PreprintWeak martingale solutions to the stochastic LandauLifschitzGilbert equation with multidimensional noise via a convergent finiteelement schemeBeniamin Goldys, Joe Grotowski, Kim NganLeAbstractWe propose an unconditionally convergent linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert (LLG) equation with multidimensional noise. By using the DossSussmann 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. Keywords: stochastic partial differential equation, Landau–Lifshitz–Gilbert equation, finite element, ferromagnetism.AMS Subject Classification: Primary 35Q40, 35K55, 35R60, 60H15, 65L60, 65L20, 65C30; Secondary 82D45.
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