Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Numerical analysis of an explicit approximation scheme for the Landau-Lifshitz-Gilbert equation

Authors: Sören Bartels, Joy Ko and Andreas Prohl
Journal: Math. Comp. 77 (2008), 773-788
MSC (2000): Primary 65N12, 65N30, 35K55
Published electronically: October 29, 2007
MathSciNet review: 2373179
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Landau-Lifshitz-Gilbert equation describes magnetic behavior in ferromagnetic materials. Construction of numerical strategies to approximate weak solutions for this equation is made difficult by its top order nonlinearity and nonconvex constraint. In this paper, we discuss necessary scaling of numerical parameters and provide a refined convergence result for the scheme first proposed by Alouges and Jaisson (2006). As an application, we numerically study discrete finite time blowup in two dimensions.

References [Enhancements On Off] (What's this?)

  • 1. François Alouges and Pascal Jaisson.
    Convergence of a finite element discretization for the Landau-Lifshitz equations in micromagnetism.
    Math. Models Methods Appl. Sci., 16(2):299-316, 2006. MR 2210092 (2007b:65091)
  • 2. François Alouges and Alain Soyeur.
    On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness.
    Nonlinear Anal., 18(11):1071-1084, 1992. MR 1167422 (93i:35148)
  • 3. John W. Barrett, Sören Bartels, Xiaobing Feng, and Andreas Prohl.
    A convergent and constraint-preserving finite finite element method for the $ p$-harmonic flow into spheres.
    SIAM J. Numer. Anal. (accepted), 2006.
  • 4. Kung-Ching Chang, Wei Yue Ding, and Rugang Ye.
    Finite-time blowup of the heat flow of harmonic maps from surfaces.
    J. Differential Geom., 36(2):507-515, 1992. MR 1180392 (93h:58043)
  • 5. Jean-Michel Coron.
    Nonuniqueness for the heat flow of harmonic maps.
    Ann. Inst. H. Poincaré Anal. Non Linéaire, 7(4):335-344, 1990. MR 1067779 (91g:58058)
  • 6. Weinan E and Xiao-Ping Wang.
    Numerical methods for the Landau-Lifshitz equation.
    SIAM J. Numer. Anal., 38(5):1647-1665 (electronic), 2000. MR 1813249 (2002g:65112)
  • 7. Josef Fidler and Thomas Schrefl.
    Micromagnetic modelling -- the current state of the art.
    33:R135-R156, 2000.
  • 8. Boling Guo and Min-Chun Hong.
    The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps.
    Calc. Var. Partial Differential Equations, 1(1):311-334, 1993. MR 1261548 (94m:58059)
  • 9. Stephen Gustafson, Kyungkeun Kang, and Tai-Peng Tsai.
    Schrödinger flow near harmonic maps.
    Comm. Pure Appl. Math., 60(4):463-499, 2007. MR 2290708
  • 10. Joy Ko.
    The construction of a partially regular solution to the Landau-Lifshitz-Gilbert equation in $ \mathbb{R}\sp 2$.
    Nonlinearity, 18(6):2681-2714, 2005. MR 2176953 (2006h:35260)
  • 11. Martin Kruzik and Andreas Prohl.
    Recent developments in the modeling, analysis, and numerics of ferromagnetism.
    SIAM Review, 48(3):439-483, 2006. MR 2278438
  • 12. Christof Melcher.
    Logarithmic lower bounds for Néel walls.
    Calc. Var. Partial Differential Equations, 21(2):209-219, 2004. MR 2085302 (2005m:49075)
  • 13. Francesca Pistella and Vanda Valente.
    Numerical study of the appearance of singularities in ferromagnets.
    Adv. Math. Sci. Appl., 12(2):803-816, 2002. MR 1943993 (2003m:82101)
  • 14. Jalal Shatah and Chongchun Zeng.
    Schrödinger maps and anti-ferromagnetic chains.
    Comm. Math. Phys., 262(2):299-315, 2006. MR 2200262 (2006m:58043)
  • 15. Michael Struwe.
    On the evolution of harmonic mappings of Riemannian surfaces.
    Comment. Math. Helv., 60(4):558-581, 1985. MR 826871 (87e:58056)
  • 16. Xiao-Ping Wang, Carlos J. García-Cervera, and Weinan E.
    A Gauss-Seidel projection method for micromagnetics simulations.
    J. Comput. Phys., 171(1):357-372, 2001. MR 1843650 (2002f:78015)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N12, 65N30, 35K55

Retrieve articles in all journals with MSC (2000): 65N12, 65N30, 35K55

Additional Information

Sören Bartels
Affiliation: Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
Address at time of publication: Institut für Numerische Simulation, University of Bonn, Wegelerstr. 6, D-53115 Bonn, Germany

Joy Ko
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912

Andreas Prohl
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Received by editor(s): May 9, 2005
Received by editor(s) in revised form: November 23, 2006
Published electronically: October 29, 2007
Additional Notes: The first author was supported by Deutsche Forschungsgemeinschaft through the DFG Research Center Matheon “Mathematics for key technologies” in Berlin
The second author was partially supported by NSF grant DMS-0402788
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society