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P. Goldenits, G. Hrkac, D. Praetorius, D. Süss:
"An effective integrator for the Landau-Lifshitz-Gilbert equation";
Talk: MATHMOD 2012 - 7th Vienna Conference on Mathematical Modelling, Wien (invited); 02-14-2012 - 02-17-2012; in: "7th Vienna International Conference on Mathematical Modelling", International Federation of Automatic Control, Mathematical Modelling, Volume 7, Part 1 (2012), ISBN: 978-3-902823-23-6; 493 - 497.

We consider a lowest-order finite element scheme for the
Landau-Lifshitz-Gilbert equation (LLG) which describes the dynamics
of micromagnetism. In contrast to previous works from the mathematics
literature, we examine LLG including the total magnetic field induced
by physical phenomena described in terms of exchange energy,
anisotropy energy, magnetostatic energy, as well as Zeeman energy.
Besides a strong non-linearity and a non-convex side constraint, the
non-local dependence of the demagnetization field from the
magnetization represents a challenging task for the numerical
integrator. In our numerical scheme, only the highest order term,
namely the exchange contribution, is treated implicitly, whereas the
remaining contributions are computed explicitly. This is, in
particular, advantageous for the computation of the demagnetization
field by means of the popular approach of Fredkin et al. (1990).
Furthermore, our scheme requires to solve only one linear system per
time-step and allows a simplified computation of the arising system
matrices by mass-lumping. Finally, the proposed integrator is
mathematically reliable in the sense that we prove unconditional
convergence for the approximation of a weak solution.

http://dx.doi.org/10.3182/20120215-3-AT-3016.00086