Beniamin Goldys

Beniamin Goldys, Kim-Ngan Le, Thanh Tran

The stochastic Landau--Lifshitz--Gilbert (LLG) equation describes the behaviour of the magnetization under the influence of the effective field consisting of random fluctuations. We first reformulate the equation into an equation the unknown of which is differentiable with respect to the time variable. We then propose a convergent $θ$-linear scheme for the numerical solution of the reformulated equation. As a consequence, we show the existence of weak martingale solutions to the stochastic LLG equation. A salient feature of this scheme is that it does not involve a nonlinear system, and that no condition on time and space steps is required when $θ\in(\frac{1}{2},1]$. Numerical results are presented to show the applicability of the method.

Beniamin Goldys, Kim-Ngan Le, Thanh Tran

The stochastic Landau--Lifshitz--Gilbert (LLG) equation coupled with the Maxwell equations (the so called stochastic MLLG system) describes the creation of domain walls and vortices (fundamental objects for the novel nanostructured magnetic memories). We first reformulate the stochastic LLG equation into an equation with time-differentiable solutions. We then propose a convergent $θ$-linear scheme to approximate the solutions of the reformulated system. As a consequence, we prove convergence of the approximate solutions, with no or minor conditions on time and space steps (depending on the value of $θ$). Hence, we prove the existence of weak martingale solutions of the stochastic MLLG system. Numerical results are presented to show applicability of the method.

Beniamin Goldys, Joseph Grotowski, Kim-Ngan Le

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau--Lifshitz--Gilbert (LLG) equation with multi-dimensional noise. By using the Doss-Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent $θ$-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.

Beniamin Goldys, Agus L. Soenjaya, Thanh Tran

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.