A finite element approximation for the stochastic Landau-Lifshitz-Gilbert equation

TL;DR

This paper introduces a convergent finite element scheme for the stochastic Landau-Lifshitz-Gilbert equation, reformulating it for differentiability, proving solution existence, and demonstrating numerical effectiveness without nonlinear systems or step restrictions.

Contribution

It proposes a novel $ heta$-linear scheme for the stochastic LLG equation that is convergent, avoids nonlinear systems, and requires no step size restrictions for certain parameter ranges.

Findings

The scheme is convergent and does not involve nonlinear systems.

Existence of weak martingale solutions is established.

Numerical results confirm the method's applicability.

Abstract

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 $\theta$-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 $\theta\in(\frac{1}{2},1]$. Numerical results are presented to show the applicability of the method.