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Structure-preserving finite element methods for stationary MHD models


Authors: Kaibo Hu and Jinchao Xu
Journal: Math. Comp. 88 (2019), 553-581
MSC (2010): Primary 65N30, 65N12
DOI: https://doi.org/10.1090/mcom/3341
Published electronically: May 29, 2018
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Abstract: We develop a class of mixed finite element schemes for stationary magnetohydrodynamics (MHD) models, using the magnetic field $ \bm B$ and the current density $ \bm j$ as discretization variables. We show that Gauss's law for the magnetic field, namely $ \nabla \cdot \bm {B}=0$, and the energy law for the entire system are exactly preserved in the finite element schemes. Based on some new basic estimates for $ H(\mathrm {div})$ finite elements, we show that the new finite element scheme is well-posed. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of the Picard iterations and the finite element methods under some conditions.


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Additional Information

Kaibo Hu
Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing 100871, People’s Republic of China
Address at time of publication: Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, Oslo, Norway
Email: kaibohu@math.uio.no

Jinchao Xu
Affiliation: Center for Computational Mathematics and Applications and Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: xu@math.psu.edu

DOI: https://doi.org/10.1090/mcom/3341
Keywords: Divergence-free, stationary, MHD equations, finite element
Received by editor(s): March 20, 2015
Received by editor(s) in revised form: January 29, 2016, November 13, 2016, September 18, 2017, and November 12, 2017
Published electronically: May 29, 2018
Additional Notes: 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-SC0014400 and by Beijing International Center for Mathematical Research of Peking University, China.
The first author was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 339643.
Article copyright: © Copyright 2018 American Mathematical Society

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