Abstract. We study two schemes for a time-fractional Fokker-Planck equation with spaceand time-dependent forcing in one space dimension. The first scheme is continuous in time and is discretized in space using a piecewise-linear Galerkin finite element method. The second is continuous in space and employs a time-stepping procedure similar to the classical implicit Euler method. We show that the space discretization is second-order accurate in the spatial L 2 -norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is O(k α ) for a uniform time step k, where α ∈ (1/2, 1) is the fractional diffusion parameter. In numerical experiments using a combined, fully-discrete method, we observe convergence behaviour consistent with these results.
The Landau-Lifshitz-Bloch (LLB) equation is a formulation of dynamic micromagnetics valid at all temperatures, treating both the transverse and longitudinal relaxation components important for high-temperature applications. We study LLB equation in case the temperature raised higher than the Curie temperature. The existence of weak solution is showed and its regularity properties are also discussed. In this way, we lay foundations for the rigorous theory of LLB equation that is currently not available.
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.
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