Error analysis of a divergence-preserving mixed finite element scheme for the incompressible Hall--magnetohydrodynamic equations
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| Format: | Preprint |
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2026
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| author | Goldys, Beniamin Soenjaya, Agus L. Tran, Thanh |
| author_facet | Goldys, Beniamin Soenjaya, Agus L. Tran, Thanh |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_01764 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Error analysis of a divergence-preserving mixed finite element scheme for the incompressible Hall--magnetohydrodynamic equations Goldys, Beniamin Soenjaya, Agus L. Tran, Thanh Numerical Analysis 65M12, 65M60, 76M10, 76W05 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. |
| title | Error analysis of a divergence-preserving mixed finite element scheme for the incompressible Hall--magnetohydrodynamic equations |
| topic | Numerical Analysis 65M12, 65M60, 76M10, 76W05 |
| url | https://arxiv.org/abs/2605.01764 |