99-9   Berichtsreihe des Mathematischen Seminars der Universität Kiel

Carsten Carstensen, Andreas Prohl:

Numerical Analysis of Relaxed Micromagnetics by Penalised Finite Elements

Micromagnetic phenomena in rigid (ferro-)magnetic materials can be modelled by a non-convex minimisation problem. Typically, minimising sequences develop finer and finer oscillations and their weak limits do not attain the infimal energy. Solutions exist in a generalised sense and the observed microstructure can be described in terms of Young measures. A relaxation by convexifying the energy density resolves the essential macroscopic information. The numerical analysis of the relaxed problem faces convex but degenerated energy functionals in a setting similar to mixed finite element formulations. The lowest order conforming finite element schemes appear instable and nonconforming finite element methods are proposed. An a~priori and a~posteriori error analysis is presented for a penalised version of the side-restriction that the modulus of the magnetic field is bounded pointwisely. Residual-based adaptive algorithms are proposed and experimentally shown to be efficient.

Mathematics Subject Classification (1991): 64M07, 65K10, 65N30, 73C50, 73S10, 65N15, 65N30, 65N50.

Keywords: micromagnetics, microstructure, relaxation, variational problems, nonconvex minimisation, degenerate problems, conforming finite elements, nonconforming finite elements, a~priori error estimates, adaptive algorithms, a~posteriori error estimates


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[Thu Feb 19 18:56:34 2009]
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