M. Aurada, J. Melenk, D. Praetorius:

"Mixed conforming elements for the large-body limit in micromagnetics (MATHMOD 09)";

in: "ASC Report 49/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

The macroscopic behavior of stationary micromagnetic phenomena can be

modeled by a relaxed version of the Landau-Lifshitz minimization problem first introduced

by DeSimone in 1993, [5], where the magnetic potential u is linked to the

magnetization m through the magnetostatic Maxwell equation. In the case of large and

soft magnets

, one neglects the exchange energy and convexifies the remaining energy

densities. In our discretization we enforce the pointwise side constraint |m| ≤ 1 by a

penalization strategy and we replace the entire space Rd in the energy functional E and

in the potential equation by a bounded Lipschitz domain b

containing

. Since conforming

elements appear to be unstable for the pure Galerkin discretization (cf. [3]) we

append to the Galerkin discretization a consistent stabilization term. We reformulate

the minimization problem in terms of the augmented Lagrangean, which leads us to a

so called saddle-point formulation for the corresponding Euler-Lagrange equations.

In this paper we discuss the well-posedness of the discrete problem and the a priori

error analysis. Furthermore we introduce a residual-based a posteriori error estimator

and comment on adaptive strategies

http://www.asc.tuwien.ac.at/preprint/2009/asc49x2009.pdf

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