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Finite element approximation of an Allen-Cahn/Cahn-Hilliard system

Barrett, J.W.; Blowey, J.F.

Authors

J.W. Barrett



Abstract

We consider an Allen–Cahn/Cahn–Hilliard system with a non-degenerate mobility and (i) a logarithmic free energy and (ii) a non-smooth free energy (the deep quench limit). This system arises in the modelling of phase separation and ordering in binary alloys. In particular we prove in each case that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation of (i) and (ii) in one and two space dimensions (and three space dimensions for constant mobility). The error bound being optimal in the deep quench limit. In addition an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments are presented.

Citation

Barrett, J., & Blowey, J. (2002). Finite element approximation of an Allen-Cahn/Cahn-Hilliard system. IMA Journal of Numerical Analysis, 22(1), 11-71. https://doi.org/10.1093/imanum/22.1.11

Journal Article Type Article
Publication Date Jan 1, 2002
Deposit Date Apr 26, 2007
Journal IMA Journal of Numerical Analysis
Print ISSN 0272-4979
Electronic ISSN 1464-3642
Publisher Oxford University Press
Peer Reviewed Peer Reviewed
Volume 22
Issue 1
Pages 11-71
DOI https://doi.org/10.1093/imanum/22.1.11
Keywords Allen–Cahn/Cahn–Hilliard, Order-disorder, Phase separation, finite elements, Error analysis.
Publisher URL http://imanum.oxfordjournals.org/cgi/content/abstract/22/1/11

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