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The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis

Published online by Cambridge University Press:  16 July 2009

J. F. Blowey  and
C. M. Elliott
J. F. Blowey
Affiliation:
School of Mathemetical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK
C. M. Elliott
Affiliation:
School of Mathemetical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK
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Abstract

In this paper we consider the numerical analysis of a parabolic variational inequality arising from a deep quench limit of a model for phase separation in a binary mixture due to Cahn and Hilliard. Stability, convergence and error bounds for a finite element approximation are proven. Numerical simulations in one and two space dimensions are presented.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 2 , June 1992 , pp. 147 - 179
Copyright
Copyright © Cambridge University Press 1992

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References

Berger, A. & Falk, R. 1977 An error estimate for the truncation method for the solution of parabolic variational inequalities. Math. Comp. 31, 619–628.CrossRefGoogle Scholar
Bellettini, G., Paolini, M. & Verdi, C. 1990 Γ-convergence of discrete approximations to interfaces with prescribed mean curvature. A Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 1, 317–328.Google Scholar
Blowey, J. F. & Elliott, C. M. 1991 The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis. Euro. J. Appl. Math. 2, 233–280.CrossRefGoogle Scholar
Cerezo, A., Godfrey, T. J., Grovenor, C. R. M., Hetherington, M. G., Hoyle, R. M., Jakubovics, J. P., Liddle, J. A., Smith, G. D. W. & Worral, G. M. 1989 Material analysis with a position-sensitive atom probe. J. Microscopy. 154 (3), 215–225.CrossRefGoogle Scholar
Chakrabarti, R. 1988 Numerical Solution of some Free Boundary Problems. PhD Thesis, Imperial College, UK.Google Scholar
Chen, Xinfu & Elliott, C. M. 1991 Asymptotics for a Parabolic Double Obstacle Problem. Preprini.Google Scholar
Ciarlet, P. G. 1978 The Finite Element Method for Elliptic Problems. North-Holland.Google Scholar
Colli, P. L. & Verdi, C. 1985 Error estimates for an approximation of a problem of percolation in gently sloping beaches. Calcolo 22, 383–390.CrossRefGoogle Scholar
Copetti, M. I. M. & Elliott, C. M. 1990 Kinetics of phase decomposition processes: numerical solutions to the Cahn-Hilliard equation. Mat. Sci. & Technol. 6, 273–283.CrossRefGoogle Scholar
Elliott, C. M. 1989 The Cahn–Hilliard model for the kinetics of phase separation. In Rodrigues, J. F. (ed.), Mathematical Models for Phase change Problems, Birkhäuser.Google Scholar
Elliott, C. M. & French, D. A. 1987 Numerical studies of the Cahn–Hilliard equation for phase separation. IMA J. Appl. Math. 35, 97–128.CrossRefGoogle Scholar
Elliott, C. M., French, D. A. & Milner, F. 1989 A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54, 575–590.CrossRefGoogle Scholar
Elliott, C. M. & Larsson, S. 1991 On a finite element method for the Cahn–Hilliard equation (to appear in Math. Comp.).Google Scholar
Fetter, A. 1987 Lx-error estimates for an approximation of a parabolic variational inequality. Numer. Math. 50, 557–565.CrossRefGoogle Scholar
French, D. A. 1990 Computations on a Cahn–Hilliard model of solidification. Appl. Math. Comp. 40 (1), 55–76.CrossRefGoogle Scholar
French, D. A. & Nicolaides, R. A. 1989 Numerical results on a Cahn-Hilliard model of phase transition. Preprint.Google Scholar
Glowinski, R., Lions, J. L. & TréMolièeres, R. 1981 Numerical Analysis of Variadonal Inequalities. North-Holland.Google Scholar
Johnson, C. 1976 A convergence estimate for an approximation of a parabolic variational inequality. SIAM J. Numer. Anal. 13, 559–606.CrossRefGoogle Scholar
Lions, P. L. & Mercier, B. 1979 Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16 (6), 964–979.CrossRefGoogle Scholar
Scarpini, F. & Vivaldi, M. A. 1978 Evaluation de lerreur d'approximation pour unc inéquation parabolique rélative aux convex dépendent du temps. Appl. Math. Opt. 4, 121–138.CrossRefGoogle Scholar
Vuik, C. 1990 An L2-error estimate for an approximation of a parabolic variational inequality. Numer. Math. 57, 453–471.CrossRefGoogle Scholar