Robust Finite Elements for linearized Magnetohydrodynamics

TL;DR

This paper presents a pressure robust finite element method for linearized magnetohydrodynamics that remains stable at high Reynolds numbers, combining non-conforming and conforming approaches with stabilization techniques.

Contribution

It introduces a novel finite element scheme that is quasi-robust for high Reynolds numbers in 3D magnetohydrodynamics, integrating BDM, DG, and CIP stabilization.

Findings

The method is proven to be quasi-robust theoretically.

Numerical experiments confirm the stability and accuracy.

The scheme effectively handles high fluid and magnetic Reynolds numbers.

Abstract

We introduce a pressure robust Finite Element Method for the linearized Magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a non-conforming BDM approach with suitable DG terms for the fluid part, combined with an $H^1$-conforming choice for the magnetic fluxes. The method introduces also a specific CIP-type stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments.