Convergence of a $B$-$E$ based finite element method for MHD models on Lipschitz domains
For researchers in computational MHD, this provides theoretical guarantees for finite element methods on non-smooth domains, though the result is incremental.
The paper proves convergence of Picard iterations and finite element schemes for MHD models on Lipschitz domains under weak regularity assumptions, enabling reliable computation of singular solutions.
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.