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L. Banas, M. Page, D. Praetorius:
"A general integrator for the LLG equation";
Talk: 9th Austrian Numerical Analysis Day, Graz; 04-11-2013 - 04-12-2013.

The understanding of magnetization dynamics, especially on a microscale, is of utter relevance, for example in the development of magnetic sensors, recording heads, and magneto-resistive storage devices. In the literature, it is well-accepted that dynamic micromagnetic phenomena are modeled best by the Landau-Lifshitz-Gilbert equation (LLG) which describes the behaviour of the magnetization under the influence of some effective field that may consist of several contributions such as the microcrystalline anisotropy or the demagnetization field. Numerical challenges for the time integration arise from strong nonlinearities, a non-convex modulus constraint and possible non-local field contributions.

Recently there has been a huge progress in the mathematical literature for weak solvers to LLG. In [Alouges, (2008)], the author introduced an integrator that requires to solve only one linear system per timestep and still guarantees unconditional convergence towards a weak solution of LLG. While this work was done for an effective field with exchange energy only, in [Alouges (2011)] and [Goldenits (2012)], the analysis was extended to cover a more general, however, linear effective field.

In our contribution, we extend the above approach to show the full potential of this solver. By exploiting an abstract framework, we can cover general field contributions that might be nonlinear, non-local, and/or time-dependent. Applications include multiscale modeling, coupling of LLG to the full Maxwell's equations, or even to the conservation of momentum equation to include magnetostrictive effects. Even in this general setting, we can still prove unconditional convergence while sustaining very little computational effort.

llg, maxwell