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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.

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\\
|\mathbf{m}|&=1\quad\text{a.e.\ in }(0,\tau)\times\Omega,
\end{split}
\end{align*}
where $\mathbf{H}(\mathbf{m})$ denotes the total magnetic field, including exchange energy, anisotropy energy, stray-field energy, as well as Zeeman energy.

In our contribution, we generalize the approach of~[\textsc{Alouges}, Discrete Contin. Dyn. Syst. Ser. S, 1 (2008)] to the total magnetic field, i.e.\ including all four energy terms. Since the computation of the non-local demagnetization field is the most time and memory consuming part of the simulation, the proposed time integrator is split into an implicit part and an explicit part. The first one deals with the higher-order term stemming from the exchange energy, whereas the lower-order terms are treated explicitly.
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 $\mathcal{P}_h$.

FEM, micromagnetism, LLG equation, convergence