science, School of Mathematics and Computational Sciences, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China
Email: yy0dd@126.com

DOI: https://doi.org/10.1090/S0025-5718-2014-02881-0
Keywords: Finite volume methods
Received by editor(s): October 4, 2012
Received by editor(s) in revised form: June 19, 2013
Published electronically: July 28, 2014
Additional Notes: This work was supported in part by Guangdong provincial government of China through the “Computational Science Innovative Research Team” program
The first author was also supported in part by the Natural Science Foundation of China under grants 10771224 and 11071264
The second author was supported in part by US Air Force Office of Scientific Research under grant FA9550-09-1-0511, by the US National Science Foundation under grants DMS-0712827, DMS-1115523, and by the Natural Science Foundation of China under grants 11071286 and 91130009. All correspondence should be sent to this author
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.