Structure-preserving Finite Element Methods for Stationary MHD Models

In this paper, we develop two classes of mixed finite element discretization for stationary MHD models, one using $\bm B$ (the magnetic field) and $\bm E$ (the electric field) as the discretization variables while the other using $\bm B$ and $\bm j$ (the current density) as the discretization variables. We show that the Gauss's law for magnetic field, namely $\nabla\cdot\bm{B}=0$, and the energy law for the entire system are preserved for both of these finite element schemes. We further establish that both of these new finite element schemes are well-posed under certain conditions, but we conclude that the finite element scheme based on $\bm B$-$\bm j$ formulation seems to be superior than the one based on $\bm B$-$\bm E$ formulation as the former requires much weaker conditions than the latter does.