Numerical methods for temperature dependent micromagnetism

Magnetization dynamics in ferromagnetic materials at low temperatures can be described by the Landau-Lifshitz-Gilbert (LLG) equation. The classical LLG model holds for constant temperatures that are sufficiently far from the Curie temperature. Due to increasing miniaturization of magnetic devices and recent applications, such as the thermally-assisted magnetic recording, it has become necessary to account for temperature effects in the model. We review two most common approaches currently used for the modelling of thermally activated dynamics in ferromagnetic materials. The first approach includes thermal fluctuations via a random term and leads to a stochastic mesoscopic model, the so-called Stochastic-LLG equation. The second approach includes temperature effects via an additional term in the LLG equation and leads to a deterministic macroscopic model. We propose structure preserving finite element based numerical approximations for the respective equations and show numerical experiments to demonstrate some interesting features and connections between the two models.