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

TL;DR

This paper proves the convergence of a finite element method for stationary MHD models on Lipschitz domains, handling singular solutions and establishing key divergence-free estimates.

Contribution

It introduces a new $L^{3}$ estimate for divergence-free finite element functions under boundary conditions, ensuring convergence of the scheme.

Findings

Proves convergence of the finite element scheme for MHD models.

Establishes a new $L^{3}$ divergence-free estimate.

Demonstrates scheme effectiveness for singular solutions.

Abstract

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.