Stabilization methods in relaxed micromagnetism

Stefan A. Funken; Andreas Prohl

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 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 ].

How to cite

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Funken, Stefan A., and Prohl, Andreas. "Stabilization methods in relaxed micromagnetism." ESAIM: Mathematical Modelling and Numerical Analysis 39.5 (2010): 995-1017. <http://eudml.org/doc/194296>.

@article{Funken2010,
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 ]. },
author = {Funken, Stefan A., Prohl, Andreas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Micromagnetics; stationary; nonstationary; microstructure; relaxation; nonconvex minimization; degenerate convexity; finite elements methods; stabilization; penalization; a priori error estimates; a posteriori error estimates.; micromagnetics; nonconvex minimization; stabilization; a priori error; estimates; a posteriori error estimates},
language = {eng},
month = {3},
number = {5},
pages = {995-1017},
publisher = {EDP Sciences},
title = {Stabilization methods in relaxed micromagnetism},
url = {http://eudml.org/doc/194296},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Funken, Stefan A.
AU - Prohl, Andreas
TI - Stabilization methods in relaxed micromagnetism
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
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 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.; micromagnetics; nonconvex minimization; stabilization; a priori error; estimates; a posteriori error estimates
UR - http://eudml.org/doc/194296
ER -

References

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  1. J. Alberty, C. Carstensen and S.A. Funken, Remarks around 50 lines of Matlab: finite element implementation. Numer. Algorithms20 (1999) 117–137.  
  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.  
  5. C. Carstensen and A. Prohl, Numerical analysis of relaxed micromagnetics by penalized finite elements. Numer. Math.90 (2001) 65–99.  
  6. A. De Simone, Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal.125 (1993) 99–143.  
  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.  
  10. M. Kružík, Maximum principle based algorithm for hysteresis in micromagnetics. Adv. Math. Sci. Appl.13 (2003) 461–485.  
  11. M. Kružík and A. Prohl, Young measure approximation in micromagnetics. Numer. Math.90 (2001) 291–307.  
  12. M. Kružík and A. Prohl, Macroscopic modeling of magnetic hysteresis. Adv. Math. Sci. Appl.14 (2004) 665–681.  
  13. M. Kružík and A. Prohl, Recent developments in modeling, analysis and numerics of ferromagnetism. SIAM Rev. (accepted, 2005).  
  14. M. Kružík and T. Roubíček, Microstructure evolution model in micromagnetics. Z. Angew. Math. Phys.55 (2004) 159–182.  
  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).  
  17. A. Prohl, Computational micromagnetism. Teubner (2001).  
  18. R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996).  

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