A finite element approximation for the stochastic Landau--Lifshitz--Gilbert equation with multi-dimensional noise
This work provides a numerical method and theoretical foundation for simulating magnetization dynamics under thermal fluctuations, which is important for micromagnetics and spintronics.
The paper proposes an unconditionally convergent linear finite element scheme for the stochastic Landau-Lifshitz-Gilbert equation with multi-dimensional noise, and proves the existence of weak martingale solutions.
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.