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An efficient multigrid preconditioner for Maxwell’s equations in micromagnetism. (English) Zbl 1197.78005

Summary: We consider a system of Maxwell’s and Landau-Lifshitz-Gilbert equations describing magnetization dynamics in micromagnetism. The problem is discretized by a convergent, unconditionally stable finite element method. A multigrid preconditioned Uzawa type method for the solution of the algebraic system resulting from the discretized Maxwell’s equations is constructed. The efficiency of the method is demonstrated on numerical experiments and the results are compared to those obtained by simplified models.

MSC:

78A25 Electromagnetic theory (general)
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
35Q60 PDEs in connection with optics and electromagnetic theory

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References:

[1] Arnold, D. N.; Falk, R. S.; Winther, R., Multigrid in H(div) and H(curl), Numer. Math., 85, 197-217 (2000) · Zbl 0974.65113
[2] Baňas, Ľ.; Bartels, S.; Prohl, A., A convergent implicit finite element discretization of the Maxwell-Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 46, 1399-1422 (2008) · Zbl 1173.35321
[3] Bartels, S.; Prohl, A., Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44, 175-199 (2006)
[4] Bossavit, A.; Rapetti, F., A prolongation/restriction operator for Whitney elements on simplicial meshes, SIAM J. Numer. Anal., 43, 2077-2097 (2005) · Zbl 1109.65108
[5] S.C. Brenner, L.R. Scott, The mathematical theory of finite element methods, in: Texts in Applied Mathematics, Springer-Verlag, New York, 1994.; S.C. Brenner, L.R. Scott, The mathematical theory of finite element methods, in: Texts in Applied Mathematics, Springer-Verlag, New York, 1994. · Zbl 0804.65101
[6] Brown, W., Micromagnetics, Tracts of Physics (1963), Wiley Interscience
[7] d’Aquino, M.; Serpico, C.; Miano, G., Geometrical integration of Landau-Lifshitz-Gilbert equation based on the mid-point rule, J. Comput. Phys., 209, 730-753 (2005) · Zbl 1066.78006
[8] Hiptmair, R., Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal., 36, 204-225 (1998) · Zbl 0922.65081
[9] Landau, L. D.; Lifshitz, E. M., On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion, 8, 153-169 (1935) · Zbl 0012.28501
[10] Monk, P. B., Finite Element Methods for Maxwell’s Equations (2003), Oxford University Press · Zbl 1024.78009
[11] Monk, P. B.; Vacus, O., Accurate discretization of a non-linear micromagnetic problem, Comput. Meth. Appl. Mech. Eng., 190, 5243-5269 (2001) · Zbl 0991.78004
[12] Nédélec, J., A new family of mixed finite elements in \(R^3\), Numer. Math., 50, 57-81 (1986) · Zbl 0625.65107
[13] Schmidt, A.; Siebert, K. G., ALBERT—software for scientific computations and applications, Acta Math. Univ. Comenian. (N.S.), 70, 105-122 (2000) · Zbl 0993.65134
[14] Sun, J.; Collino, F.; Monk, P. B.; Wang, L., An eddy-current and micromagnetism model with applications to disk write heads, Int. J. Numer. Meth. Eng., 60, 1673-1698 (2004) · Zbl 1109.78325
[15] Visintin, A., On Landau-Lifshitz’ equations for ferromagnetism, Jpn. J. Appl. Math., 2, 69-84 (1985) · Zbl 0613.35018
[16] webpage address: http://www.ctcms.nist.gov/ \( \sim;\) rdm/mumag.org.html.; webpage address: http://www.ctcms.nist.gov/ \( \sim;\) rdm/mumag.org.html.
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