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Unified construction and L2 analysis for the finite volume element method over tensorial meshes

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Abstract

We provide a unified construction for high order finite volume element (FVE) schemes with the optimal L2 convergence rate over tensorial meshes in any d-dimension (d ≥ 2). Under this framework, one can choose the k dual parameters for each direction on a (k − 1)-dimensional surface in the k-dimensional parameter space, which provides us more choices than the existing Gaussian point-based dual strategy. This opens up more possibilities for applying the FVE method to some complex problems (see Example 3 for a convection-dominated problem as an example). Theoretically, we present the L2 estimate of the FVE method with general dual meshes in arbitrary d-dimension. The tensorial orthogonal condition (TOC) is proposed and proved to be a sufficient condition for a tensorial FVE scheme to maintain the optimal L2 convergence rate. Numerical experiments illustrate the above results.

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Acknowledgements

The authors thank the anonymous referees for their suggestions and comments which are very helpful in improving the quality and readability of this paper.

Funding

This work is supported in part by the National Natural Science Foundation of China (11701211, 12071177) and the China Postdoctoral Science Foundation (2021M690437).

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Qianjin Steet, Changchun, 130012, China

    Yuqing Zhang & Xiang Wang

  2. School of Mathematical Sciences, Beijing Normal University, Xinjiekouwai Steet, Beijing, 100875, China

    Xiang Wang

  3. Research Center for Mathematics, Beijing Normal University, Jinfeng Road, Zhuhai, 519087, China

    Xiang Wang

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  1. Yuqing Zhang
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  2. Xiang Wang
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Correspondence to Xiang Wang.

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Communicated by: Long Chen

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Zhang, Y., Wang, X. Unified construction and L2 analysis for the finite volume element method over tensorial meshes. Adv Comput Math 49, 2 (2023). https://doi.org/10.1007/s10444-022-10004-0

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