## A construction of higher-order finite volume methods

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- by Zhongying Chen, Yuesheng Xu and Yuanyuan Zhang PDF
- Math. Comp.
**84**(2015), 599-628 Request permission

## 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.## References

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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
- MR Author ID: 214352
- 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
- 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 - © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**84**(2015), 599-628 - MSC (2010): Primary 65N30, 65N12
- DOI: https://doi.org/10.1090/S0025-5718-2014-02881-0
- MathSciNet review: 3290957