Error analysis of a divergence-preserving mixed finite element scheme for the incompressible Hall--magnetohydrodynamic equations

TL;DR

This paper develops and analyzes a structure-preserving finite element scheme for a regularised Hall--MHD system, ensuring divergence-free magnetic fields and energy stability, with proven convergence and error estimates.

Contribution

It introduces a novel, fully discrete finite element method for the regularised Hall--MHD system that preserves divergence-free magnetic fields and provides rigorous error analysis.

Findings

Optimal convergence rates for the regularised problem are established.

Error estimates for the original Hall--MHD system are derived using regularisation.

Numerical simulations confirm theoretical stability and accuracy.

Abstract

The incompressible Hall-magnetohydrodynamics (Hall--MHD) system presents substantial analytical and computational challenges due to its stiff, highly nonlinear Hall term and the strict requirement that the magnetic field remains solenoidal. In this paper, we study a Voigt-regularised Hall--MHD system, which is of independent analytical interest and provides a physically consistent, well-posed regularisation of the original model. We propose, analyse, and implement a structure-preserving, linear, fully discrete finite element method for this regularised problem. Using finite element exterior calculus and a mixed formulation, the spatial discretisation enforces the divergence-free condition on the magnetic field exactly, while a skew-symmetric, linearly implicit time discretisation yields unconditional energy stability. We establish optimal convergence rates for the Voigt-regularised problem and, additionally, derive error estimates for the unregularised Hall--MHD system, with the Voigt regularisation playing a crucial role in the non-resistive regime. Finally, numerical simulations in both 2.5D and 3D corroborate the theoretical results and demonstrate the physical fidelity of the scheme.