Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. (English) Zbl 0947.65109

Barrett, John W.; Blowey, James F.; Garcke, Harald

SIAM J. Numer. Anal. 37, No. 1, 286-318 (1999).

A finite element approximation of the Cahn-Hilliard equation with degenerate mobility \[ {\partial u\over\partial t}= \nabla(b(u) \nabla(-\gamma\Delta u+ \Psi'(u))) \] is considered, where \(b(u)\geq 0\) is a diffusional mobility and \(\Psi(u)\) is a homogeneous free energy. Well-posedness and stability bounds for this approximation is shown and convergence in one space dimension is proved.
An iterative scheme for solving the resulting nonlinear discrete system is analyzed and a method is discussed how this approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy. Some numerical experiments are presented.

Reviewer: Z.Dżygadło (Warszawa)

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
82C26	Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
35K65	Degenerate parabolic equations
35K55	Nonlinear parabolic equations

Keywords:

fourth-order degenerate parabolic equation; phase separation; finite elements; convergence; Cahn-Hilliard equation; stability; iterative scheme; nonlinear discrete system; numerical experiments