Weak martingale solutions to the stochastic Landau-Lifschitz-Gilbert equation with multi-dimensional noise via a convergent finite-element scheme

Beniamin Goldys, Joe Grotowski, Kim Ngan-Le

Abstract

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 $\theta$-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.

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

AMS Subject Classification: Primary 35Q40, 35K55, 35R60, 60H15, 65L60, 65L20, 65C30; Secondary 82D45.