Abstract

Thermally activated magnetization dynamics is modelled by the stochastic Landau-Lifshitz-Gilbert equation (SLLG). A finite element based space-time discretization is proposed, where iterates conserve the unit-length constraint at nodal points of the mesh, satisfy an energy inequality, and construct weak martingale solutions of the limiting problem for vanishing discretization parameters.

Then, we study long-time dynamics of the space discretization of SLLG . The system is shown to relax exponentially fast to the unique invariant measure (Boltzmann), as well as the convergent space-time discretization.

Computational results for SLLG will be discussed to evidence the role of noise, including avoidance of finite time blow-up behavior of solutions of the related deterministic problem, and the study of long-time dynamics.

This is joint work with L. Banas (Edinburgh), Z. Brzezniak (York), and M. Neklyudov (Tuebingen).