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Optimal Error Estimates of Penalty Based Iterative Methods for Steady Incompressible Magnetohydrodynamics Equations with Different Viscosities

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Abstract

In this paper, we consider the penalty based finite element methods for the 2D/3D stationary incompressible magnetohydrodynamics (MHD) equations with different Reynolds numbers. Penalty method is applied to address the incompressible constraint “\(div \,\mathbf{u }=0\)” based on two different finite element pairs \(P_{1}{-}P_{0}{-}P_{1}\) and \(P_{1}b{-}P_{1}{-}P_{1}b\). Furthermore, the proposed methods are the interesting combination of three different iterations and two-level finite element algorithm such that the uniqueness condition holds. Besides, the rigorous analysis of stability and optimal error estimate with respect to the penalty parameter \(\epsilon \) for the proposed methods are given. Extensive 2D/3D numerical tests demonstrated the competitive performance of penalty methods.

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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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Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Ürümqi, 830046, People’s Republic of China

    Haiyan Su & Xinlong Feng

  2. LSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Shipeng Mao

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  1. Haiyan Su
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  2. Shipeng Mao
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Correspondence to Xinlong Feng.

Additional information

This work is in part supported by The National Key Research and Development Program of China (2016YFB0201304), National Magnetic Confinement Fusion Science Program of China (No. 2015GB110003), the NSF of China (Nos. 11271313, 11471329, 11401511, 11701493, 11871467 and 11461068) and the Natural Science Foundation of Xinjiang Province (No. 2016D01C073).

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Su, H., Mao, S. & Feng, X. Optimal Error Estimates of Penalty Based Iterative Methods for Steady Incompressible Magnetohydrodynamics Equations with Different Viscosities. J Sci Comput 79, 1078–1110 (2019). https://doi.org/10.1007/s10915-018-0883-7

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  • DOI: https://doi.org/10.1007/s10915-018-0883-7

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