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

  • Haiyan Su
  • Shipeng Mao
  • Xinlong Feng
Article
  • 46 Downloads

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.

Keywords

Magnetohydrodynamics equations Penalty finite element method Two-level method Inf-sup condition Error estimate 

Notes

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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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and System SciencesXinjiang UniversityÜrümqiPeople’s Republic of China
  2. 2.LSEC, Academy of Mathematics and System Sciences, Chinese Academy of SciencesSchool of Mathematical Science, University of Chinese Academy of SciencesBeijingPeople’s Republic of China

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