A finite element approximation for the stochastic Maxwell--Landau--Lifshitz--Gilbert system
This work provides a rigorous numerical analysis for a stochastic model of magnetic nanostructures, addressing a known bottleneck in convergence guarantees for such systems.
The authors propose a convergent θ-linear scheme for the stochastic Maxwell-Landau-Lifshitz-Gilbert system, proving convergence with minimal step-size conditions and establishing existence of weak martingale solutions. Numerical results demonstrate applicability.
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 $θ$-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 $θ$). Hence, we prove the existence of weak martingale solutions of the stochastic MLLG system. Numerical results are presented to show applicability of the method.