We consider the numerical approximation of the inertial Landau–Lifshitz–Gilbert equation (iLLG), which describes the dynamics of the magnetisation in ferromagnetic materials at subpicosecond time scales. We propose and analyse two fully discrete numerical schemes: The first method is based on a reformulation of the problem as a linear constrained variational formulation for the linear velocity. The second method exploits a reformulation of the problem as a first order system in time for the magnetisation and the angular momentum. Both schemes are implicit, based on first-order finite elements, and generate approximations satisfying the unit-length constraint of iLLG at the vertices of the underlying mesh. For both methods, we prove convergence of the approximations towards a weak solution of the problem. Numerical experiments validate the theoretical results and show the applicability of the methods for the simulation of ultrafast magnetic processes.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2022043
Keywords: Finite element method, inertial Landau–Lifshitz–Gilbert equation, micromagnetics
@article{M2AN_2022__56_4_1199_0,
author = {Ruggeri, Michele},
title = {Numerical analysis of the {Landau{\textendash}Lifshitz{\textendash}Gilbert} equation with inertial effects},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1199--1222},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {4},
doi = {10.1051/m2an/2022043},
mrnumber = {4444535},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022043/}
}
TY - JOUR AU - Ruggeri, Michele TI - Numerical analysis of the Landau–Lifshitz–Gilbert equation with inertial effects JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1199 EP - 1222 VL - 56 IS - 4 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022043/ DO - 10.1051/m2an/2022043 LA - en ID - M2AN_2022__56_4_1199_0 ER -
%0 Journal Article %A Ruggeri, Michele %T Numerical analysis of the Landau–Lifshitz–Gilbert equation with inertial effects %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1199-1222 %V 56 %N 4 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022043/ %R 10.1051/m2an/2022043 %G en %F M2AN_2022__56_4_1199_0
Ruggeri, Michele. Numerical analysis of the Landau–Lifshitz–Gilbert equation with inertial effects. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 4, pp. 1199-1222. doi: 10.1051/m2an/2022043
[1] , , , , and , Spin-polarized transport in ferromagnetic multilayers: an unconditionally convergent FEM integrator. Comput. Math. Appl. 68 (2014) 639–654. | MR
[2] , A new finite element scheme for Landau-Lifchitz equations. Discrete Contin. Dyn. Syst. Ser. S 1 (2008) 187–196. | MR | Zbl
[3] and , Convergence of a finite element discretization for the Landau-Lifshitz equation in micromagnetism. Math. Models Methods Appl. Sci. 16 (2006) 299–316. | MR | Zbl
[4] and , On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness. Nonlinear Anal. 18 (1992) 1071–1084. | MR | Zbl
[5] , , and , A convergent and precise finite element scheme for Landau–Lifschitz–Gilbert equation. Numer. Math. 128 (2014) 407–430. | MR | Zbl
[6] , Optimal error estimates of linearized Crank-Nicolson Galerkin method for Landau-Lifshitz equation. J. Sci. Comput. 69 (2016) 1–27. | MR
[7] , Stability and convergence of finite-element approximation schemes for harmonic maps. SIAM J. Numer. Anal. 43 (2005) 220–238. | MR | Zbl
[8] , Semi-implicit approximation of wave maps into smooth or convex surfaces. SIAM J. Numer. Anal. 47 (2009) 3486–3506. | MR | Zbl
[9] , Fast and accurate finite element approximation of wave maps into spheres. ESAIM: Math. Model. Numer. Anal. 49 (2015) 551–558. | MR | Zbl | Numdam
[10] , Numerical Methods for Nonlinear Partial Differential Equations. Vol. 47 of Springer Series in Computational Mathematics. Springer (2015). | MR | Zbl
[11] , Projection-free approximation of geometrically constrained partial differential equations. Math. Comp. 85 (2016) 1033–1049. | MR
[12] and , Convergence of an implicit finite element method for the Landau–Lifshitz–Gilbert equation. SIAM J. Numer. Anal. 44 (2006) 1405–1419. | MR | Zbl
[13] , and , Finite element approximations of wave maps into spheres. SIAM J. Numer. Anal. 46 (2007) 61–87. | MR | Zbl
[14] , and , Numerical analysis of an explicit approximation scheme for the Landau–Lifshitz–Gilbert equation. Math. Comp. 77 (2008) 773–788. | MR | Zbl
[15] , and , Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers. Math. Comp. 78 (2009) 1269–1292. | MR | Zbl
[16] , , and , Ultrafast spin dynamics in ferromagnetic nickel. Phys. Rev. Lett. 76 (1996) 4250–4253.
[17] , , , , , , and , Multiscale modeling in micromagnetics: Existence of solutions and numerical integration. Math. Models Methods Appl. Sci. 24 (2014) 2627–2662. | MR | Zbl
[18] , Thin layers in micromagnetism. Math. Models Methods Appl. Sci. 11 (2001) 1529–1546. | MR | Zbl
[19] , and , Magnetization dynamics in the inertial regime: nutation predicted at short time scales. Phys. Rev. B 83 (2011) 020410.
[20] , Micromagnetics of curved thin films, Z. Angew. Math. Phys. 71 (2020) 111. | MR
[21] , , , and , Linear second order IMEX-type integrator for the (eddy current) Landau–Lifshitz–Gilbert equation. IMA J. Numer. Anal. 40 (2020) 2802–2838. | MR
[22] and , The Eddy Current–LLG equations: FEM-BEM coupling and a priori error estimates. SIAM J. Numer. Anal. 55 (2017) 1786–1819. | MR
[23] , Optimal error estimates of a linearized backward Euler FEM for the Landau-Lifshitz equation. SIAM J. Numer. Anal. 52 (2014) 2574–2593. | MR | Zbl
[24] and , Micromagnetics of very thin films. Proc. Roy. Soc. Lond. A 453 (1997) 213–223.
[25] and , On magnetization dynamics with inertial effects. J. Eng. Math. 88 (2014) 197–206. | MR
[26] , , , , and , Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics. Adv. Comput. Math. 45 (2019) 1329–1368. | MR
[27] and , A new angular momentum method for computing wave maps into spheres. SIAM J. Numer. Anal. 52 (2014) 2073–2091. | MR | Zbl
[28] and , Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation. Quart. Appl. Math. 76 (2018) 383–405. | MR
[29] , and , Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle. Math. Comput. 70 (2001) 107–119. | MR | Zbl
[30] and , The Analysis of Harmonic Maps and Their Heat Flows. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2008). | MR | Zbl
[31] and , A finite-difference scheme for a model of magnetization dynamics with inertial effects. J. Eng. Math. 100 (2016) 95–106. | MR
[32] , , , , , , , , , , , , , , , , , , and , Inertial spin dynamics in ferromagnets. Nat. Phys. 17 (2021) 245–250.
[33] NIST, Micromagnetic modeling activity group website. http://www.ctcms.nist.gov/~rdm/mumag.html. Accessed on November 25, 2020.
[34] , and , Convergence of an implicit–explicit midpoint scheme for computational micromagnetics. Comput. Math. Appl. 75 (2018) 1719–1738. | MR
[35] , The wave maps equation. Bull. Amer. Math. Soc. (N.S.) 41 (2004) 185–204. | MR | Zbl
[36] and , Perspective: ultrafast magnetism and THz spintronics. J. Appl. Phys. 120 (2016) 140901.
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