Structure-preserving and helicity-conserving finite element approximations and preconditioning for the Hall MHD equations

TL;DR

This paper introduces structure-preserving finite element methods for Hall MHD equations, ensuring physical invariants are maintained and providing effective preconditioning, with verified numerical experiments demonstrating their effectiveness.

Contribution

The paper develops novel finite element schemes for Hall MHD that preserve key physical properties and introduces an augmented Lagrangian preconditioner for improved computational efficiency.

Findings

Exact enforcement of magnetic Gauss's law in discretization

Energy and helicity preservation in time discretizations

Numerical experiments confirm theoretical properties

Abstract

We develop structure-preserving finite element methods for the incompressible, resistive Hall magnetohydrodynamics (MHD) equations. These equations incorporate the Hall current term in Ohm's law and provide a more appropriate description of fully ionized plasmas than the standard MHD equations on length scales close to or smaller than the ion skin depth. We introduce a stationary discrete variational formulation of Hall MHD that enforces the magnetic Gauss's law exactly (up to solver tolerances) and prove the well-posedness and convergence of a Picard linearization. For the transient problem, we present time discretizations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. Additionally, we present an augmented Lagrangian preconditioning technique for both the stationary and transient cases. We confirm our findings with several numerical experiments.