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An unconditionally stable and \(L^2\) optimal quadratic finite volume scheme over triangular meshes for anisotropic elliptic equations

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Abstract

In this paper, we propose an unconditionally stable and \(L^2\) optimal quadratic finite volume (FV) scheme for solving the two-dimensional anisotropic elliptic equation on triangular meshes. In quadratic FV schemes, the construction of the dual partition is closely related to the \(L^2\) error estimate. While many dual partitions over triangular meshes have been investigated in the literature, only the one proposed by Wang and Li (SIAM J. Numer. Anal. 54:2729–2749, 2016) has been proven to achieve optimal \(L^2\) norm convergence rate. This paper introduces a novel approach for constructing the dual partition using multiblock control volumes, which is also shown to optimally converge in the \(L^2\) norm (\(O(h^3)\)). Furthermore, we present a new mapping from the trial space to the test space, which enables us to demonstrate that the inf-sup condition of the scheme holds independently of the minimal angle of the underlying mesh. To the best of our knowledge, this is the first unconditionally stable quadratic FV scheme over triangular meshes that achieves optimal \(L^2\) norm convergence rate. We provide numerical experiments to validate our findings.

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Data Availibility Statement

The data that support the findings of this study are available on request.

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Acknowledgements

We would like to thank the Editor and two anonymous reviewers for their valuable suggestions and careful reading which have helped us to improve the paper.

Funding

Kejia Pan was supported by the National Natural Science Foundation of China (No. 42274101). Weifeng Qiu was partially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (Nos. CityU 11302219, CityU 11300621). Xiaoxin Wu was supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2022ZZTS0147).

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Author notes

  1. Kejia Pan contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, People’s Republic of China

    Xiaoxin Wu & Kejia Pan

  2. Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China

    Xiaoxin Wu & Weifeng Qiu

Authors
  1. Xiaoxin Wu
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  2. Weifeng Qiu
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Correspondence to Kejia Pan.

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Communicated by: Long Chen

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Wu, X., Qiu, W. & Pan, K. An unconditionally stable and \(L^2\) optimal quadratic finite volume scheme over triangular meshes for anisotropic elliptic equations. Adv Comput Math 49, 83 (2023). https://doi.org/10.1007/s10444-023-10085-5

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