Numerical analysis of relaxed micromagnetics by penalised finite elements |
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Authors: | Carsten Carstensen Andreas Prohl |
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Institution: | Mathematisches Seminar, Christian-Albrechts-Universit?t zu Kiel, Ludewig-Meyn-Strasse 4, 24098 Kiel, Germany; e-mail: {cc, apr}@numerik.uni-kiel.de, DE
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Abstract: | Summary. Some 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 pointwise. Residual-based adaptive algorithms are proposed and experimentally shown to be efficient.
Received June 24, 1999 / Revised version received August 24, 2000 / Published online May 4, 2001 |
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Keywords: | Mathematics Subject Classification (1991): 64M07 65K10 65N30 73C50 73S10 65N15 65N30 65N50 |
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