Goldys, Beniamin; Grotowski, Joseph F.; Le, Kim-Ngan Weak martingale solutions to the stochastic Landau-Lifshitz-Gilbert equation with multi-dimensional noise via a convergent finite-element scheme. (English) Zbl 1433.35380 Stochastic Processes Appl. 130, No. 1, 232-261 (2020). Summary: 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 \(\theta \)-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. Cited in 4 Documents MSC: 35Q60 PDEs in connection with optics and electromagnetic theory 35K55 Nonlinear parabolic equations 35R60 PDEs with randomness, stochastic partial differential equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65C30 Numerical solutions to stochastic differential and integral equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35D30 Weak solutions to PDEs 82D40 Statistical mechanics of magnetic materials 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35R09 Integro-partial differential equations 78A25 Electromagnetic theory (general) Keywords:stochastic partial differential equation; Landau-Lifshitz-Gilbert equation; finite element; ferromagnetism PDFBibTeX XMLCite \textit{B. 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