  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.
