Open Access
Issue
ESAIM: M2AN
Volume 60, Number 3, May-June 2026
Page(s) 1451 - 1501
DOI https://doi.org/10.1051/m2an/2026028
Published online 05 June 2026
  1. C. Abert, G. Hrkac, M. Page, D. Praetorius, M. Ruggeri and D. Suess, Spin-polarized transport in ferromagnetic multilayers: an unconditionally convergent FEM integrator. Comput. Math. Appl. 68 (2014) 639–654. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  2. Y. Acremann, Magnetization dynamics: from the Landau-Lifschitz equation to spintronics. Struct. Dyn. 12 (2025) 024502. [Google Scholar]
  3. M.S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C.N. Richardson, J. Ring, M.E. Rognes and G.N. Wells, The FEniCS project version 1.5. ANS 3 (2015). [Google Scholar]
  4. J.-P. Ansermet, Classical description of spin wave excitation by currents in bulk ferromagnets. IEEE Trans. Magn. 40 (2004) 358–360. [Google Scholar]
  5. U. Atxitia, O. Chubykalo-Fesenko, N. Kazantseva, D. Hinzke, U. Nowak and R.W. Chantrell, Micromagnetic modeling of laser-induced magnetization dynamics using the Landau-Lifshitz-Bloch equation. Appl. Phys. Lett. 91 (2007) 232507. [Google Scholar]
  6. U. Atxitia, D. Hinzke and U. Nowak, Fundamentals and applications of the Landau-Lifshitz-Bloch equation. J. Phys. D Appl. Phys. 50 (2016) 033003. [Google Scholar]
  7. C. Ayouch, K.S. Nisar, M. Tilioua and M. Zakarya, On the Landau-Lifshitz-Bloch equation with spin torque effects. Alex. Eng. J. 60 (2021) 4433–4439. [Google Scholar]
  8. S. Bartels and A. Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation. SIAM J. Numer. Anal. 44 (2006) 1405–1419. [CrossRef] [MathSciNet] [Google Scholar]
  9. M. Benmouane, E.-H. Essoufi and C. Ayouch, A finite element scheme for the Landau-Lifshitz-Bloch equation. Comput. Appl. Math. 43 (2024) Paper No. 394. [Google Scholar]
  10. S.C. Brenner, Discrete Sobolev and Poincaré inequalities for piecewise polynomial functions. Electron. Trans. Numer. Anal. 18 (2004) 42–48. [Google Scholar]
  11. S.C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, in Vol. 15 of Texts in Applied Mathematics, 3rd edition. Springer, New York (2008). [Google Scholar]
  12. C. Burrowes, A. Mihai, D. Ravelosona, J. Kim, C. Chappert, L. Vila, A. Marty, Y. Samson, F. Garcia-Sanchez, L. Buda-Prejbeanu, I. Tudosa, E.E. Fullerton and J.-P. Attané, Non-adiabatic spin-torques in narrow magnetic domain walls. Nat. Phys. 6 (2010) 17–21. [Google Scholar]
  13. O. Chubykalo-Fesenko and P. Nieves, Landau-Lifshitz-Bloch Approach for Magnetization Dynamics Close to Phase Transition. Springer International Publishing, Cham (2020) 867–893. [Google Scholar]
  14. O. Chubykalo-Fesenko, U. Nowak, R.W. Chantrell and D. Garanin, Dynamic approach for micromagnetics close to the Curie temperature. Phys. Rev. B 74 (2006) 094436. [Google Scholar]
  15. L. Cimrák. A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism. Arch. Comput. Method. Eng. 15 (2008) 277–309. [Google Scholar]
  16. M. Crouzeix and V. Thomée, The stability in Lp and Wp1 of the L2. projection onto finite element function spaces. Math. Comput. 48 (1987) 521–532. [Google Scholar]
  17. E. Davoli, G. Di Fratta, D. Praetorius and M. Ruggeri, Micromagnetics of thin films in the presence of Dzyaloshinskii- Moriya interaction. Math. Models Method. Appl. Sci. 32 (2022) 911–939. [Google Scholar]
  18. A. Demlow, D. Leykekhman, A.H. Schatz and L.B. Wahlbin, Best approximation property in the W1 norm for finite element methods on graded meshes. Math. Comput. 81 (2012) 743–764. [Google Scholar]
  19. J. Douglas, Jr., T. Dupont and L. Wahlbin, The stability in Lq of the L2-projection into finite element function spaces. Numer. Math. 23 (1974/75) 193–197. [Google Scholar]
  20. M. Eltschka, M. Wötzel, J. Rhensius, S. Krzyk, U. Nowak, M. Kläui, T. Kasama, R.E. Dunin-Borkowski, L.J. Hey- derman, H.J. van Driel and R.A. Duine, Nonadiabatic spin torque investigated using thermally activated magnetic domain wall dynamics. Phys. Rev. Lett. 105 (2010) 056601. [Google Scholar]
  21. E. Emmrich, Discrete versions of Gronwall's lemma and their application to the numerical analysis of parabolic problems. Issue 637 of Preprint Reihe Mathematik. TU, Fachbereich (1999). [Google Scholar]
  22. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, in Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). [Google Scholar]
  23. L.C. Evans, Partial Differential Equations, in Vol. 19 of Graduate Studies in Mathematics, 2nd edition. American Mathematical Society, Providence (2010). [Google Scholar]
  24. M. Feischl, and T. Tran, The eddy current-LLG equations: FEM-BEM coupling and a priori error estimates. SIAM J. Numer. Anal. 55 (2017) 1786–1819. [CrossRef] [MathSciNet] [Google Scholar]
  25. D.A. Garanin, Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets. Phys. Rev. B 55 (1997) 3050–3057. [Google Scholar]
  26. L.D. Geng and Y.M. Jin, Magnetic vortex racetrack memory. J. Magn. Magn. Mater. 423 (2017) 84–89. [Google Scholar]
  27. B. Goldys, C. Jiao and K.-N. Le, Numerical method and error estimate for stochastic Landau-Lifshitz-Bloch equation. IMA J. Numer. Anal. 45 (2025) 1821–1867. [Google Scholar]
  28. G. Grinstein and R.H. Koch, Coarse graining in micromagnetics. Phys. Rev. Lett. 90 (2003) 207201. [Google Scholar]
  29. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2011). [Google Scholar]
  30. X. Gui, B. Li and J. Wang, Convergence of renormalized finite element methods for heat flow of harmonic maps. SIAM J. Numer. Anal. 60 (2022) 312–338. [Google Scholar]
  31. B. Guo and S. Ding, Landau-Lifshitz Equations, Vol. 1 of Frontiers of Research with the Chinese Academy of Sciences. World Scientific Publishing Co. Pte. Ltd., Hackensack (2008). [Google Scholar]
  32. J.G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990) 353–384. [CrossRef] [MathSciNet] [Google Scholar]
  33. X. Jiao, Y. Ren, Z. Zhang, Q. Jin and Y. Liu, Modeling of the laser-heating induced ultrafast demagnetization dynamics in ferrimagnetic thin films. IEEE Trans. Magn. 49 (2013) 3191–3194. [Google Scholar]
  34. L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 8 (1935) 153–168. [Google Scholar]
  35. K.N. Le, Weak solutions of the Landau-Lifshitz-Bloch equation. J. Differ. Equ. 261 (2016) 6699–6717. [Google Scholar]
  36. K.-N. Le, A.L. Soenjaya and T. Tran, The Landau-Lifshitz-Bloch equation in polytopal domains: unique existence and finite element approximation. Preprint arXiv:2406.05808 (2026). To appear in IMA Journal of Numerical Analysis. [Google Scholar]
  37. D. Leykekhman and B. Li, Weak discrete maximum principle of finite element methods in convex polyhedra. Math. Comput. 90 (2021) 1–18. [Google Scholar]
  38. B. Li, Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh. Math. Comput. 91 (2022) 1533–1585. [Google Scholar]
  39. Y. P. Lin, V. Thomée and L.B. Wahlbin, Ritz-Volterra projections to finite-element spaces and applications to integrodifferential and related equations. SIAM J. Numer. Anal. 28 (1991) 1047–1070. [Google Scholar]
  40. V. Maz'ya, Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain. C. R. Math. Acad. Sci. Paris 347 (2009) 517–520. [Google Scholar]
  41. C. Melcher and M. Ptashnyk, Landau–Lifshitz–Slonczewski equations: global weak and classical solutions. SIAM J. Math. Anal. 45 (2013) 407–429. [Google Scholar]
  42. A. Meo, W. Pantasri, W. Daeng-am, S.E. Rannala, S.I. Ruta, R.W. Chantrell, P. Chureemart and J. Chureemart, Magnetization dynamics of granular heat-assisted magnetic recording media by means of a multiscale model. Phys. Rev. B 102 (2020) 174419. [Google Scholar]
  43. R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38 (1982) 437–445. [Google Scholar]
  44. M. Ruggeri, Numerical analysis of the Landau–Lifshitz–Gilbert equation with inertial effects. ESAIM Math. Model. Numer. Anal. 56 (2022) 1199–1222. [Google Scholar]
  45. R. Scott, Optimal L estimates for the finite element method on irregular meshes. Math. Comput. 30 (1976) 681–697. [Google Scholar]
  46. J. Slonczewski, Excitation of spin waves by an electric current. J. Magn. Magn. Mater. 195 (1999) L261–L268. [Google Scholar]
  47. A.L. Soenjaya, Mixed finite element methods for the Landau–Lifshitz–Baryakhtar and the regularised Landau–Lifshitz–Bloch equations in micromagnetics. J. Sci. Comput. 103 (2025) Paper No. 65. [Google Scholar]
  48. A.L. Soenjaya and T. Tran, Global solutions of the Landau–Lifshitz–Baryakhtar equation. J. Differ. Equ. 371 (2023) 191–230. [Google Scholar]
  49. M.D. Stiles, W.M. Saslow, M.J. Donahue and A. Zangwill, Adiabatic domain wall motion and Landau–Lifshitz damping. Phys. Rev. B 75 (2007) 214423. [Google Scholar]
  50. A. Thiaville, Y. Nakatani, J. Miltat and Y. Suzuki, Micromagnetic understanding of current-driven domain wall motion in patterned nanowires. Europhys. Lett. 69 (2005) 990. [Google Scholar]
  51. Y. Tserkovnyak, A. Brataas and G.E. Bauer, Theory of current-driven magnetization dynamics in inhomogeneous ferromagnets. J. Magn. Magn. Mater. 320 (2008) 1282–1292. [Google Scholar]
  52. N. Vinod and T. Tran, Well-posedness and finite element approximation for the Landau–Lifshitz–Gilbert equation with spin torques. Appl. Anal. 104 (2025) 1876–1900. [Google Scholar]
  53. C. Vogler, C. Abert, F. Bruckner, D. Suess and D. Praetorius, Heat-assisted magnetic recording of bit-patterned media beyond 10 Tb/in2. Appl. Phys. Lett. 108 (2016) 102406. [Google Scholar]
  54. I.A. Yastremsky, J. Fassbender, B.A. Ivanov and D. Makarov, Enhanced longitudinal relaxation of magnetic solitons in ultrathin films. Phys. Rev. Appl. 17 (2022) L061002. [Google Scholar]
  55. J.-G. Zhu and H. Li, Understanding signal and noise in heat assisted magnetic recording. IEEE Trans. Magn. 49 (2013) 765–772. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you