Abstract
We discuss in this article the numerical solution of the Cahn-Hilliard equation modelling the spinodal decomposition of binary alloys. The numerical methodology combines a second-order finite difference time discretization with a mixed finite element space approximation and a least squares formulation based on an approximate factorization of a fourth-order elliptic operator which appears in the numerical model. The least squares problem—which is linear—is solved by a preconditioned conjugate gradient algorithm. The results of numerical experiments illustrate the possibilities of the methods discussed in this article.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys.,28 (1958), 258–267.
J.W. Cahn, Phase separation by spinodal decomposition in isotropic systems. J. Chem. Phys.,42 (1965), 93–99.
J.S. Langer, Theory of spinodal decomposition in alloys. Ann. Physics.,65 (1971), 53–86.
A. Novick-Cohen and L.A. Segal, Nonlinear aspects of the Cahn-Hilliard equation. Physica,D10 (1984), 277–298.
C.M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation. Mathematical Models for Phase Change Problems (ed. J.F. Rodrigues), International Series of Numerical Mathematics,88, Birkhäuser, Basel, 1989, 35–73.
C.M. Elliott and A. Mikelic, Existence for the Cahn-Hilliard phase separation model with a nondifferentiable energy. Ann. Mat. Pura Appl.,158 (1991), 181–203.
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin, 1988.
C.M. Elliott and D.A. French, Numerical studies of the Cahn-Hilliard equation for phase separation. IMA J. Appl. Math.,38 (1987), 97–128.
C.M. Elliott and D.A. French, A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal.,26 (1989), 884–903.
C.M. Elliott, D.A. French and F.A. Milner, A second-order splitting method for the Cahn-Hilliard equation. Numer. Math.,54 (1989), 575–590.
M.I.M. Copetti and C.M. Elliott, Kinetics of phase decomposition processes: Numerical solution to the Cahn-Hilliard equation. Material Sci. Tech.,6 (1990), 273–283.
J.F. Blowey and C.F. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part II: Numerical analysis. European J. appl. Math.,3 (1992), 147–179.
M.I.M. Copetti and C.M. Elliott, Numerical analysis of the Cahn-Hilliard equation. Numer. Math.,63 (1992), 39–65.
C.M. Elliott and S. Larsson, Error estimates with smooth and non-smooth data for a finite element method for the Cahn-Hilliard equation. Math. Comp.,58 (1992), 603–630.
F. Bai, C.M. Elliott, A Gardiner, A. Spence and A.M. Stuart, The Viscous Cahn-Hilliard Equation. Part I: Computations. Manuscript SCCM-93-12, Computer Science Department, Stanford Univ., November 1993.
T. Onda, Numerical analysis of phase separation based on the Cahn-Hilliard equation. IACM WCCM II, The Third World Congress in Computational Mechanics Vol. I, August 1–5, 1994, Chiba, Japan, Intern. Assoc. for Comp. Mech., 745–746.
R. Glowinski, H.B. Keller and L. Reinhart, Continuation conjugate gradient method for the least squares solution of nonlinear boundary value problems. SIAM J. Sci. Statist. Comp.,6 (1985), 793–832.
E.J. Dean, R. Glowinski and O. Pironneau, Iterative solution of the stream function— vorticity formulation of the Stokes problem. Applications to the Numerical Simulation of Incompressible Viscous Flow. Comput. Methods Appl. Mech. Engrg.,87 (1991), 117–155.
R. Glowinski, J.L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam 1981.
M. Fortin and R. Glowinski, Lagrangiens Augmentés. Dunod, Paris, 1982.
M. Fortin and R. Glowinski, Augmented Lagrangians. North-Hollnd, Amsterdam, 1983.
R. Glowinski and P. Le Tallec, Augmented Lagrangians and Operator Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia, 1989.
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, New York, 1984.
L. Reinhart, On the numerical analysis of the Von Karman equations: Mixed finite element approximation and continuation techniques. Numer. Math.,39 (1982), 371–404.
A. Nicolas-Carrizosa, A finite difference approach to the Kuramoto-Sivashinsky equation. Advances in Numerical Partial Differential Equations and Optimization (eds S. Gómez, J.P. Hennart, R.A. Tapia), SIAM, Philadelphia, 1991, 262–272.
P.G. Ciarlet, The Finite Element Method for Elliptic problems. North-Holland, Amsterdam, 1978.
P.G. Ciarlet, Basic error estimates for elliptic problems. Handbook of Numerical Analysis, Vol. 2, Finite Element Methods (Part 1), (eds. P.G. Ciarlet and J.L. Lions), North-Holland, Amsterdam, 1991, 17–351.
P.A. Raviart and J.M. Thomas, Introduction à l’Analyse Numérique des Equations aux Dérivées Partielles. Masson, Paris, 1983.
S.C. Brenner and L.R. Scott, The Mathematical Theory of the Finite Element Method. Springer-Verlag, New York, 1994.
J.E. Roberts and J.M. Thomas, Mixed and hybrid methods. Handbook of Numerical Analysis, Vol. 2, Finite Element Methods (Part 1), (eds. P.G. Ciarlet and J.L. Lions), North-Holland, Amsterdam, 1991, 523–639.
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991.
J.L. Lions, Controle Optimal des Systèmes Gouvernés par des Equations aux Dérivées Partielles. Dunod, Paris, 1968.
J. Daniel, The Approximate Minimization of Functionals. Prentice-Hall, Englewood Cliffs, New Jersey, 1970.
Author information
Authors and Affiliations
About this article
Cite this article
Dean, E.J., Glowinski, R. & Trevas, D.A. An approximate factorization/least squares solution method for a mixed finite element approximation of the Cahn-Hilliard equation. Japan J. Indust. Appl. Math. 13, 495–517 (1996). https://doi.org/10.1007/BF03167260
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03167260