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

Technical Summary: Error Analysis of a Divergence-Preserving Mixed Finite Element Scheme for the Incompressible Hall–MHD Equations

Problem Statement
The incompressible Hall-magnetohydrodynamics (Hall–MHD) system models the interaction between fluid motion and magnetic fields in electrically conducting fluids, incorporating the Hall effect which becomes significant at scales comparable to the ion skin depth. This effect introduces a stiff, highly nonlinear term ($\eta \text{curl}(\text{curl} B \times B)$) into the induction equation, leading to dispersive behavior, steep gradients, and high-frequency oscillations (whistler waves). These features pose substantial analytical and computational challenges. Specifically, the global existence of strong solutions in three dimensions remains an open problem, and standard numerical methods often struggle with stability and the strict requirement that the magnetic field remains solenoidal ($\text{div } B = 0$). Violations of this divergence-free condition can induce nonphysical magnetic monopoles and trigger severe numerical instabilities. Furthermore, existing numerical approaches, such as those based on Scalar Auxiliary Variable (SAV) methods or spectral techniques, often dissipate modified energy functionals rather than the true physical energy, fail to enforce the divergence-free condition exactly at the discrete level, or lack rigorous error analysis and geometric flexibility.

Methodology
To address these challenges, the authors propose a structure-preserving, linear, fully discrete mixed finite element method (FEM) for a Voigt-regularised Hall–MHD system. The Voigt regularisation replaces the time derivative in the momentum equation with a spatially filtered version $(I - \alpha_1 \Delta)\partial_t u$ and retains an electron inertia term $\alpha_2 \partial_t J$ in the generalised Ohm's law. This modification improves well-posedness and numerical stability while preserving the essential physical structure and energy laws of the original system.

The numerical scheme is constructed using the following components:

1. Mixed Formulation: The problem is reformulated using the electric field $E$ and current density $J$ as primary variables, leading to a coupled system of momentum, induction, Ohm's law, and Ampère's law equations.
2. Spatial Discretisation: The scheme employs Finite Element Exterior Calculus (FEEC) to ensure the divergence-free condition is satisfied exactly.
   • The velocity $u$ and pressure $p$ are approximated using the MINI finite element pair (lowest-order inf-sup stable).
   • The magnetic field $B$ and current density $J$ are approximated using lowest-order Raviart–Thomas ($RT_0$) and Nédélec ($X_0$) spaces, respectively.
   • These spaces form a discrete de Rham complex, ensuring that $\text{div } B_h = 0$ is preserved exactly at the discrete level.
3. Temporal Discretisation: A skew-symmetric, linearly implicit time-stepping scheme is used. The convective and Lorentz terms are treated semi-implicitly using values from the previous time step to maintain linearity and unconditional energy stability.
4. Error Analysis: The authors establish optimal convergence rates by decomposing the error into approximation errors (using Stokes projections and orthogonal projections) and discrete errors. They utilize discrete Gaffney inequalities and energy estimates to handle the nonlinear coupling terms.

Key Contributions

• Structure-Preserving Scheme: The paper presents a linear, fully discrete FEM that enforces the solenoidal constraint on the magnetic field exactly, avoiding the need for projection steps or penalty methods that can degrade accuracy or stability.
• Unconditional Energy Stability: The proposed scheme satisfies a discrete energy dissipation law unconditionally, mirroring the continuous physical energy law, even in the presence of the stiff Hall term.
• Rigorous Error Estimates: The authors provide a complete error analysis for the Voigt-regularised problem, establishing optimal convergence rates of order $O(h^s + \tau)$ under suitable regularity assumptions.
• Extension to Unregularised System: Crucially, the analysis extends to the unregularised Hall–MHD system (where $\alpha_1 = \alpha_2 = 0$). The authors demonstrate that convergence can still be proven for the unregularised system, provided the resistivity $\sigma > 0$. In the non-resistive regime ($\sigma = 0$), the Voigt regularisation is shown to play a crucial role in the error estimates.
• 2.5D and 3D Implementation: The framework is validated in both 2.5D (translational invariance) and fully 3D settings, demonstrating the method's applicability to complex geometries and physical phenomena.

Results
Numerical simulations in 2.5D and 3D confirm the theoretical findings:

• Convergence: The scheme achieves the predicted spatial and temporal convergence rates for the velocity, magnetic field, and current density. While convergence rates for velocity and current density are slightly reduced in some cases (attributed to limited solution regularity due to the Hall term), the magnetic field maintains optimal convergence.
• Divergence Preservation: The maximum divergence error ($\max | \text{div } B_h |$) remains at the level of solver tolerance throughout the simulations, confirming the exact preservation of the solenoidal constraint.
• Energy Stability: The total energy of the system decays monotonically over time, consistent with the theoretical energy dissipation law.
• Physical Fidelity: Simulations of the Arnold–Beltrami–Childress (ABC) flow, Orszag–Tang vortex, and Hall-driven reconnection in a Harris sheet successfully capture complex physical phenomena, including magnetic reconnection, the formation of magnetic islands, flux ropes, and the characteristic quadrupolar current patterns associated with Hall–MHD dynamics. The scheme resolves these features accurately even on relatively coarse meshes.

Significance
The paper claims that this work provides a mathematically rigorous and computationally robust framework for simulating Hall–MHD systems. By combining Voigt regularisation with a divergence-preserving mixed finite element formulation, the authors overcome the dual challenges of the Hall term's nonlinearity and the strict divergence-free requirement. The method offers a physically consistent approximation that is stable, convergent, and capable of handling complex geometries, filling a gap left by previous methods that lacked rigorous error analysis or exact divergence preservation. The results suggest that this approach is a viable tool for studying astrophysical and laboratory plasma phenomena where Hall effects are dominant.