Structure-preserving Finite Element Methods for Stationary MHD Models

TL;DR

This paper introduces a novel mixed finite element scheme for stationary MHD models that exactly preserves magnetic divergence and energy laws, ensuring well-posedness and convergence.

Contribution

It develops a new finite element method for stationary MHD that maintains physical laws exactly and proves its mathematical well-posedness and convergence.

Findings

Exact preservation of divergence-free magnetic field law

Proof of well-posedness of the finite element scheme

Convergence of solutions and iterative methods

Abstract

In this paper, we develop a class of mixed finite element scheme for stationary magnetohydrodynamics (MHD) models, using magnetic field $\bm B$ and current density $\bm j$ as the discretization variables. We show that the Gauss's law for the magnetic field, namely $\nabla\cdot\bm{B}=0$, and the energy law for the entire system are exactly preserved in the finite element schemes. Based on some new basic estimates for $H^{h}(\mathrm{div})$, we show that the new finite element scheme is well-posed. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of Picard iterations and finite element methods under some conditions.