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An Arrow-Hurwicz iterative method based on charge-conservation for the stationary inductionless magnetohydrodynamic system

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Abstract

In this paper, an Arrow-Hurwicz iterative finite element method based on charge-conservation for solving the stationary inductionless magnetohydrodynamic equations is proposed and analyzed. The current density and electric potential are discretized by \(\textbf{H}(\textrm{div},\varOmega )\times L^2(\varOmega )\)-conforming finite element pairs, respectively, which maintains charge-conservation. The hydrodynamic unknowns are discretized by stable velocity-pressure finite element pairs. Furthermore, we use the Arrow-Hurwicz iterative method to decouple the velocity and pressure, which leads to that there is no large-scale saddle point system that needs to be solved at each iteration step except for the determination of the initial functions. It has been proven that the proposed method converges geometrically with a contraction number independent of the mesh width. Numerical results are presented to verify the results of theoretical analysis and the effectiveness of the proposed method.

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Acknowledgements

The authors would like to thank the editor and anonymous reviewers for reviewing this paper and giving some comments and suggestions.

Funding

This work was supported by the Natural Science Foundation of China (NSFC) under grant 12161095, the Basic Research Program Project of Yunnan Province (No. 202001AU070068, 202201AT070032) and Yunnan Key Laboratory of Modern Analytical Mathematics and Applications (no. 202302AN360007).

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Authors and Affiliations

  1. Department of Mathematics, Yunnan Normal University, 650500, Kunming, People’s Republic of China

    Yande **a & Yun-Bo Yang

  2. Yunnan Key Laboratory of Model Analytical Mathematics and Application, Yunnan Normal University, 650500, Kunming, People’s Republic of China

    Yun-Bo Yang

  3. Key Laboratory of Complex System Modeling and Application for Universities in Yunnan, Yunnan Normal University, 650500, Kunming, People’s Republic of China

    Yun-Bo Yang

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  1. Yande **a
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Contributions

Y.B. Yang given out conceptualization, methodology, and formal analysis; Y.B. Yang and Y.D. **a wrote the main manuscript. Y.D. **a has done the numerical experiment. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Yun-Bo Yang.

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**a, Y., Yang, YB. An Arrow-Hurwicz iterative method based on charge-conservation for the stationary inductionless magnetohydrodynamic system. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01825-9

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