We develop a class of mixed finite element schemes for stationary magnetohydrodynamics (MHD) models, using the magnetic field B and the current density j as discretization variables. We show that Gauss's law for the magnetic field, namely ∇· 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(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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Received by the editor March 20, 2015, and, in revised form, January 29, 2016, November 13, 2016, September 18, 2017, and November 12, 2017. 2010 Mathematics Subject Classification. Primary 65N30, 65N12. Key words and phrases. Divergence-free, stationary, MHD equations, finite element. 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.
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- Finite element
- MHD equations