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Talks and Poster Presentations (without Proceedings-Entry):

P. Goldenits, D. Praetorius, D. Süss:
"Convergent Geometric Integrator for the Landau-Lifshitz-Gilbert equation in Micromagnetics";
Talk: 7th Austrian Numerical Analysis Day, Klagenfurt; 05-05-2011 - 05-06-2011.

English abstract:
The understanding and development of magnetic materials is of utter relevance for example in magneto-resistive storage devices. In the literature it is well-accepted that dynamic micromagnetic phenomena are described best by the Landau-Lifshitz-Gilbert equation, i.e.
\begin{align*}
\begin{split}
\mathbf{m}_t &= \frac{-1}{1+\alpha^2}\mathbf{m}\times\mathbf{H}(\mathbf{m}) -
\frac{\alpha}{1 + \alpha^2}\mathbf{m}\times(\mathbf{m}\times\mathbf{H}(\mathbf{m}))\\
\mathbf{m}(0) &= \mathbf{m}_0 \quad\text{in } H^1(\Omega;\mathbb{S}^2)\\
\partial_n \mathbf{m} &= 0 \quad\text{on } (0,\tau)\times\partial\Omega\\
where $\mathbf{H}(\mathbf{m})$ denotes the total magnetic field, including exchange energy, anisotropy energy, stray-field energy, as well as Zeeman energy.
As the original algorithm, our extension guarantees the non-convex side constraint $|\mathbf{m}|=1$ to be fulfilled as well as unconditional convergence. In contrast to previous works, another benefit of our scheme is that only one linear system per time-step has to be solved. Finally, our analysis allows to replace the operator $\mathcal{P}$ which maps $\mathbf{m}$ onto the corresponding demagnetization field, by a discrete operator $\mathcal{P}_h$.