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Stable finite element methods preserving \(\nabla \cdot \varvec{B}=0\) exactly for MHD models

Abstract

This paper is devoted to the design and analysis of some structure-preserving finite element schemes for the magnetohydrodynamics (MHD) system. The main feature of the method is that it naturally preserves the important Gauss’s law, namely \(\nabla \cdot \varvec{B}=0\). In contrast to most existing approaches that eliminate the electrical field variable \(\varvec{E}\) and give a direct discretization of the magnetic field, our new approach discretizes the electric field \(\varvec{E}\) by Nédélec type edge elements for \(H(\mathrm {curl})\), while the magnetic field \(\varvec{B}\) by Raviart–Thomas type face elements for \(H(\mathrm {div})\). As a result, the divergence-free condition on the magnetic field holds exactly on the discrete level. For this new finite element method, an energy stability estimate can be naturally established in an analogous way as in the continuous case. Furthermore, well-posedness is rigorously established in the paper for the Picard linearization of the fully nonlinear systems by using the Brezzi theory. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.

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Acknowledgments

The authors would like to thank Long Chen, Xiaozhe Hu, Maximilian Metti, Shuo Zhang, Ludmil Zikatanov for useful discussions and suggestions. And the authors appreciate many valuable suggestions from the anonymous referees.

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Correspondence to Jinchao Xu.

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This material is based upon work supported in part by the US Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-SC-0014400 and by Beijing International Center for Mathematical Research of Peking University, China.

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Hu, K., Ma, Y. & Xu, J. Stable finite element methods preserving \(\nabla \cdot \varvec{B}=0\) exactly for MHD models. Numer. Math. 135, 371–396 (2017). https://doi.org/10.1007/s00211-016-0803-4

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Mathematics Subject Classification

  • 65N30
  • 65M60
  • 76W05