A finite element approximation for the stochastic Maxwell–Landau–LIfschitz–Gilbert system

Beniamin Goldys, Kim-Ngan Le and Thanh Tran

Abstract

The stochastic Landau–Lifshitz–Gilbert (LLG) equation coupled with the Maxwell equations describes creation of domain walls and vortices (fundamental objects for the novel nanostructured magnetic memories). We first reformulate the stochastic LLG equation as 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 the convergence of our numerical scheme, with no or minor conditions on time and space steps (depending on the value of $\theta$) to a weak martingale solutions of the stochastic MLLG system. Numerical results are presented to show applicability of the method.

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

AMS Subject Classification: Primary 35R60; secondary 60H15, 65L60, 65L20, 82D45.