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C. Carstensen, D. Praetorius:
"Effective Simulation of a Macroscopic Model in Micromagnetics";
Talk: MAFELAP 2003 Mathematics of Finite Elements and Applications, Brunel, UK; 06-21-2003 - 06-24-2003.

The large body limit in the Landau-Lifshitz equations of
micromagnetics [1] yields a macroscopic model without
exchange energy and con\-ve\-xi\-fied side conditions for the macroscopic magnetisation vectors. Its Euler Lagrange equations \pmb{$(P)$} read: Given a magnetic body $\Omega\subseteq{\mathbb R}^d$, $d=2,3$, an exterior field ${\bf f}\in L^2(\Omega)^d$ and the con\-vexi\-fied anisotropy density $\phi^{**}:{\mathbb R}^d\to{\mathbb R}_{\ge0}$, find a magnetization ${\bf m}\in L^2(\Omega)^d$ and a Lagrange multiplier $\lambda\in L^2(\Omega)$ such that a.e. in $\Omega$
\begin{equation*}
\begin{array}{l}
\nabla u+D\phi^{**}({\bf m})+\lambda{\bf m}={\bf f},\\
|{\bf m}|\le1,\quad\lambda\ge0,\\
\lambda(1-|{\bf m}|)=0.
\end{array}
\end{equation*}
The potential $u\in H^1_{\ell oc}({\mathbb R}^d)$ solves the (Maxwell) equations
\begin{equation*}
\nabla u\in L^2({\mathbb R}^d)^d\text{ and }{\rm div}(-\nabla u+{\bf m})=0
\text{ in }{\mathcal D}'({\mathbb R}^d)
\end{equation*}
in the entire space. It therefore appears natural to recast the associated far field energy into an integral operator ${\mathcal L}$ which maps ${\bf m}$ to the corresponding potential $u$.

The proposed numerical scheme involves the operator ${\mathcal L}$
and replaces pointwise side-condition $|{\bf m}|\le1$ by a penalization strategy. Given a triangulation ${\mathcal T}$, the induced space of piecewise constant functions $P_0({\mathcal T})$ on $\Omega$, and a penalization parameter $\varepsilon>0$, the discrete penalized problem \pmb{$(P_{\varepsilon,h})$}
reads: Find ${\bf m}_h\in P_0({\mathcal T})^d$ such that for all $\pmb\nu_h\in P_0({\mathcal T})^d$
\begin{equation*}
\begin{array}{l}
\langle\nabla({\mathcal L}{\bf m}_h)+D\phi^{**}({\bf m}_h)+\lambda_h{\bf m}_h
\,;\pmb\nu_h\rangle_{L^2(\Omega)}=\langle{\bf
f}\,;\pmb\nu_h\rangle_{L^2(\Omega)}
% \quad\text{for all }\pmb\nu_h\in P_0({\mathcal T})^d,\\
% \lambda_h:=\varepsilon^{-1}\frac{\max\{0,|{\bf m}_h|-1\}{|{\bf m}_h|}.
\end{array}
\end{equation*}
with $\displaystyle\lambda_h:=\varepsilon^{-1}\,\frac{\max\{0,|{\bf m}_h|-1\}}{|
{\bf m}_h|}
\in P_0({\mathcal T})$.

Numerical aspects addressed in the presentation include the
integration of the matrices with quadrature rules and hierarchical
matrices as well as a ~priori and a~posteriori error control
with a reliability-efficiency gap and adaptive mesh-design.
Surprisingly, a comparison with finite element approximations [2,3]
indicates that mesh-adaptation is not really required for
\pmb{$(P_{\varepsilon,h})$} in many numerical examples.


REFERENCES:

[1] {\sc A. DeSimone}:
{\it Energy Minimizers for Large Ferromagnetic Bodies},
Arch. Rational Mech. Anal. 125 (1993), 99-143.

[2] {\sc C. Carstensen, A. Prohl}:
{\it Numerical Analysis of Relaxed Micrmagnetics by Penalised
Finite Elements},
Numer. Math. 90 (2001), 65-99.

[3] {\sc D. Praetorius}:
{\it Analysis, Numerik und Simulation eines relaxierten Modellproblems zum Mikromagnetismus},
Doctorial thesis, Vienna University of Technology 2003.