C. Carstensen, D. Praetorius:

"Numerical Analysis for a Macroscopic Model in Micromagnetics";

SIAM Journal on Numerical Analysis,42(2005), 6; 2633 - 2651.

The macroscopic behaviour of stationary micromagnetic phenomena

can be modelled by a relaxed version of the Landau-Lifshitz

minimization problem. In the limit of large and soft magnets

$\Omega$, it is reasonable to exclude the exchange energy and

convexify the remaining energy densities. The numerical analysis

of the resulting minimization problem

\begin{align*}

\min E_0^{**}(\m)

\text{ amongst }\m:\Omega\to\R^d

\text{ with } |\m(x)|\le1 \text{ for a.e. }x\in\Omega,

\end{align*}

for $d=2,3$, faces difficulties caused by the pointwise

side-constraint $|\m|\le1$ and an integral over the whole

space $\R^d$ for the stray field energy. This paper involves

a penalty method to model the side-constraint and reformulates

the exterior Maxwell equation via a nonlocal integral operator

$\PP$ acting on functions exclusively defined on $\Omega$. The

discretization with piecewise constant discrete magnetizations

leads to edge-oriented boundary integrals. The implementation

of which and related numerical quadrature is discussed as well

as adaptive algorithms for automatic mesh-refinement.A~priori

and a~posteriori error estimates provide a thorough rigorous

error control of certain quantities. Three classes of numerical

experiments study the penalization, empirical convergence rates,

and the performance of the uniform and adaptive mesh-refining

algorithms.

http://www.anum.tuwien.ac.at/~dirk/download/published/ccdpr02_sinum42.pdf

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