# A finite element approximation for the stochastic Landau--Lifshitz--Gilbert equation with multi-dimensional noise

## TL;DR

This paper introduces a new finite element method for solving the stochastic Landau--Lifshitz--Gilbert equation with multi-dimensional noise, ensuring convergence and enabling the proof of weak martingale solutions.

## Contribution

It develops an unconditionally convergent linear finite element scheme using the Doss-Sussmann technique for the stochastic LLG equation with multi-dimensional noise.

## Key findings

- The scheme is unconditionally convergent.
- Existence of weak martingale solutions is established.
- The method effectively handles multi-dimensional stochastic noise.

## 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.