Stabilization methods in relaxed micromagnetism
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 5, pp. 995-1017.

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 ].

Classification : 65K10,  65N15,  65N30,  65N50,  73C50,  73S10
Mots clés : micromagnetics, stationary, nonstationary, microstructure, relaxation, nonconvex minimization, degenerate convexity, finite elements methods, stabilization, penalization, a priori error estimates, a posteriori error estimates
     author = {Funken, Stefan A. and Prohl, Andreas},
     title = {Stabilization methods in relaxed micromagnetism},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {995--1017},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {5},
     year = {2005},
     doi = {10.1051/m2an:2005043},
     zbl = {1079.78031},
     mrnumber = {2178570},
     language = {en},
     url = {}
Funken, Stefan A.; Prohl, Andreas. Stabilization methods in relaxed micromagnetism. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 5, pp. 995-1017. doi : 10.1051/m2an:2005043.

[1] J. Alberty, C. Carstensen and S.A. Funken, Remarks around 50 lines of Matlab: finite element implementation. Numer. Algorithms 20 (1999) 117-137. | Zbl 0938.65129

[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. | Zbl 1088.78009

[5] C. Carstensen and A. Prohl, Numerical analysis of relaxed micromagnetics by penalized finite elements. Numer. Math. 90 (2001) 65-99. | Zbl 1004.78006

[6] A. De Simone, Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal. 125 (1993) 99-143. | Zbl 0811.49030

[7] S.A. Funken and A. Prohl, On stabilized finite element methods in relaxed micromagnetism. Preprint 99-18, University of Kiel (1999).

[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. | Zbl 0572.65008

[10] M. Kružík, Maximum principle based algorithm for hysteresis in micromagnetics. Adv. Math. Sci. Appl. 13 (2003) 461-485. | Zbl 1093.82020

[11] M. Kružík and A. Prohl, Young measure approximation in micromagnetics. Numer. Math. 90 (2001) 291-307. | Zbl 0994.65078

[12] M. Kružík and A. Prohl, Macroscopic modeling of magnetic hysteresis. Adv. Math. Sci. Appl. 14 (2004) 665-681. | Zbl 1105.74034

[13] M. Kružík and A. Prohl, Recent developments in modeling, analysis and numerics of ferromagnetism. SIAM Rev. (accepted, 2005). | MR 2278438 | Zbl 1126.49040

[14] M. Kružík and T. Roubíček, Microstructure evolution model in micromagnetics. Z. Angew. Math. Phys. 55 (2004) 159-182. | Zbl 1059.82047

[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). | MR 1452107 | Zbl 0879.49017

[17] A. Prohl, Computational micromagnetism. Teubner (2001). | MR 1885923 | Zbl 0988.78001

[18] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996). | Zbl 0853.65108