Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation
Authors:
Eugenia Kim and Jon Wilkening
Journal:
Quart. Appl. Math. 76 (2018), 383-405
MSC (2010):
Primary 65M60, 35Q60; Secondary 78M10
DOI:
https://doi.org/10.1090/qam/1485
Published electronically:
September 21, 2017
MathSciNet review:
3769900
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Abstract: The dynamics of the magnetic distribution in a ferromagnetic material is governed by the Landau-Lifshitz equation, which is a nonlinear geometric dispersive equation with a nonconvex constraint that requires the magnetization to remain of unit length throughout the domain. In this article, we present a mass-lumped finite element method for the Landau-Lifshitz equation. This method preserves the nonconvex constraint at each node of the finite element mesh, and is energy nonincreasing. We show that the numerical solution of our method for the Landau-Lifshitz equation converges to a weak solution of the Landau-Lifshitz-Gilbert equation using a simple proof technique that cancels out the product of weakly convergent sequences. Numerical tests for both explicit and implicit versions of the method on a unit square with periodic boundary conditions are provided for structured and unstructured meshes.
References
- Claas Abert, Lukas Exl, Gunnar Selke, André Drews, and Thomas Schrefl, Numerical methods for the stray-field calculation: A comparison of recently developed algorithms, Journal of Magnetism and Magnetic Materials 326 (2013), 176–185.
- Claas Abert, Gunnar Selke, Benjamin Krüger, and André Drews, A fast finite-difference method for micromagnetics using the magnetic scalar potential, IEEE Transactions on Magnetics 48 (2012), no. 3, 1105–1109.
- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- François Alouges, A new finite element scheme for Landau-Lifchitz equations, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 2, 187–196. MR 2379897, DOI https://doi.org/10.3934/dcdss.2008.1.187
- François Alouges, Sergio Conti, Antonio DeSimone, and Yvo Pokern, Energetics and switching of quasi-uniform states in small ferromagnetic particles, M2AN Math. Model. Numer. Anal. 38 (2004), no. 2, 235–248. MR 2069145, DOI https://doi.org/10.1051/m2an%3A2004011
- François Alouges and Pascal Jaisson, Convergence of a finite element discretization for the Landau-Lifshitz equations in micromagnetism, Math. Models Methods Appl. Sci. 16 (2006), no. 2, 299–316. MR 2210092, DOI https://doi.org/10.1142/S0218202506001169
- François Alouges, Evaggelos Kritsikis, Jutta Steiner, and Jean-Christophe Toussaint, A convergent and precise finite element scheme for Landau-Lifschitz-Gilbert equation, Numer. Math. 128 (2014), no. 3, 407–430. MR 3268842, DOI https://doi.org/10.1007/s00211-014-0615-3
- François Alouges, Evaggelos Kritsikis, and Jean-Christophe Toussaint, A convergent finite element approximation for Landau-Lifschitz-Gilbert equation, Physica B: Condensed Matter 407 (2012), no. 9, 1345–1349.
- François Alouges and Alain Soyeur, On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness, Nonlinear Anal. 18 (1992), no. 11, 1071–1084. MR 1167422, DOI https://doi.org/10.1016/0362-546X%2892%2990196-L
- Sören Bartels, Stability and convergence of finite-element approximation schemes for harmonic maps, SIAM J. Numer. Anal. 43 (2005), no. 1, 220–238. MR 2177142, DOI https://doi.org/10.1137/040606594
- Sören Bartels and Andreas Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal. 44 (2006), no. 4, 1405–1419. MR 2257110, DOI https://doi.org/10.1137/050631070
- Sören Bartels and Andreas Prohl, Constraint preserving implicit finite element discretization of harmonic map flow into spheres, Math. Comp. 76 (2007), no. 260, 1847–1859. MR 2336271, DOI https://doi.org/10.1090/S0025-5718-07-02026-1
- James L Blue and MR Scheinfein, Using multipoles decreases computation time for magnetostatic self-energy, IEEE Transactions on Magnetics 27 (1991), no. 6, 4778–4780.
- Dietrich Braess, Finite elements, 3rd ed., Cambridge University Press, Cambridge, 2007. Theory, fast solvers, and applications in elasticity theory; Translated from the German by Larry L. Schumaker. MR 2322235
- W. F. Brown, Magnetostatic Principles, North-Holland Amsterdam, 1962.
- Xavier Brunotte, Gérard Meunier, and Jean-François Imhoff, Finite element modeling of unbounded problems using transformations: a rigorous, powerful and easy solution, IEEE Transactions on Magnetics 28 (1992), no. 2, 1663–1666.
- Snorre H. Christiansen, A div-curl lemma for edge elements, SIAM J. Numer. Anal. 43 (2005), no. 1, 116–126. MR 2177137, DOI https://doi.org/10.1137/S0036142903433807
- Ivan Cimrák, A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism, Arch. Comput. Methods Eng. 15 (2008), no. 3, 277–309. MR 2430351, DOI https://doi.org/10.1007/s11831-008-9021-2
- Ivan Cimrák, Convergence result for the constraint preserving mid-point scheme for micromagnetism, J. Comput. Appl. Math. 228 (2009), no. 1, 238–246. MR 2514283, DOI https://doi.org/10.1016/j.cam.2008.09.017
- F. de la Hoz, C. J. García-Cervera, and L. Vega, A numerical study of the self-similar solutions of the Schrödinger map, SIAM J. Appl. Math. 70 (2009), no. 4, 1047–1077. MR 2546352, DOI https://doi.org/10.1137/080741720
- L. Exl, W. Auzinger, S. Bance, M. Gusenbauer, F. Reichel, and T. Schrefl, Fast stray field computation on tensor grids, J. Comput. Phys. 231 (2012), no. 7, 2840–2850. MR 2882102, DOI https://doi.org/10.1016/j.jcp.2011.12.030
- Josef Fidler and Thomas Schrefl, Micromagnetic modelling-the current state of the art, Journal of Physics D: Applied Physics 33 (2000), no. 15, R135.
- DR Fredkin and TR Koehler, Hybrid method for computing demagnetizing fields, IEEE Transactions on Magnetics 26 (1990), no. 2, 415–417.
- Atsushi Fuwa, Tetsuya Ishiwata, and Masayoshi Tsutsumi, Finite difference scheme for the Landau-Lifshitz equation, Jpn. J. Ind. Appl. Math. 29 (2012), no. 1, 83–110. MR 2890356, DOI https://doi.org/10.1007/s13160-011-0054-9
- Carlos J. García-Cervera, Numerical micromagnetics: a review, Bol. Soc. Esp. Mat. Apl. SeMA 39 (2007), 103–135. MR 2406975
- Carlos J García-Cervera and Weinan E, Improved Gauss-Seidel projection method for micromagnetics simulations, IEEE transactions on magnetics 39 (2003), no. 3, 1766–1770.
- Carlos J García-Cervera and Alexandre M Roma, Adaptive mesh refinement for micromagnetics simulations, IEEE Transactions on Magnetics 42 (2006), no. 6, 1648–1654.
- Bo Ling Guo and Min Chun Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. Partial Differential Equations 1 (1993), no. 3, 311–334. MR 1261548, DOI https://doi.org/10.1007/BF01191298
- Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai, Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on $\Bbb R^2$, Comm. Math. Phys. 300 (2010), no. 1, 205–242. MR 2725187, DOI https://doi.org/10.1007/s00220-010-1116-6
- E. Hairer and G. Wanner, Solving ordinary differential equations. II, 2nd ed., Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1996. Stiff and differential-algebraic problems. MR 1439506
- J. Samuel Jiang, Hans G. Kaper, and Gary K. Leaf, Hysteresis in layered spring magnets, Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 2, 219–232. MR 1822533, DOI https://doi.org/10.3934/dcdsb.2001.1.219
- Stavros Komineas, Rotating vortex dipoles in ferromagnets, Physical Review Letters 99 (2007), no. 11, 117202.
- Stavros Komineas and Nikos Papanicolaou, Skyrmion dynamics in chiral ferromagnets, Physical Review B 92 (2015), no. 6, 064412.
- Perinkulam S Krishnaprasad and Xiaobo Tan, Cayley transforms in micromagnetics, Physica B: Condensed Matter 306 (2001), no. 1, 195–199.
- E. Kritsikis, A. Vaysset, L. D. Buda-Prejbeanu, F. Alouges, and J.-C. Toussaint, Beyond first-order finite element schemes in micromagnetics, J. Comput. Phys. 256 (2014), 357–366. MR 3117413, DOI https://doi.org/10.1016/j.jcp.2013.08.035
- Martin Kružík and Andreas Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Rev. 48 (2006), no. 3, 439–483. MR 2278438, DOI https://doi.org/10.1137/S0036144504446187
- Lev D Landau and E Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion 8 (1935), no. 153, 101–114.
- D. Lewis and N. Nigam, Geometric integration on spheres and some interesting applications, J. Comput. Appl. Math. 151 (2003), no. 1, 141–170. MR 1950234, DOI https://doi.org/10.1016/S0377-0427%2802%2900743-4
- Boris Livshitz, Amir Boag, H Neal Bertram, and Vitaliy Lomakin, Nonuniform grid algorithm for fast calculation of magnetostatic interactions in micromagnetics, Journal of Applied Physics 105 (2009), no. 7, 07D541.
- HH Long, ET Ong, ZJ Liu, and EP Li, Fast fourier transform on multipoles for rapid calculation of magnetostatic fields, IEEE Transactions on Magnetics 42 (2006), no. 2, 295–300.
- Jacques E Miltat and Michael J Donahue, Numerical micromagnetics: Finite difference methods, Handbook of magnetism and advanced magnetic materials (2007).
- Per-Olof Persson and Gilbert Strang, A simple mesh generator in Matlab, SIAM Rev. 46 (2004), no. 2, 329–345. MR 2114458, DOI https://doi.org/10.1137/S0036144503429121
- N. Popović and D. Praetorius, Applications of $\scr H$-matrix techniques in micromagnetics, Computing 74 (2005), no. 3, 177–204. MR 2139412, DOI https://doi.org/10.1007/s00607-004-0098-7
- Thomas Schrefl, Finite elements in numerical micromagnetics: Part i: granular hard magnets, Journal of magnetism and magnetic materials 207 (1999), no. 1, 45–65.
- Luc Tartar, The general theory of homogenization, Lecture Notes of the Unione Matematica Italiana, vol. 7, Springer-Verlag, Berlin; UMI, Bologna, 2009. A personalized introduction. MR 2582099
- Igor Tsukerman, Alexander Plaks, and H Neal Bertram, Multigrid methods for computation of magnetostatic fields in magnetic recording problems, Journal of Applied Physics 83 (1998), no. 11, 6344–6346.
- Xiao-Ping Wang, Carlos J. García-Cervera, and Weinan E, A Gauss-Seidel projection method for micromagnetics simulations, J. Comput. Phys. 171 (2001), no. 1, 357–372. MR 1843650, DOI https://doi.org/10.1006/jcph.2001.6793
- Samuel W Yuan and H Neal Bertram, Fast adaptive algorithms for micromagnetics, IEEE Transactions on Magnetics 28 (1992), no. 5, 2031–2036.
References
- Claas Abert, Lukas Exl, Gunnar Selke, André Drews, and Thomas Schrefl, Numerical methods for the stray-field calculation: A comparison of recently developed algorithms, Journal of Magnetism and Magnetic Materials 326 (2013), 176–185.
- Claas Abert, Gunnar Selke, Benjamin Krüger, and André Drews, A fast finite-difference method for micromagnetics using the magnetic scalar potential, IEEE Transactions on Magnetics 48 (2012), no. 3, 1105–1109.
- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- François Alouges, A new finite element scheme for Landau-Lifchitz equations, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 2, 187–196. MR 2379897, DOI https://doi.org/10.3934/dcdss.2008.1.187
- François Alouges, Sergio Conti, Antonio DeSimone, and Yvo Pokern, Energetics and switching of quasi-uniform states in small ferromagnetic particles, M2AN Math. Model. Numer. Anal. 38 (2004), no. 2, 235–248. MR 2069145, DOI https://doi.org/10.1051/m2an%3A2004011
- François Alouges and Pascal Jaisson, Convergence of a finite element discretization for the Landau-Lifshitz equations in micromagnetism, Math. Models Methods Appl. Sci. 16 (2006), no. 2, 299–316. MR 2210092, DOI https://doi.org/10.1142/S0218202506001169
- François Alouges, Evaggelos Kritsikis, Jutta Steiner, and Jean-Christophe Toussaint, A convergent and precise finite element scheme for Landau-Lifschitz-Gilbert equation, Numer. Math. 128 (2014), no. 3, 407–430. MR 3268842, DOI https://doi.org/10.1007/s00211-014-0615-3
- François Alouges, Evaggelos Kritsikis, and Jean-Christophe Toussaint, A convergent finite element approximation for Landau-Lifschitz-Gilbert equation, Physica B: Condensed Matter 407 (2012), no. 9, 1345–1349.
- François Alouges and Alain Soyeur, On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness, Nonlinear Anal. 18 (1992), no. 11, 1071–1084. MR 1167422, DOI https://doi.org/10.1016/0362-546X%2892%2990196-L
- Sören Bartels, Stability and convergence of finite-element approximation schemes for harmonic maps, SIAM J. Numer. Anal. 43 (2005), no. 1, 220–238. MR 2177142, DOI https://doi.org/10.1137/040606594
- Sören Bartels and Andreas Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal. 44 (2006), no. 4, 1405–1419. MR 2257110, DOI https://doi.org/10.1137/050631070
- Sören Bartels and Andreas Prohl, Constraint preserving implicit finite element discretization of harmonic map flow into spheres, Math. Comp. 76 (2007), no. 260, 1847–1859. MR 2336271, DOI https://doi.org/10.1090/S0025-5718-07-02026-1
- James L Blue and MR Scheinfein, Using multipoles decreases computation time for magnetostatic self-energy, IEEE Transactions on Magnetics 27 (1991), no. 6, 4778–4780.
- Dietrich Braess, Finite elements, 3rd ed., Cambridge University Press, Cambridge, 2007. Theory, fast solvers, and applications in elasticity theory; Translated from the German by Larry L. Schumaker. MR 2322235
- W. F. Brown, Magnetostatic Principles, North-Holland Amsterdam, 1962.
- Xavier Brunotte, Gérard Meunier, and Jean-François Imhoff, Finite element modeling of unbounded problems using transformations: a rigorous, powerful and easy solution, IEEE Transactions on Magnetics 28 (1992), no. 2, 1663–1666.
- Snorre H. Christiansen, A div-curl lemma for edge elements, SIAM J. Numer. Anal. 43 (2005), no. 1, 116–126. MR 2177137, DOI https://doi.org/10.1137/S0036142903433807
- Ivan Cimrák, A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism, Arch. Comput. Methods Eng. 15 (2008), no. 3, 277–309. MR 2430351, DOI https://doi.org/10.1007/s11831-008-9021-2
- Ivan Cimrák, Convergence result for the constraint preserving mid-point scheme for micromagnetism, J. Comput. Appl. Math. 228 (2009), no. 1, 238–246. MR 2514283, DOI https://doi.org/10.1016/j.cam.2008.09.017
- F. de la Hoz, C. J. García-Cervera, and L. Vega, A numerical study of the self-similar solutions of the Schrödinger map, SIAM J. Appl. Math. 70 (2009), no. 4, 1047–1077. MR 2546352, DOI https://doi.org/10.1137/080741720
- L. Exl, W. Auzinger, S. Bance, M. Gusenbauer, F. Reichel, and T. Schrefl, Fast stray field computation on tensor grids, J. Comput. Phys. 231 (2012), no. 7, 2840–2850. MR 2882102, DOI https://doi.org/10.1016/j.jcp.2011.12.030
- Josef Fidler and Thomas Schrefl, Micromagnetic modelling-the current state of the art, Journal of Physics D: Applied Physics 33 (2000), no. 15, R135.
- DR Fredkin and TR Koehler, Hybrid method for computing demagnetizing fields, IEEE Transactions on Magnetics 26 (1990), no. 2, 415–417.
- Atsushi Fuwa, Tetsuya Ishiwata, and Masayoshi Tsutsumi, Finite difference scheme for the Landau-Lifshitz equation, Jpn. J. Ind. Appl. Math. 29 (2012), no. 1, 83–110. MR 2890356, DOI https://doi.org/10.1007/s13160-011-0054-9
- Carlos J. García-Cervera, Numerical micromagnetics: a review, Bol. Soc. Esp. Mat. Apl. SeMA 39 (2007), 103–135. MR 2406975
- Carlos J García-Cervera and Weinan E, Improved Gauss-Seidel projection method for micromagnetics simulations, IEEE transactions on magnetics 39 (2003), no. 3, 1766–1770.
- Carlos J García-Cervera and Alexandre M Roma, Adaptive mesh refinement for micromagnetics simulations, IEEE Transactions on Magnetics 42 (2006), no. 6, 1648–1654.
- Bo Ling Guo and Min Chun Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. Partial Differential Equations 1 (1993), no. 3, 311–334. MR 1261548, DOI https://doi.org/10.1007/BF01191298
- Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai, Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on $\mathbb {R}^2$, Comm. Math. Phys. 300 (2010), no. 1, 205–242. MR 2725187, DOI https://doi.org/10.1007/s00220-010-1116-6
- E. Hairer and G. Wanner, Solving ordinary differential equations. II, 2nd ed., Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1996. Stiff and differential-algebraic problems. MR 1439506
- J. Samuel Jiang, Hans G. Kaper, and Gary K. Leaf, Hysteresis in layered spring magnets, Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 2, 219–232. MR 1822533, DOI https://doi.org/10.3934/dcdsb.2001.1.219
- Stavros Komineas, Rotating vortex dipoles in ferromagnets, Physical Review Letters 99 (2007), no. 11, 117202.
- Stavros Komineas and Nikos Papanicolaou, Skyrmion dynamics in chiral ferromagnets, Physical Review B 92 (2015), no. 6, 064412.
- Perinkulam S Krishnaprasad and Xiaobo Tan, Cayley transforms in micromagnetics, Physica B: Condensed Matter 306 (2001), no. 1, 195–199.
- E. Kritsikis, A. Vaysset, L. D. Buda-Prejbeanu, F. Alouges, and J.-C. Toussaint, Beyond first-order finite element schemes in micromagnetics, J. Comput. Phys. 256 (2014), 357–366. MR 3117413, DOI https://doi.org/10.1016/j.jcp.2013.08.035
- Martin Kružík and Andreas Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Rev. 48 (2006), no. 3, 439–483. MR 2278438, DOI https://doi.org/10.1137/S0036144504446187
- Lev D Landau and E Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion 8 (1935), no. 153, 101–114.
- D. Lewis and N. Nigam, Geometric integration on spheres and some interesting applications, J. Comput. Appl. Math. 151 (2003), no. 1, 141–170. MR 1950234, DOI https://doi.org/10.1016/S0377-0427%2802%2900743-4
- Boris Livshitz, Amir Boag, H Neal Bertram, and Vitaliy Lomakin, Nonuniform grid algorithm for fast calculation of magnetostatic interactions in micromagnetics, Journal of Applied Physics 105 (2009), no. 7, 07D541.
- HH Long, ET Ong, ZJ Liu, and EP Li, Fast fourier transform on multipoles for rapid calculation of magnetostatic fields, IEEE Transactions on Magnetics 42 (2006), no. 2, 295–300.
- Jacques E Miltat and Michael J Donahue, Numerical micromagnetics: Finite difference methods, Handbook of magnetism and advanced magnetic materials (2007).
- Per-Olof Persson and Gilbert Strang, A simple mesh generator in Matlab, SIAM Rev. 46 (2004), no. 2, 329–345. MR 2114458, DOI https://doi.org/10.1137/S0036144503429121
- N. Popović and D. Praetorius, Applications of $\mathcal {H}$-matrix techniques in micromagnetics, Computing 74 (2005), no. 3, 177–204. MR 2139412, DOI https://doi.org/10.1007/s00607-004-0098-7
- Thomas Schrefl, Finite elements in numerical micromagnetics: Part i: granular hard magnets, Journal of magnetism and magnetic materials 207 (1999), no. 1, 45–65.
- Luc Tartar, The general theory of homogenization, Lecture Notes of the Unione Matematica Italiana, vol. 7, Springer-Verlag, Berlin; UMI, Bologna, 2009. A personalized introduction. MR 2582099
- Igor Tsukerman, Alexander Plaks, and H Neal Bertram, Multigrid methods for computation of magnetostatic fields in magnetic recording problems, Journal of Applied Physics 83 (1998), no. 11, 6344–6346.
- Xiao-Ping Wang, Carlos J. García-Cervera, and Weinan E, A Gauss-Seidel projection method for micromagnetics simulations, J. Comput. Phys. 171 (2001), no. 1, 357–372. MR 1843650, DOI https://doi.org/10.1006/jcph.2001.6793
- Samuel W Yuan and H Neal Bertram, Fast adaptive algorithms for micromagnetics, IEEE Transactions on Magnetics 28 (1992), no. 5, 2031–2036.
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Additional Information
Eugenia Kim
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720
MR Author ID:
895340
Email:
kim107@math.berkeley.edu
Jon Wilkening
Affiliation:
Department of Mathematics and Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720
MR Author ID:
725944
Email:
wilkening@berkeley.edu
Keywords:
Landau-Lifshitz equation,
Landau-Lifshitz-Gilbert equation,
micromagnetics,
finite element methods,
mass-lumped method,
convergence,
weak solutions
Received by editor(s):
August 8, 2017
Published electronically:
September 21, 2017
Additional Notes:
The first author was supported in part by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists, Office of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education for the DOE under contract number DE-AC05-06OR23100.
The second author was supported in part by the U.S. Department of Energy, Office of Science, Applied Scientific Computing Research, under award number DE-AC02-05CH11231.
Article copyright:
© Copyright 2017
Brown University
