Structure-preserving finite element methods for stationary MHD models

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.