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

The stochastic Landau-Lifshitz-Gilbert (LLG) equation describes the behaviour of the magnetisation under the influence of the effective field containing random fluctuations. We first transform the stochastic LLG equation into a partial differential equation with random coefficients (without the Itô term). The resulting equation has time-differentiable solutions. 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 solving a system of nonlinear algebraic equations, and that no condition on time and space steps is required when θ∈(1/2,1]. Numerical results are presented to show the applicability of the method.