Weak martingale solutions to the stochastic Landau–Lifshitz–Gilbert equation with multi-dimensional noise via a convergent finite-element scheme

Beniamin Goldys, Joseph F. Grotowski, Kim Ngan Le

Research output: Contribution to journalArticleResearchpeer-review

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 θ-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.

Original languageEnglish
Pages (from-to)232-261
Number of pages30
JournalStochastic Processes and their Applications
Volume130
Issue number1
DOIs
Publication statusPublished - 1 Jan 2020
Externally publishedYes

Keywords

  • Ferromagnetism
  • Finite element
  • Landau–Lifshitz–Gilbert equation
  • Stochastic partial differential equation

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