Structure-preserving finite element methods for stationary MHD models

Hu, Kaibo; Xu, Jinchao

doi:10.1090/mcom/3341

Structure-preserving finite element methods for stationary MHD models
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by Kaibo Hu and Jinchao Xu;

Math. Comp. 88 (2019), 553-581

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 $\mathbfit {B}$ and the current density $\mathbfit {j}$ as discretization variables. We show that Gauss’s law for the magnetic field, namely $\nabla \cdot \mathbfit {B}=0$, 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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