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A Convergent Finite Element Method for the Compressible Magnetohydrodynamics System

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Abstract

This paper is devoted to the study of finite element method for the isentropic compressible magnetohydrodynamics system. We employ quadratic finite elements to approximate the velocity and Nédélec edge elements to approximate the magnetic induction. The continuity equation is approximated by Discontinuous Galerkin method. Based on the renormalized scheme, we derive the stability of the proposed numerical scheme for compressible magnetohydrodynamics equations. With the help of the theory of the topological degree, the existence of solution to the numerical scheme is proved. Some techniques have to be adopted to improve the integrability of density so as to achieve strong convergence of the discrete density. As both meshwidth and timestep size tend to zero, we show that finite element solution converges to a global weak solution of the continuous problem. The results of this paper can be regarded as a numerical version of the existence analysis of the compressible MHD system.

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Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this paper.

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  1. NCMIS, LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100190, China

    Qianqian Ding & Shipeng Mao

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  1. Qianqian Ding
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  2. Shipeng Mao
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Correspondence to Shipeng Mao.

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The research was in part supported by the Major State Research Development Program of China (No. 2016YFB0201304), National Natural Science Foundation of China (No. 11871467), National Magnetic Confinement Fusion Science Program of China (No. 2015GB110003) and Youth Innovation Promotion Association of CAS (2016003).

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Ding, Q., Mao, S. A Convergent Finite Element Method for the Compressible Magnetohydrodynamics System. J Sci Comput 82, 21 (2020). https://doi.org/10.1007/s10915-020-01129-9

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