Robust H(curl)-based finite element methods for the incompressible MHD system
This work addresses computational challenges in magnetohydrodynamics simulations for nonconvex domains, representing an incremental improvement with specific stabilization mechanisms.
The authors tackled the problem of simulating time-dependent incompressible magnetohydrodynamics on nonconvex domains by proposing finite element methods using H(curl)-conforming discretizations, resulting in three stabilized formulations that are pressure-robust and quasi-robust with respect to Reynolds numbers, as validated through numerical experiments.
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.