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New structure-preserving mixed finite element method for the stationary MHD equations with magnetic-current formulation

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Abstract

In this paper, we propose and analyze a new structure-preserving finite element method for the stationary magnetohydrodynamic equations with magnetic-current formulation on Lipschitz domains. Using a mixed finite element approach, we discretize the hydrodynamic unknowns by inf-sup stable velocity-pressure finite element pairs, and the current density, the induced electric field and the magnetic field by using the edge-edge-face elements from a discrete de-Rham complex pair. To deal with the divergence-free condition of the magnetic field, we introduce an augmented term to the discrete scheme rather than a Lagrange multiplier in the existing schemes. Thanks to discrete differential forms and finite element exterior calculus, the proposed scheme preserves the divergence-free property exactly for the magnetic induction on the discrete level. The well-posedness of the discrete problem is further proved under the small data condition. Under weak regularity assumptions, we rigorously establish the error estimates of the finite element schemes. Numerical results are provided to illustrate the theoretical results and demonstrate the efficiency of the proposed method.

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Acknowledgements

The authors wish to thank the anonymous referees and the associate editor for many constructive comments that improved the paper.

Funding

The first author’s research is supported in part by the National Natural Science Foundation of China under Grant 12201575 and China Postdoctoral Science Foundation under Grant 2022M722878. The second author’s research is supported in part by Xinjiang Graduate Student Research Innovation Project under Grant XJ2023G081.

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Authors and Affiliations

  1. Henan Academy of Big Data, Zhengzhou University, Zhengzhou, 450052, China

    Xiaodi Zhang

  2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Xiaodi Zhang

  3. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Xiaodi Zhang

  4. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, China

    Shitian Dong

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  1. Xiaodi Zhang
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  2. Shitian Dong
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XZ: conceptualization, methodology, formal analysis, writing, review. SD: visualization, validation, review. All authors reviewed the manuscript.

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Correspondence to Xiaodi Zhang.

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Zhang, X., Dong, S. New structure-preserving mixed finite element method for the stationary MHD equations with magnetic-current formulation. Bit Numer Math 63, 55 (2023). https://doi.org/10.1007/s10543-023-00995-7

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