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P. Goldenits, G. Hrkac, D. Praetorius, D. Süss:
"An Effective Integrator for the Landau-Lifshitz-Gilbert Equation";
in: "ASC Report 02/2012", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2012, ISBN: 978-3-902627-05-6.

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://www.asc.tuwien.ac.at/preprint/2012/asc02x2012.pdf