The stochastic Landau-Lifshitz-Gilbert equation describes the thermally induced dynamics of magnetic moments in ferromagnetic materials. Solutions of this highly nonlinear stochastic PDE are unit vector fields and satisfy an energy estimate. These are crucial properties to construct a convergent discretization in space and time. We propose a convergent finite element approximation of the problem based on the midpoint rule. The numerical scheme preserves the underlying properties of the continuous problem. Further, we construct a robust and efficient Newton-multigrid solver for the solution of the nonlinear systems associated with the discretized problems at each time level. Computational studies show the optimal convergence behavior of the scheme in the case of smooth solutions. Long-time dynamics for finite ensembles of spins evidence the ergodicity of an invariant measure of the continuum model. Numerical experiments in two dimensions demonstrate pathwise finite time blow-up behavior of the solution which goes together with a smooth evolution of expectations. Finally, thermally activated switching of spherical magnetic nanoparticles in three dimensions with space-time white noise is examined for a range of physical parameters.
- stochastic partial differential equations
- finite element methods