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Flexible Ultra-convergence Structures for the Finite Volume Element Method

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Abstract

We introduce a novel class of ultra-convergent structures for the Finite Volume Element (FVE) method. These structures are characterized by asymmetric and optional superconvergent points. We establish a crucial relationship between ultra-convergence properties and the orthogonality condition. Remarkably, within this framework, certain FVE schemes achieve simultaneous superconvergence of both derivatives and function values at designated points, as demonstrated in Example 2. This is a phenomenon rarely observed in other numerical methods. Theoretical validation of these findings is provided through the proposed Generalized M-Decomposition (GMD). Numerical experiments effectively substantiate our results.

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Data Availability

The code used in this work will be made available upon request to the authors.

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Funding

This work is supported in part by the National Natural Science Foundation of China (No.12371396) and the National Key Research and Development Program of China (No.2020YFA0713602, 2022YFB3707301).

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Authors and Affiliations

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

    Xiang Wang & Yuqing Zhang

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

    Zhimin Zhang

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  1. Xiang Wang
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Correspondence to Yuqing Zhang.

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Wang, X., Zhang, Y. & Zhang, Z. Flexible Ultra-convergence Structures for the Finite Volume Element Method. J Sci Comput 101, 15 (2024). https://doi.org/10.1007/s10915-024-02654-7

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