# A finite element approximation for the stochastic Maxwell--Landau--Lifshitz--Gilbert system

## TL;DR

This paper introduces a finite element method for approximating solutions to the stochastic Maxwell--Landau--Lifshitz--Gilbert system, enabling the simulation of magnetic phenomena relevant for nanostructured memories.

## Contribution

It reformulates the stochastic LLG equation for time-differentiable solutions and proposes a convergent $ heta$-linear scheme with proven convergence and weak solution existence.

## Key findings

- Convergent numerical scheme for stochastic MLLG system
- Proof of existence of weak martingale solutions
- Numerical results demonstrating method applicability

## Abstract

The stochastic Landau--Lifshitz--Gilbert (LLG) equation coupled with the Maxwell equations (the so called stochastic MLLG system) describes the creation of domain walls and vortices (fundamental objects for the novel nanostructured magnetic memories). We first reformulate the stochastic LLG equation into an equation with time-differentiable solutions. We then propose a convergent $\theta$-linear scheme to approximate the solutions of the reformulated system. As a consequence, we prove convergence of the approximate solutions, with no or minor conditions on time and space steps (depending on the value of $\theta$). Hence, we prove the existence of weak martingale solutions of the stochastic MLLG system. Numerical results are presented to show applicability of the method.