A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation

Zdzislaw Brzezniak; Andreas Prohl; Lubomír  Banas; Mikhail Neklyudov

doi:10.1093/imanum/drt020

A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation

Zdzislaw Brzezniak, Andreas Prohl, Lubomír Banas, Mikhail Neklyudov

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We propose a convergent finite-element-based discretization of the stochastic Landau–Lifshitz–Gilbert equation. The main difficulties in the convergence analysis for this nonlinear stochastic partial differential equation are to properly address the pointwise sphere condition and, in the fully discrete scheme, the Stratonovich noise. Approximations of the scheme proposed here satisfy a sphere constraint at nodal points of the spatial discretization and have finite energies, and their increments may be controlled uniformly with respect to discretization parameters. Sequences of corresponding continuified processes may then be generated which construct weak martingale solutions of the limiting equations.

Original language	English
Pages (from-to)	502-549
Number of pages	48
Journal	IMA Journal of Numerical Analysis
Volume	34
Issue number	2
Early online date	30 Jul 2013
DOIs	https://doi.org/10.1093/imanum/drt020
Publication status	Published - Apr 2014

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10.1093/imanum/drt020

Cite this

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abstract = "We propose a convergent finite-element-based discretization of the stochastic Landau–Lifshitz–Gilbert equation. The main difficulties in the convergence analysis for this nonlinear stochastic partial differential equation are to properly address the pointwise sphere condition and, in the fully discrete scheme, the Stratonovich noise. Approximations of the scheme proposed here satisfy a sphere constraint at nodal points of the spatial discretization and have finite energies, and their increments may be controlled uniformly with respect to discretization parameters. Sequences of corresponding continuified processes may then be generated which construct weak martingale solutions of the limiting equations. ",

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AB - We propose a convergent finite-element-based discretization of the stochastic Landau–Lifshitz–Gilbert equation. The main difficulties in the convergence analysis for this nonlinear stochastic partial differential equation are to properly address the pointwise sphere condition and, in the fully discrete scheme, the Stratonovich noise. Approximations of the scheme proposed here satisfy a sphere constraint at nodal points of the spatial discretization and have finite energies, and their increments may be controlled uniformly with respect to discretization parameters. Sequences of corresponding continuified processes may then be generated which construct weak martingale solutions of the limiting equations.

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