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A Family of Finite Volume Schemes of Arbitrary Order on Rectangular Meshes

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Abstract

In this paper, we analyze vertex-centered finite volume method (FVM) of any order for elliptic equations on rectangular meshes. The novelty is a unified proof of the inf-sup condition, based on which, we show that the FVM approximation converges to the exact solution with the optimal rate in the energy norm. Furthermore, we discuss superconvergence property of the FVM solution. With the help of this superconvergence result, we find that the FVM solution also converges to the exact solution with the optimal rate in the \(L^2\)-norm. Finally, we validate our theory with numerical experiments.

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Acknowledgments

The authors would like to thank a Ph.D. student, Waixiang Cao for her assistance in the numerical experiments.

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Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100084, People’s Republic of China

    Zhimin Zhang

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

  3. College of Mathematics and Scientific Computing, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Qingsong Zou

  4. Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Qingsong Zou

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  1. Zhimin Zhang
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Correspondence to Qingsong Zou.

Additional information

Zhimin Zhang was supported in part by the US National Science Foundation through grant DMS-1115530, the Ministry of Education of China through the Changjiang Scholars program, and Guangdong Provincial Government of China through the “Computational Science Innovative Research Team” program.

Qingsong Zou was supported in part by the National Natural Science Foundation of China under the grant 11171359 and in part by the Fundamental Research Funds for the Central Universities of China.

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Zhang, Z., Zou, Q. A Family of Finite Volume Schemes of Arbitrary Order on Rectangular Meshes. J Sci Comput 58, 308–330 (2014). https://doi.org/10.1007/s10915-013-9737-5

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