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Maximum-norms error estimates for high-order finite volume schemes over quadrilateral meshes

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Abstract

In this paper, we perform \(L^\infty \) and \(W^{1,\infty }\) error estimates for a class of bi-k finite volume schemes on a quadrilateral mesh for elliptic equations, where \(k\ge 2\) is arbitrary. We show that the errors of the finite volume solution in both the \(L^\infty \) and \(W^{1,\infty }\) norms converge to zero with optimal orders, provided the solution \(u\in W^{k+2,\infty }\). Our analysis is based mainly on an estimate of the difference between the finite volume and the corresponding finite element bilinear forms, as well as some techniques derived for \(L^\infty \) and \(W^{1,\infty }\) estimates of the finite element method. Our theoretical findings are supported by several numerical examples.

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Acknowledgements

The authors are grateful to Dr. Li Guo for her careful reading and corrections of the paper, and to Dr. Waixiang Cao for her help in numerical experiments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, 524048, Guangdong, China

    Wenming He

  2. Department of Mathematics, Wenzhou University, Wenzhou, 320035, Zhejiang, China

    Wenming He

  3. Beijing Computational Science Research Center, Beijing, 100193, China

    Zhimin Zhang

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

    Zhimin Zhang

  5. School of Data and Computer Science and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

    Qingsong Zou

Authors
  1. Wenming He
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  2. Zhimin Zhang
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  3. Qingsong Zou
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Corresponding author

Correspondence to Qingsong Zou.

Additional information

W. He: Research supported in part by NSFC Grant 11671304, and the Zhejiang Provincial Natural Science Foundation, China LY15A010015. Z. Zhang: Research supported in part by NSFC Grants 11471031 and 91430216, NASF Grant U1530401, and NSF Grant DMS-1419040. Q. Zou: Research supported in part by the following Grants: the special project High performance computing of National Key Research and Development Program 2016YFB0200604, NSFC 11571384, Guangdong Provincial NSF 2014A030313179, the Fundamental Research Funds for the Central Universities 16lgjc80.

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He, W., Zhang, Z. & Zou, Q. Maximum-norms error estimates for high-order finite volume schemes over quadrilateral meshes. Numer. Math. 138, 473–500 (2018). https://doi.org/10.1007/s00211-017-0912-8

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  • DOI: https://doi.org/10.1007/s00211-017-0912-8

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