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

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