Numerische Mathematik

, Volume 122, Issue 1, pp 101–131 | Cite as

Numerical quadratic energy minimization bound to convex constraints in thin-film micromagnetics

  • Samuel Ferraz-Leite
  • Jens Markus Melenk
  • Dirk Praetorius
Open Access
Article

Abstract

We analyze the reduced model for thin-film devices in stationary micromagnetics proposed in DeSimone et al. (R Soc Lond Proc Ser A Math Phys Eng Sci 457(2016):2983–2991, 2001). We introduce an appropriate functional analytic framework and prove well-posedness of the model in that setting. The scheme for the numerical approximation of solutions consists of two ingredients: The energy space is discretized in a conforming way using Raviart–Thomas finite elements; the non-linear but convex side constraint is treated with a penalty method. This strategy yields a convergent sequence of approximations as discretization and penalty parameter vanish. The proof generalizes to a large class of minimization problems and is of interest beyond the scope of thin-film micromagnetics.

Mathematics Subject Classification (2010)

65K05 65K15 49M20 

Notes

Acknowledgments

S. Ferraz-Leite acknowledges a Grant of the graduate school “Differential Equations—Models in Science and Engineering”, funded by the Austrian Science Fund (FWF) under Grant W800-N05. The research of SFL and the last author DP is supported through the FWF project “Adaptive Boundary Element Method”, funded by the Austrian Science Fund (FWF) under Grant P21732.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

References

  1. 1.
    Adams, R.: Sobolev spaces. In: Pure and Applied Mathematics, vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1975)Google Scholar
  2. 2.
    Carstensen C., Prohl A.: Numerical analysis of relaxed micromagnetics by penalised finite elements. Numer. Math. 90(1), 65–99 (2001)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Carstensen C., Praetorius D.: Numerical analysis for a macroscopic model in micromagnetics. SIAM J. Numer. Anal. 42(6), 2633–2651 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Carstensen C., Praetorius D.: Effective simulation of a macroscopic model for stationary micromagnetics. Comput. Methods Appl. Mech. Eng. 194(2–5), 531–548 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Dacorogna, B.: Direct methods in the calculus of variations. In: Applied Mathematical Sciences, vol. 78. Springer, Berlin (1989)Google Scholar
  6. 6.
    Deny J., Lions J.L.: Les espaces du type de Beppo Levi. Ann. Inst. Fourier, Grenoble 5, 305–370 (1954)MathSciNetCrossRefGoogle Scholar
  7. 7.
    DeSimone A., Kohn R., Müller S., Otto F., Schäfer R.: Two-dimensional modelling of soft ferromagnetic films. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457(2016), 2983–2991 (2001)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    DeSimone A., Kohn R., Müller S., Otto F.: A reduced theory for thin-film micromagnetics. Commun. Pure Appl. Math. 55(11), 1408–1460 (2002)MATHCrossRefGoogle Scholar
  9. 9.
    DeSimone A., Kohn R., Müller S., Otto F.: Recent analytical developments in micromagnetics. In: Bertotti, G., Mayergoyz, I.D. (eds) The science of hysteresis, vol. II, pp. 269–381. Elsevier/Academic Press, Amsterdam (2006)Google Scholar
  10. 10.
    Drwenski, J.: Numerical methods for a reduced model in thin-film micromagnetics. Dissertation, Rheinische Friedrichs-Wilhelms-Universität Bonn (2008)Google Scholar
  11. 11.
    Ferraz-Leite, S.: Quadratic minimization with non-local operators and non-linear constraints. Dissertation, Vienna University of Technology (2011)Google Scholar
  12. 12.
    Ferraz-Leite, S., Melenk, J.M., Praetorius, D.: Reduced Model in Thin-Film Micromagnetics, Proceedings MATHMOD 09 Vienna, I. Troch, F. Breitenecker (eds.) Argesim/Asim, ARGESIM Report no. 35, 2009Google Scholar
  13. 13.
    Girault, V., Raviart, P.: Finite element methods for Navier–Stokes equations. In: Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)Google Scholar
  14. 14.
    Hackbusch W.: A sparse matrix arithmetic based on H-matrices. Part I: introduction to H-matrices. Computing 62, 89–108 (1999)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE constraints. In: Mathematical Modelling: Theory and Applications, vol. 23. Springer, New York (2009)Google Scholar
  16. 16.
    Ito, K., Kunisch, K.: Lagrange Multiplier approach to variational problems and applications. In: Advances in Design and Control, vol. 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)Google Scholar
  17. 17.
    Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. In: Pure and Applied Mathematics, vol. 88. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1980)Google Scholar
  18. 18.
    Landau L.D., Lifschits E.M.: On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Zeitsch. der Sow. 8, 153–169 (1935)MATHGoogle Scholar
  19. 19.
    Maischak M., Stephan E.P.: A priori error estimates for hp penalty BEM for contact problems in elasticity. Comput. Methods Appl. Mech. Eng. 196(37–40), 3871–3880 (2007)MathSciNetMATHGoogle Scholar
  20. 20.
    McLean W.: Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  21. 21.
    Nocedal, J., Wright, S.: Numerical optimization. In: Springer Series in Operations Research. Springer, New York (1999)Google Scholar
  22. 22.
    Praetorius, D.: Analysis, Numerik und Simulation eines relaxierten Modellproblems zum Mikromagnetismus. Dissertation, Vienna University of Technology (2003)Google Scholar
  23. 23.
    Raviart P.-A., Thomas J.M.: Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp. 31(138), 391–413 (1977)MathSciNetMATHGoogle Scholar
  24. 24.
    Sauter S., Schwab C.: Boundary Element Methods. Springer Verlag, Berlin (2011)MATHCrossRefGoogle Scholar
  25. 25.
    Stephan E.P.: Boundary integral equations for screen problems in \({\mathbb{R}^3}\) . Integral Equ. Oper. Theory 10(2), 236–257 (1987)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  • Samuel Ferraz-Leite
    • 1
  • Jens Markus Melenk
    • 2
  • Dirk Praetorius
    • 2
  1. 1.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Vienna University of TechnologyViennaAustria

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