Finite element approximation of the Cahn-Hilliard equation with degenerate mobility.

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

Abstract

We consider a fully practical finite element approximation of the Cahn--Hilliard equation with degenerate mobility $$ \textstyle \frac{\partial u}{\partial t}= \del .(\,b(u)\, \del (-\gamma\lap u+\Psi'(u))) , $$ where $b(\cdot)\geq 0$ is a diffusional mobility and $\Psi(\cdot)$ is a homogeneous free energy. 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 analyzed. We also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy. Finally, some numerical experiments are presented.

Citation

Blowey, J., Barrett, J., & Garcke, H. (1999). Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM Journal on Numerical Analysis, 37(1), 286-318. https://doi.org/10.1137/s0036142997331669

• Journal Article Type: Article
• Publication Date: 1999
• Journal: SIAM Journal on Numerical Analysis
• Print ISSN: 0036-1429
• Electronic ISSN: 1095-7170
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 37
• Issue: 1
• Pages: 286-318
• DOI: https://doi.org/10.1137/s0036142997331669
• Public URL: https://durham-repository.worktribe.com/output/1599393