Energy minimization using Sobolev gradients: application to phase separation and ordering. (English) Zbl 1097.49002

Summary: A common problem in physics and engineering is the calculation of the minima of energy functionals. The theory of Sobolev gradients provides an efficient method for seeking the critical points of such a functional. We apply the method to functionals describing coarse-grained Ginzburg–Landau models commonly used in pattern formation and ordering processes.


49J10 Existence theories for free problems in two or more independent variables
65K10 Numerical optimization and variational techniques
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
74G65 Energy minimization in equilibrium problems in solid mechanics
74N99 Phase transformations in solids
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[1] Neuberger, J.W., ()
[2] Hohenberg, P.C.; Halperin, B.I., Theory of dynamic critical phenomena, Rev. mod. phys., 49, 435-479, (1977)
[3] Richardson, W.B., Steepest descent using smoothed gradients, Appl. math. comput., 112, 241-254, (2000) · Zbl 1023.65053
[4] Courant, R.; Friedrichs, K.O.; Lewy, H., Uber die partiellen differenzengleichungen der mathematisches physik, Math. ann., 100, 32-74, (1928) · JFM 54.0486.01
[5] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[6] Rogers, T.M.; Elder, K.R.; Desai, R.C., Numerical study of the late stages of spinodal decomposition, Phys. rev. B, 37, 9638-9649, (1988)
[7] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial free energy, J. chem. phys., 28, 258, (1958)
[8] Gompper, G.; Schick, M., Correlation between structural and interfacial properties of amphiphilic systems, Phys. rev. lett., 65, 1116, (1990)
[9] Shenoy, S.R.; Lookman, T.; Saxena, A.; Bishop, A.R., Martensitic textures: multiscale consequences of elastic compatibility, Phys. rev. B, 60, 18, R12537-R12541, (1999)
[10] Mahavier, W.T., A numerical method utilizing weighted Sobolev descent to solve singular differential equations, Nonlinear world, 4, 4, (1997) · Zbl 0908.65060
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