Sobolev gradient approach for the time evolution related to energy minimization of Ginzburg-Landau functionals. (English) Zbl 1166.65348

Summary: The Sobolev gradient technique has been discussed previously in this journal as an efficient method for finding energy minima of certain Ginzburg-Landau type functionals [S. Sial, J. Neuberger, T. Lookman and A. Saxena, J. Comput. Phys. 189, No. 1, 88–97 (2003; Zbl 1097.49002)]. In this article, a Sobolev gradient method for the related time evolution is discussed.


65K10 Numerical optimization and variational techniques
49M25 Discrete approximations in optimal control
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)


Zbl 1097.49002
Full Text: DOI


[1] Sial, S.; Neuberger, J.; Lookman, T.; Saxena, A., Energy minimization using Sobolev gradients: application to phase separation and ordering, J. comput. phys., 189, 88-97, (2003) · Zbl 1097.49002
[2] Neuberger, J.W., Sobolev gradients and differential equations, Springer lecture notes in mathematics, 1670, (1997), Springer-Verlag New York · Zbl 0870.47029
[3] Mahavier, W.T., A numerical method utilizing weighted Sobolev descent to solve singular differential equations, Nonlinear world, 4, 4, (1997) · Zbl 0908.65060
[4] C. Beasley, Finite Element Solution to Nonlinear Partial Differential Equations, Ph.D. Thesis, University of North Texas, Denton, TX, 1981.
[5] S. Sial, J. Neuberger, T. Lookman, A. Saxena, Energy minimization using Sobolev gradients finite element setting, in: Chaudhary, Bhatti (Eds.), Proceedings of the World Conference on 21st Century Mathematics, Lahore, Pakistan, March 2005. · Zbl 1097.49002
[6] Sial, S., Sobolev gradient algorithm for minimum energy states of s-wave superconductors: finite element setting, Supercond. sci. technol., 18, 675-677, (2005)
[7] Karatson, J.; Loczi, L., Sobolev gradient preconditioning for the electrostatic potential equation, Comput. math. appl., 50, 1093-1104, (2005) · Zbl 1098.65113
[8] Karatson, J.; Farago, I., Preconditioning operators and Sobolev gradients for nonlinear elliptic problems, Comput. math. appl., 50, 1077-1092, (2005) · Zbl 1118.65122
[9] Karatson, J., Constructive Sobolev gradient preconditioning for semilinear elliptic systems, Electron. J. differ. equations, 75, 1-26, (2004) · Zbl 1109.65308
[10] Garcia-Ripoll, J.; Konotop, V.; Malomed, B.; Perez-Garcia, V., A quasi-local gross – pitaevskii equation for attractive bose – einstein condensates, Math. comput. simulat., 62, 21-30, (2003) · Zbl 1034.35126
[11] Brown, B.; Jais, M.; Knowles, I., A variational approach to an elastic inverse problem, Inverse probl., 21, 1953-1973, (2005) · Zbl 1274.35407
[12] Knowles, I.; Yan, A., On the recovery of transport parameters in groundwater modelling, J. comput. appl. math., 171, 277-290, (2004) · Zbl 1044.86009
[13] Hohenberg, P.C.; Halperin, B.I., Theory of dynamic critical phenomena, Rev. mod. phys., 49, 435-479, (1977)
[14] Rogers, T.M.; Elder, K.E.; Desai, R.C., Numerical study of the late stages of spinodal decomposition, Phys. rev. B, 37, 9638-9649, (1988)
[15] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I: interfacial free energy, J. chem. phys., 28, 258, (1958)
[16] Courant, R.; Friedrichs, K.O.; Lewy, H., Uber die partiellen differenzengleichungen der mathematisches physik, Math. ann., 100, 32-74, (1928) · JFM 54.0486.01
[17] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
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