Stabilization methods in relaxed micromagnetism

Stefan A. Funken; Andreas Prohl

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

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

Abstract

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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 P 1 - ( P 0 ) 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 ].

How to cite

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Funken, 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

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  13. [13] M. Kružík and A. Prohl, Recent developments in modeling, analysis and numerics of ferromagnetism. SIAM Rev. (accepted, 2005). Zbl1126.49040MR2278438
  14. [14] M. Kružík and T. Roubíček, Microstructure evolution model in micromagnetics. Z. Angew. Math. Phys. 55 (2004) 159–182. Zbl1059.82047
  15. [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. [16] P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser (1997). Zbl0879.49017MR1452107
  17. [17] A. Prohl, Computational micromagnetism. Teubner (2001). Zbl0988.78001MR1885923
  18. [18] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996). Zbl0853.65108

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