A finite element approximation for the quasi-static Maxwell,Landau, Lifshitz,Gilbert equations

The quasi-static Maxwell–Landau–Lifshitz–Gilbert equations which describe the electromagnetic behaviour of a ferromagnetic material are highly nonlinear. Sophisticated numerical schemes are required to solve the equations, given their nonlinearity and the constraint that the solution stays on a sphere. We propose an implicit finite element solution to the problem. The resulting system of algebraic equations is linear which facilitates the solution process compared to nonlinear methods. We present numerical results to show the efficacy of the proposed method. References F. Alouges. A new finite element scheme for Landau–Lifchitz equations. Discrete Contin. Dyn. Syst. Ser. S , 1:187–196 (2008). doi:10.3934/dcdss.2008.1.187 L. Banas, S. Bartels, and A. Prohl. A convergent implicit finite element discretization of the Maxwell–Landau–Lifshitz–Gilbert equation. SIAM J. Numer. Anal. , 46:1399–1422 (2008). doi:10.1137/070683064 I. Cimrak. Error analysis of a numerical scheme for 3D Maxwell–Landau–Lifshitz system. Math. Methods Appl. Sci. , 30:1667–1683 (2008). doi:10.1002/mma.863 I. Cimrak. Existence, regularity and local uniqueness of the solutions to the Maxwell–Landau–Lifshitz system in three dimensions. J. Math. Anal. Appl. , 329:1080–1093 (2007). doi:10.1016/j.jmaa.2006.06.080 Micromagnetic Modeling Activity Group, Center for Theoretical and Computational Materials Science, National Institute of Standards and Technology, USA. http://www.ctcms.nist.gov/ rdm/mumag.org.html T. L. Gilbert. A Lagrangian formulation of the gyromagnetic equation of the magnetic field. Phys Rev , 100:1243–1255 (1955). Abstract only; full report, Armor Research Foundation Project No. A059, Supplementary Report, May 1, 1956 (unpublished). B. Guo and S. Ding. Landau–Lifshitz Equations , volume 1 of Frontiers of Research with the Chinese Academy of Sciences . World Scientific, Hackensack NJ (2008). doi:10.1142/9789812778765 L. Landau and E. Lifschitz. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys Z Sowjetunion , 8:153–168 (1935); Perspectives in Theoretical Physics , pp. 51–65, Pergamon, Amsterdam (1992). doi:10.1016/B978-0-08-036364-6.50008-9 K.-N. Le and T. Tran. A convergent finite element approximation for the quasi-static Maxwell–Landau–Lifshitz–Gilbert equations. Research Report, arXiv:1212.3369, 2012, submitted to Computers and Mathematics with Applications . http://arxiv.org/abs/1212.3369 P. Monk. Finite Element Methods for Maxwell's equations . Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003).