Stable Finite Element Methods Preserving $\nabla \cdot \boldsymbol{B} = 0$ Exactly for MHD Models

TL;DR

This paper introduces structure-preserving finite element schemes for MHD that exactly maintain the divergence-free condition of the magnetic field, ensuring stability and well-posedness through novel discretization techniques.

Contribution

The paper presents a new finite element approach discretizing electric and magnetic fields separately, preserving divergence-free magnetic fields exactly and establishing stability and well-posedness.

Findings

Exact divergence-free magnetic field preservation

Energy stability similar to continuous models

Well-posedness for nonlinear systems

Abstract

This paper is devoted to the design and analysis of some structure-preserving finite element schemes for the magnetohydrodynamics (MHD) system. The main feature of the method is that it naturally preserves the important Gauss law, namely $\nabla\cdot\boldsymbol{B}=0$. In contrast to most existing approaches that eliminate the electrical field variable $\boldsymbol{E}$ and give a direct discretization of the magnetic field, our new approach discretizes the electric field $\boldsymbol{E}$ by N\'{e}d\'{e}lec type edge elements for $H(\mathrm{curl})$, while the magnetic field $\boldsymbol{B}$ by Raviart-Thomas type face elements for $H(\mathrm{div})$. As a result, the divergence-free condition on the magnetic field holds exactly on the discrete level. For this new finite element method, an energy stability estimate can be naturally established in an analogous way as in the continuous case. Furthermore, well-posedness is rigorously established in the paper for both the Picard and Newton linearization of the fully nonlinear systems by using the Brezzi theory for both the continuous and discrete cases. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.