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

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

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.
• Public URL: https://durham-repository.worktribe.com/output/1567294
• Publisher URL: http://imanum.oxfordjournals.org/cgi/content/abstract/22/1/11