Convergence of a B-E based finite element method for MHD models on Lipschitz domains

We discuss a class of magnetic-electric fields based finite element schemes for stationary magnetohydrodynamics (MHD) systems with two types of boundary conditions. We establish a key $L^{3}$ estimate for divergence-free finite element functions for a new type of boundary conditions. With this estimate and a similar one in [Hu&Xu,2018], we rigorously prove the convergence of Picard iterations and the finite element schemes with weak regularity assumptions. These results demonstrate the convergence of the finite element methods for singular solutions.