A finite element approximation for the stochastic Landau-Lifshitz-Gilbert equation

Beniamin Goldys, Kim Ngan Le, Thanh Tran

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The stochastic Landau-Lifshitz-Gilbert (LLG) equation describes the behaviour of the magnetisation under the influence of the effective field containing random fluctuations. We first transform the stochastic LLG equation into a partial differential equation with random coefficients (without the Itô term). The resulting equation has time-differentiable solutions. We then propose a convergent θ-linear scheme for the numerical solution of the reformulated equation. As a consequence, we show the existence of weak martingale solutions to the stochastic LLG equation. A salient feature of this scheme is that it does not involve solving a system of nonlinear algebraic equations, and that no condition on time and space steps is required when θ∈(1/2,1]. Numerical results are presented to show the applicability of the method.

Original languageEnglish
Pages (from-to)937-970
Number of pages34
JournalJournal of Differential Equations
Issue number2
Publication statusPublished - 1 Jan 2016
Externally publishedYes


  • Ferromagnetism
  • Finite element
  • Landau-Lifshitz-Gilbert equation
  • Primary
  • Secondary
  • Stochastic partial differential equation

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