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Weak martingale solutions to the stochastic Landau-Lifshitz-Gilbert equation with multi-dimensional noise via a convergent finite-element scheme. (English) Zbl 1433.35380

Summary: We propose an unconditionally convergent linear finite element scheme for the stochastic Landau-Lifshitz-Gilbert (LLG) equation with multi-dimensional noise. By using the Doss-Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent \(\theta \)-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35K55 Nonlinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65C30 Numerical solutions to stochastic differential and integral equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35D30 Weak solutions to PDEs
82D40 Statistical mechanics of magnetic materials
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35R09 Integro-partial differential equations
78A25 Electromagnetic theory (general)
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References:

[1] Alouges, F., A new finite element scheme for landau-lifshitz equations, Discrete Continuous Dyn. Syst. Ser. S, 1, 187-196 (2008) · Zbl 1152.35304
[2] Alouges, F.; de Bouard, A.; Hocquet, A., A semi-discrete scheme for the stochastic landau-lifshitz equation, Stoch. Partial Differ. Equ. Anal. Comput., 2, 281-315 (2014) · Zbl 1332.35348
[3] Alouges, F.; Jaisson, P., Convergence of a finite element discretization for the landau-lifshitz equations in micromagnetism, Math. Models Methods Appl. Sci., 16, 299-316 (2006) · Zbl 1102.35333
[4] Bartels, S., Stability and convergence of finite-element approximation schemes for harmonic maps, SIAM J. Numer. Anal., 43, 220-238 (2005), electronic · Zbl 1090.35014
[5] Baňas, L.; Brzeźniak, Z.; Prohl, A.; Neklyudov, M., A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation, IMA J. Numer. Anal. (2013)
[6] Brown, W. F., Thermal fluctuations of a single-domain particle, Phys. Rev., 130, 1677-1686 (1963)
[7] Brown, W., Thermal fluctuation of fine ferromagnetic particles, IEEE Trans. Magn., 15, 1196-1208 (1979)
[8] Brzeźniak, Z.; Goldys, B.; Jegaraj, T., Weak solutions of a stochastic landau-lifshitz-gilbert equation, Appl. Math. Res. eXpress, 1-33 (2012) · Zbl 1272.60041
[9] Cimrák, I., A survey on the numerics and computations for the landau-lifshitz equation of micromagnetism, Arch. Comput. Methods Eng., 15, 277-309 (2008) · Zbl 1206.78008
[10] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (Encyclopedia of Mathematics and its Applications (2014), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1317.60077
[11] Doss, H., Liens entre équations différentielles stochastiques et ordinaires, (Ann. Probab. Stat., vol. 13 (1977)), 99-125 · Zbl 0359.60087
[12] Gilbert, T., A lagrangian formulation of the gyromagnetic equation of the magnetic field, Phys. Rev., 100, 1243-1255 (1955)
[13] Goldys, B.; Le, K.-N.; Tran, T., A finite element approximation for the stochastic landau-lifshitz-gilbert equation, J. Differential Equations, 260, 937-970 (2016) · Zbl 1364.35449
[14] Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0628.65098
[15] Ju, G.; Peng, Y.; Chang, E. K.C.; Ding, Y.; Wu, A. Q.; Zhu, X.; Kubota, Y.; Klemmer, T. J.; Amini, H.; Gao, L.; Fan, Z.; Rausch, T.; Subedi, P.; Ma, M.; Kalarickal, S.; Rea, C. J.; Dimitrov, D. V.; Huang, P. W.; Wang, K.; Chen, X.; Peng, C.; Chen, W.; Dykes, J. W.; Seigler, M. A.; Gage, E. C.; Chantrell, R.; Thiele, J. U., High density heat-assisted magnetic recording media and advanced characterization –progress and challenges, IEEE Trans. Magn., 51, 1-9 (2015)
[16] Kallenberg, O., Foundations of Modern Probability Probability and Its Applications (2002), Springer-Verlag New York: Springer-Verlag New York Cambridge · Zbl 0996.60001
[17] Kunita, H., Stochastic flows and stochastic differential equations, (Cambridge Studies in Advanced Mathematics, vol. 24 (1990), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0743.60052
[18] Landau, L.; Lifshitz, E., On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion, 8, 153-168 (1935) · Zbl 0012.28501
[19] Néel, L., Bases d’une nouvelle théorie générale du champ coercitif, Ann. Univ. Grenoble, 22, 299-343 (1946)
[20] Sussmann, H. J., An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Amer. Math. Soc., 83, 296-298 (1977) · Zbl 0367.60060
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