A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation.(English)Zbl 1298.65012

The Landau-Lifshitz-Gilbert (LLG) equation describes the magnetization of a ferromagnetic material occupying a bounded region $$D\subset\mathbb{R} ^{n},\;n=2$$ or $$3$$. The stochastic version of the LLG equation with respect to the magnetization $$\mathbf{M}:$$ $D_{T}:=(0,T)\times D\rightarrow\mathbb{S}^{2}:=\{\mathbf{y\in}\mathbb{R} ^{3}:\left| \mathbf{y}\right| =1\}$ has the form \begin{aligned} d\mathbf{M}(t,\mathbf{x}) & =-\alpha\mathbf{M}(t,\mathbf{x})\times (\mathbf{M}(t,\mathbf{x})\times\Delta\mathbf{M}(t,\mathbf{x}))dt + \mathbf{M}(t,\mathbf{x})\times\Delta\mathbf{M}(t,\mathbf{x})dt\\ &+\mathbf{M}(t,\mathbf{x})\times\;\circ d\mathbf{W}(t,\mathbf{x} ),\;(t,\mathbf{x})\in D_{T},\\ \partial_{\mathbf{n}}\mathbf{M}(t,\mathbf{x}) & =\mathbf{0,\;}(t,\mathbf{x} )\in \partial D_{T},\\ \mathbf{M}(0,\mathbf{x}) & =\mathbf{M}_{0}(\mathbf{x}),\;\mathbf{x}\in D. \end{aligned} This equation is a Stratonovich nonlinear stochastic partial differential equation, $$\mathbf{W}=\{\mathbf{W}(t,\cdot)$$, $$t\in[0,T]\}$$ is a Wiener process with values in a Hilbert space. The authors propose a convergent finite-element-based discretization of the stochastic LLG equation.

MSC:

 65C30 Numerical solutions to stochastic differential and integral equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 78A30 Electro- and magnetostatics
Full Text: