# Stabilization methods in relaxed micromagnetism

Stefan A. Funken; Andreas Prohl

- Volume: 39, Issue: 5, page 995-1017
- ISSN: 0764-583X

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topFunken, Stefan A., and Prohl, Andreas. "Stabilization methods in relaxed micromagnetism." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.5 (2005): 995-1017. <http://eudml.org/doc/245087>.

@article{Funken2005,

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 $\{\bf m\}$. 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 ].},

author = {Funken, Stefan A., Prohl, Andreas},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {micromagnetics; stationary; nonstationary; microstructure; relaxation; nonconvex minimization; degenerate convexity; finite elements methods; stabilization; penalization; a priori error estimates; a posteriori error estimates; a priori error; estimates},

language = {eng},

number = {5},

pages = {995-1017},

publisher = {EDP-Sciences},

title = {Stabilization methods in relaxed micromagnetism},

url = {http://eudml.org/doc/245087},

volume = {39},

year = {2005},

}

TY - JOUR

AU - Funken, Stefan A.

AU - Prohl, Andreas

TI - Stabilization methods in relaxed micromagnetism

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2005

PB - EDP-Sciences

VL - 39

IS - 5

SP - 995

EP - 1017

AB - 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 ${\bf m}$. 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 ].

LA - eng

KW - micromagnetics; stationary; nonstationary; microstructure; relaxation; nonconvex minimization; degenerate convexity; finite elements methods; stabilization; penalization; a priori error estimates; a posteriori error estimates; a priori error; estimates

UR - http://eudml.org/doc/245087

ER -

## References

top- [1] J. Alberty, C. Carstensen and S.A. Funken, Remarks around 50 lines of Matlab: finite element implementation. Numer. Algorithms 20 (1999) 117–137. Zbl0938.65129
- [2] W.F. Brown, Micromagnetics. Interscience, New York (1963).
- [3] C. Carstensen and S. Funken, Adaptive coupling of penalised finite element methods and boundary element methods for relaxed micromagnetics. In preparation.
- [4] C. Carstensen and D. Praetorius, Numerical analysis for a macroscopic model in micromagnetics. SIAM J. Numer. Anal. 42 (2005) 2633–2651, electronic. Zbl1088.78009
- [5] C. Carstensen and A. Prohl, Numerical analysis of relaxed micromagnetics by penalized finite elements. Numer. Math. 90 (2001) 65–99. Zbl1004.78006
- [6] A. De Simone, Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal. 125 (1993) 99–143. Zbl0811.49030
- [7] S.A. Funken and A. Prohl, On stabilized finite element methods in relaxed micromagnetism. Preprint 99-18, University of Kiel (1999). Zbl1079.78031
- [8] A. Hubert and R. Schäfer, Magnetic Domains. Springer (1998).
- [9] P. Keast, Moderate-degree tetrahedral quadrature formulas. Comput. Methods Appl. Mech. Engrg. 55 (1986) 339–348. Zbl0572.65008
- [10] M. Kružík, Maximum principle based algorithm for hysteresis in micromagnetics. Adv. Math. Sci. Appl. 13 (2003) 461–485. Zbl1093.82020
- [11] M. Kružík and A. Prohl, Young measure approximation in micromagnetics. Numer. Math. 90 (2001) 291–307. Zbl0994.65078
- [12] M. Kružík and A. Prohl, Macroscopic modeling of magnetic hysteresis. Adv. Math. Sci. Appl. 14 (2004) 665–681. Zbl1105.74034
- [13] M. Kružík and A. Prohl, Recent developments in modeling, analysis and numerics of ferromagnetism. SIAM Rev. (accepted, 2005). Zbl1126.49040MR2278438
- [14] M. Kružík and T. Roubíček, Microstructure evolution model in micromagnetics. Z. Angew. Math. Phys. 55 (2004) 159–182. Zbl1059.82047
- [15] M. Kružík and T. Roubíček, Interactions between demagnetizing field and minor-loop development in bulk ferromagnets. J. Magn. Magn. Mater. 277 (2004) 192–200.
- [16] P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser (1997). Zbl0879.49017MR1452107
- [17] A. Prohl, Computational micromagnetism. Teubner (2001). Zbl0988.78001MR1885923
- [18] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996). Zbl0853.65108

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