Robust H(curl)-based finite element methods for the incompressible MHD system

TL;DR

This paper develops and analyzes $H( ext{curl})$-conforming finite element methods for the incompressible MHD system, emphasizing stability, robustness, and applicability to nonconvex domains.

Contribution

It introduces three stabilized $H( ext{curl})$ finite element formulations tailored for nonconvex domains and analyzes their structural and stability properties.

Findings

Methods are suitable for nonconvex polyhedral domains.

Stabilization mechanisms improve pressure-robustness.

Numerical experiments demonstrate practical effectiveness.

Abstract

We propose and analyze a class of finite element methods for the time-dependent incompressible magnetohydrodynamics system based on $H(\mathrm{curl})$-conforming discretizations for both the velocity and the magnetic field. This choice is guided by the aim of developing methods that are also suitable for the types of solutions arising in problems posed on nonconvex domains. Within this framework, we introduce three stabilized formulations, and study how the stabilization mechanisms employed influence their structural properties. In particular, we focus on suitability for nonconvex polyhedral domains, the need for Lagrange multipliers for the magnetic field, pressure-robustness, and quasi-robustness with respect to both the fluid and magnetic Reynolds numbers. The proposed formulations are further assessed through numerical experiments, highlighting their practical performance.