A finite element approximation for the stochastic Landau–Lifschitz–Gilbert equation

Ben Goldys, Kim-Ngan Le and Thanh Tran

Abstract

The stochastic Landau–Lifshitz–Gilbert (LLG) equation describes 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 \(\theta\in (1/2,1]\). Numerical results are presented to show the applicability of the method.

Keywords: stochastic partial differential equation, Landau–Lifshitz–Gilbert equation, finite element, ferromagnetism.

AMS Subject Classification: Primary 35Q40; secondary 35K55, 35R60, 60H15, 65L60, 65L20, 65C30.