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

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.

Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

“…For proof of the saturation condition (20), we refer to Lemma 3.15 of [8]. Repeating the same algebraic calculations as done in [15], i.e. using the definition of G and the identity (17), we obtain…”

“…We also introduce new processes m and m (n) gained by the Doss-Sussman transformation from the corresponding processes M and M (n) . We refer to [4,8,15] for further details about the properties of these processes.…”

“…To prove the equivalence of weak solution between m and M, we have to apply Itô formula, see [15]. Therefore, we present the following lemma to handle our case.…”

Strong solvability of regularized stochastic Landau–Lifshitz–Gilbert equation

“…However, from the probabilistic point of view the solution is weak in the sense that the solution and the driving Brownian motion are constructed simultaneously on some probability space and are thus interconnected. The approach, however, is also appropriate to design numerical schemes, and indeed stimulated activity in numerical analysis [3,7,6,17,21]. We are interested in the possibility of solving (1.4) in the stochastically strong sense, aiming to construct a solution for any probability space and any Brownian motion prescribed in advance.…”

Weak solutions of the Stochastic Landau–Lifshitz–Gilbert Equations with nonzero anisotrophy energy

“…There is also some research about the numerical schemes of equation (1.5), such as Baňas, Brzeźniak, and Prohl [5], Baňas, Brzeźniak, Neklyudov, and Prohl [6], Baňas, Brzeźniak, Neklyudov, and Prohl [7], Goldys, Le, and Tran [16] and Alouges, de Bouard and Hocquet [4]. The last paper differs from all previous papers as it deals with the LLGEs in the so called Gilbert form, see [15] and [3] for some related deterministic results, and with an infinite dimensional noise (correlated in space).…”