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Stabilization methods in relaxed micromagnetism

Published online by Cambridge University Press:  15 September 2005

Stefan A. Funken  and
Andreas Prohl
Stefan A. Funken
Affiliation:
Department of Numerical Analysis, University of Ulm, 89069 Ulm, Germany. stefan.funken@uni-ulm.de
Andreas Prohl
Affiliation:
Department of Mathematics, ETHZ, 8092 Zürich, Switzerland.
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Abstract

The magnetization of a ferromagnetic sample solves a non-convex variational problem, where its relaxation by convexifying the energy density resolves relevant macroscopic information. The numerical analysis of the relaxed model has to deal with a constrained convex but degenerated, nonlocal energy functional in mixed formulation for magnetic potential u and magnetization m. In [C. Carstensen and A. Prohl, Numer. Math.90 (2001) 65–99], the conforming P1 - (P0)d-element in d=2,3 spatial dimensions is shown to lead to an ill-posed discrete problem in relaxed micromagnetism, and suboptimal convergence. This observation motivated a non-conforming finite element method which leads to a well-posed discrete problem, with solutions converging at optimal rate. In this work, we provide both an a priori and a posteriori error analysis for two stabilized conforming methods which account for inter-element jumps of the piecewise constant magnetization. Both methods converge at optimal rate; the new approach is applied to a macroscopic nonstationary ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ].

Keywords

Micromagneticsstationarynonstationarymicrostructurerelaxationnonconvex minimizationdegenerate convexityfinite elements methodsstabilizationpenalizationa priori error estimatesa posteriori error estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 5 , September 2005 , pp. 995 - 1017
Copyright
© EDP Sciences, SMAI, 2005

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References

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