Abstract
The Landau-Lifshitz (LL)equation of micromagnetism governs rich variety of the evolution of magnetization patterns in ferromagnetic media. This is due to the complexity of physical quantities appearing in the LL equation. This complexity causes also an interesting mathematical properties of the LL equation: nonlocal character for some quantities,nonconvex side-constraints, strongly nonlinear terms. These effects influence also the numerical approximations. In this work, recent developments on the approximation of weak solutions, together with the overview of well-known methods for strong solutions,are addressed.
Similar content being viewed by others
References
Kružík, M. and Prohl, A.(2006) Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism.SIAM Review,48, 439–483.
Bertotti, G. (1998)Hysteresis in Magnetism. Academic Press.
Landau, L. and Lifshitz, E. (1935) On the theory of the dispersion of magnetic permeability in ferromagnetic bodies.Phys. Z. Sowjetunion,8, 153–169.
Gilbert, T. (1955) A Lagrangian formulation of gyromagnetic equation of the magnetization field.Phys. Rev.,100, 1243–1255.
d’Aquino, M., Serpico, C., and Miano, G. (2005) Geometrical integration of Landau-Lifshitz-Gilbert equation based on the mid-point rule.Journal of Computational Physics,209, 730–753.
Prohl, A. (2001)Computational Micromagnetism. Advances in Numerical Mathematics, B.G. Teubner.
Cimrák, I. (2005) Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation with an exchange field.IMA J. Numer. Anal.,25, 611–634.
Podio-Guidugli, P. (2001) On dissipation mechanisms in micromagnetics.The European Physical Journal B,19, 417–424.
Monk, P. and Vacus, O. (2001) Accurate discretization of a non-linear micromagnetic problem.Comput. Methods Appl. Mech. Engrg.,190, 5243–5269.
Baňas, L., Bartels, S., and Prohl, A. A convergent implicit discretization of the Maxwell-Landau-Lifshitz-Gilbert equation.Preprint,http://na.unituebingen.de/pub/prohl/papers/mllg.pdf.
Bartels, S. and Prohl, A. (2006) Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation.SIAM J. Numer. Anal.,44, 1405–1419.
Brown, W. (1963) Thermal fluctuations of a single domain particle.Phys. Rev.,130, 1677–1686.
Kubo, R., Toda, M., and Hashitsume, N. (1991)Nonequilibrium statistical mechanics, vol.2 of Statistical Physics. Springer.
Tsiantos, V.D., Suess, D., Scholz, W., Schreft, T., and Fidler, J. Effect of spatial correlation length in Langevin micromagnetic simulations.J. Magn. Magn. Mater., in press.
Garcia-Palacios, L. and Lazaro, F. J. (1998) Langevin-dynamics study of the dynamical properties of small magnetic particles.Phys. Rev. B,58, 14937–14958.
Scholz, W., Schrefl, T., and Fidler, J. (2001) Micromagnetic simulation of thermally activated switching in fine particles.J. Magn. Magn. Mater.,233, 296–304.
Cheng, X.Z., Jalil, M. B. A., Lee, H.K., and Okabe, Y. (2006) Mapping the Monte-Carlo Scheme to Langevin Dynamics: A Fokker-Planck Approach.Phys. Rev. Lett.,96, 067208.
Baňas, L., Prohl, A., and Slodiĉka, M. Modeling od thermally assisted magnetodynamics. Preprint, http://na.unituebingen.de/pub/prohl/papers/thermagA5.pdf.
Cimrák, I. and Melicher, V. Determination of precession and dissipation parameters in the micromagnetics. Preprint.
Cimrák, I. and Melicher, V. (2007) Sensitivity analysis framework for micromagnetism with application to optimal shape design of MRAM memories.Inverse Problems,23, 563–588.
Cimrák, I. and Melicher, V. (2006) The Landau-Lifshitz model for shape optimization of MRAM memories.PAMM 6, 23–26.
Tai, X. and Chan, T. (2004) A survey on multipole level set methods with applications for identifying piecewise constant functions.International Journal of Numerical Analysis and Modeling,1, 25–47.
Melicher, V., Cimrák, I., and Van Keer, R. Level Set Method for optimal shape design of MRAM core. Micromagnetic approach.Physica B, accepted.
Carbou, G. and Fabrie, P. (2001) Regular solutions for Landau-Lifshitz Equation in a bounded domain.Differential Integral Equations,14, 213–229.
Guo, B. and Ding, S. (2001) Neumann problem for the Landau-Lifshitz-Maxwell system in two dimensions.Chin. Ann.Math., Ser. B,22, 529–540.
Lewis, D. and Nigam, N. (2003) Geometric integration on spheres and some interesting applications.J. Comput. Appl. Math.,151, 141–170.
E, W. and Wang, X.-P (2000) Numerical methods for the Landau-Lifshitz equation.SIAM J. Numer. Anal.,38, 1647–1665.
Krishnaprasad, P. and Tan, X. (2001) Cayley transforms in micromagnetics.Physica B,306, 195–199.
Lewis, D. and Nigam, N. (2000), A geometric integration algorithm with applications to micromagnetics. Technical Report 1721, IMA preprint series.
Monk, P. and Vacus, O. (1999) Error estimates for a numerical scheme for ferromagnetic problems.SIAM J. Numer. Anal.,36, 696–718.
Serpico, C., Mayergoyz, I., and Bertotti, G. (2001) Numerical technique for integration of the Landau-Lifshitz equation.J. of Appl.Phys.,89, 6991–6993.
Cimrák, I. (2007) Error analysis of numerical scheme for 3d Maxwell-Landau-Lifshitz system. Mathematical Methods in the Applied Sciences, accepted for publication.
Cimrák, I. Existence, regularity and local uniqueness of the solutions to the Maxwell-Landau-Lifshitz system in three dimensions. Journal of Mathematical Analysis and Applications, to appear.
Cimrák, I.(2007), Regularity properties of the solutions to the 3d Maxwell-Landau-Lifshitz system in weighted sobolev spaces. doi: 10.1016/j.cam.2006.03.044.
Alouges, F. and Soyeur, A. (1992) On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness.Nonlinear Anal.,18, 1071–1094.
Guo, B. and Hong, M. (1993) The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps.Calc. Var.,1, 311–334.
Slodička, M. and Cimrák, I. (2003) Numerical study of nonlinear ferromagnetic materials.Appl. Numer. Math.,46, 95–111.
Cimrák, I. and Slodička, M. (2004) An iterative approximation scheme for the Landau-Lifshitz-Gilbert equation.J. Comput. Appl. Math.,169, 17–32.
Cimrák, I. and Slodička, M. (2004) Optimal convergence rate for Maxwell-Landau-Lifshitz system.Physica B,343, 236–240.
Slodička, M. and Baňas, L. (2004) A numerical scheme for a Maxwell-Landau-Lifshitz-Gilbert System.Appl. Math. Comput.,158, 79–99.
Slodička, M. and Cimrák, I. (2004) Improved error estimates for a Maxwell-Landau-Lifschitz system.PAMM,4, 71–74.
Alouges, F. and Jaisson, P. (2006) Convergence of a finite element discretization for the Landau-Lifshitz equations in micromagnetism.Math. Models Methods Appl. Sci.,16, 299–316.
Bartels, S., Ko, J., and Prohl, A. Numerical approximation of the Landau-Lifshitz-Gilbert equation and finite time blow-up of weak solutions. Preprint.
Barret, J., Bartels, X., S. Feng, and Prohl, A. A convergent and constraint-preserving finite element method for the p-harmonic flow into spheres. SIAM J. Numer. Anal., accepted.
Bartels, S. (2006) Constraint preserving, inexact solution of implicit discretizations of Landau-Lifshitz-Gilbert equations and consequences for convergence.PAMM,6, 19–22.
Bartels, S. and Prohl, A. Fully practical, constraint preserving, implicit approximation of harmonic map heat flow into spheres. Preprint.
Bartels, S. (2004) Stability and convergence of finite element approximation schemes for harmonic maps.SIAM J. Numer. Anal.,43, 220–238.
Bartels, S., Prohl, A., and Lubich, C. Convergent discretization of heat and wave map flows to spheres using approximate discrete lagrange multipliers. Preprint.
Bartels, S. and Prohl, A. Convergence of an implicit, constraint preserving finite element discretization of p-harmonic heat flow into spheres. Preprint.
Serpico, C., Mayergoyz, I., Bertotti, G., d’Aquino, M., and Bonin, R. (2007) Generalized Landau-Lifshitz-Gilbert equation for uniformly magnetized bodies. Physica B, doi:10.1016/j.physb.2007.08.029.
Bertotti, G., Serpico, C., and Mayergoyz, I. (2001) Nonlinear magnetization dynamics under circularly polarized field.Phys. Rev. Lett.,86, 724–727.
Cimrák, I. Convergence result for the constraint preserving mid-point scheme for micromagnetics. Preprint.
Monk, P. (2003) Finite Element Methods for Maxwell’s Equations. Numerical Mathematics and Scientific Computation, Oxford University Press.
Carbou, G. and Fabrie, P. (2001) Time average in micromagnetism.J. Differential Equations,147, 383–409.
Baňas, L’.(2005) Numerical methods for the Landau-Lifshitz-Gilbert equation. Li, Z.,Vulkov, L., and Wasniewski, J. (eds.),Numerical Analysis and Its Applications: Third International Conference, NAA 2004, Rousse, Bulgaria, vol.3401, p.158, Springer.
Mayergoyz, I., Serpico, C., and Shimizu, Y. (2000) Coupling between eddy currents and Landau-Lifshitz dynamics.J. of Appl.Phys.,87, 5529–5531.
webpage address: http://www.ctcms.nist.gov/~rdm/mumag.org.html.
Baňas, L’ (2005) Adaptive Methods for Dynamical Micromagnetics. Bermúdez de Castro, A., Gómez, D., Quintela, P., and Salgado, P. (eds.), Proceedings of ENUMATH 2005, Santiago de Compostela, Spain, Springer.
Baňas, L’. (2004) On dynamical Micromagnetism with Magnetostriction. Ghent University, Belgium, PhD thesis, http://cage.ugent.be/~lubo/.
Author information
Authors and Affiliations
Corresponding author
Additional information
Author is supported by the Fund for Scientific Research - Flanders FWO (Belgium).
Rights and permissions
About this article
Cite this article
Cimrák, I. A Survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism. ARCO 15, 1–37 (2007). https://doi.org/10.1007/BF03024947
Issue Date:
DOI: https://doi.org/10.1007/BF03024947