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Vertex-centered finite volume schemes of any order over quadrilateral meshes for elliptic boundary value problems

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Abstract

A family of any order finite volume (FV) schemes over quadrilateral meshes is analyzed under the framework of Petrov–Galerkin method. By constructing a special mapping from the trial space to the test space, a unified proof for the inf–sup condition of any order FV schemes is provided under a weak condition that the underlying mesh is an \(h^{1+\gamma },\gamma >0\) parallelogram mesh. The optimal convergence rate of FV solutions is then obtained with well-known techniques.

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Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100084, China

    Zhimin Zhang

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

    Zhimin Zhang

  3. College of Mathematics and Scientific Computing, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

    Qingsong Zou

Authors
  1. Zhimin Zhang
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  2. Qingsong Zou
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Correspondence to Qingsong Zou.

Additional information

Z. Zhang is supported in part by the US National Science Foundation through Grant DMS-111530, 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. Q. Zou is 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. Vertex-centered finite volume schemes of any order over quadrilateral meshes for elliptic boundary value problems. Numer. Math. 130, 363–393 (2015). https://doi.org/10.1007/s00211-014-0664-7

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  • DOI: https://doi.org/10.1007/s00211-014-0664-7

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