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On fully practical finite element approximations of degenerate Cahn-Hilliard systems

Published online by Cambridge University Press:  15 April 2002

John W. Barrett ,
James F. Blowey  and
Harald Garcke
John W. Barrett
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2BZ, UK. (j.barrett@ic.ac.uk)
James F. Blowey
Affiliation:
Department of Mathematical Sciences, University of Durham, DH1 3LE, UK.
Harald Garcke
Affiliation:
Institut für Angewandte Mathematik, Wegelerstraße 6, 53115 Bonn, Germany.
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Abstract

We consider a model for phase separation of a multi-component alloy with non-smooth free energy and a degenerate mobility matrix. In addition to showing well-posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. We discuss also how our approximation has to be modified in order to be applicable to a logarithmic free energy. Finally numerical experiments with three components in one and two space dimensions are presented.

Keywords

Phase separationmulti-component systemsdegenerate parabolic systems of fourth orderfinite element methodconvergence analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 4 , July 2001 , pp. 713 - 748
Copyright
© EDP Sciences, SMAI, 2001

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References

Adams, R.A. and Fournier, J., Cone conditions and properties of Sobolev spaces. J. Math. Anal. Appl. 61 (1977) 713-734. CrossRef
Barrett, J.W. and Blowey, J.F., An error bound for the finite element approximation of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 16 (1996) 257-287. CrossRef
Barrett, J.W. and Blowey, J.F., Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. Numer. Math. 77 (1997) 1-34. CrossRef
Barrett, J.W. and Blowey, J.F., Finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix. IMA J. Numer. Anal. 18 (1998) 287-328. CrossRef
J.W. Barrett and J.F. Blowey, Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy and a concentration dependent mobility matrix. M 3 AS 9 (1999) 627-663.
Barrett, J.W. and Blowey, J.F., An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix. Numer. Math. 88 (2001) 255-297. CrossRef
Barrett, J.W., Blowey, J.F. and Garcke, H., Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80 (1998) 525-556. CrossRef
Barrett, J.W., Blowey, J.F. and Garcke, H., Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286-318. CrossRef
Blowey, J.F., Copetti, M.I.M. and Elliott, C.M., The numerical analysis of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 16 (1996) 111-139. CrossRef
Blowey, J.F. and Elliott, C.M., The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy, part i: Mathematical analysis. European J. Appl. Math. 2 (1991) 233-279. CrossRef
Blowey, J.F. and Elliott, C.M., 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. CrossRef
L. Bronsard, H. Garcke and B. Stoth, A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem, in Proc. Roy. Soc. Edinburgh 128 A (1998) 481-506.
Cialvaldini, J.F., Analyse numérique d'un problème de Stefan à deux phases par une méthode d'éléments finis. SIAM J. Numer. Anal. 12 (1975) 464-487. CrossRef
P.G. Ciarlet, Introduction to numerical linear algebra and optimisation. C.U.P., Cambridge (1988).
de Fontaine, D., An analysis of clustering and ordering in multicomponent solid solutions - I. Stability criteria. J. Phys. Chem. Solids 33 (1972) 297-310. CrossRef
de Gennes, P.G., Dynamics of fluctuations and spinodal decomposition in polymer blends. J. Chem. Phys. 72 (1980) 4756-4763. CrossRef
C.M. Elliott, The Cahn-Hilliard model for the kinetics of phase transitions, in Mathematical models for phase change problems, J.F. Rodrigues Ed., Internat. Ser. Numer. Math. 88, Birkhäuser-Verlag, Basel (1989) 35-73.
Elliott, C.M. and Garcke, H., On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (1996) 404-423. CrossRef
Elliott, C.M. and Garcke, H., Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix. Physica D 109 (1997) 242-256. CrossRef
C.M. Elliott and S. Luckhaus, A generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy. SFB256 University Bonn, Preprint 195 (1991).
Eyre, D.J., Systems of Cahn-Hilliard equations. SIAM J. Appl. Math. 53 (1993) 1686-1712. CrossRef
Garcke, H., Nestler, B. and Stoth, B., Anisotropy in multi phase systems: a phase field approach. Interfaces Free Bound. 1 (1999) 175-198. CrossRef
Garcke, H. and Novick-Cohen, A., A singular limit for a system of degenerate Cahn-Hilliard equations. Adv. Diff. Eq. 5 (2000) 401-434.
Grün, G. and Rumpf, M., Nonnegativity preserving numerical schemes for the thin film equation. Numer. Math. 87 (2000) 113-152.
Ito, K. and Kohsaka, Y., Three-phase boundary motion by surface diffusion: stability of a mirror symmetric stationary solution. Interfaces Free Bound. 3 (2001) 45-80. CrossRef
Lions, P.L. and Mercier, B., Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16 (1979) 964-979. CrossRef
Morral, J.E. and Cahn, J.W., Spinodal decomposition in ternary systems. Acta Metall. 19 (1971) 1037-1045. CrossRef
Novick-Cohen, A., The Cahn-Hilliard equation: mathematical and modelling perspectives. Adv. Math. Sci. Appl. 8 (1998) 965-985.
Otto, F. and Thermodynamically, W. E driven incompressible fluid mixtures. J. Chem. Phys. 107 (1997) 10177-10184. CrossRef
Zhornitskaya, L. and Bertozzi, A.L., Positivity preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000) 523-555. CrossRef