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

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
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