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Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. (English) Zbl 0762.65074
The Cahn-Hilliard equation describing phase separation in a binary mixture is investigated by means of a finite element approach based on the backward Euler method. The authors prove existence, uniqueness and stability estimates for the solution of the discrete problem. Then, using embedding theorems of Sobolev type and the stability property, they establish the convergence of discrete solutions to the solution of the continuous problem.
The rest of the paper is devoted to the one-dimensional case. Two iterative procedures to solve the resulting algebraic problem are proposed, analyzed and applied to test calculations. In particular, a comparison is made with the case of quartic free energy earlier studied by other authors.
Reviewer: O.Titow (Berlin)

MSC:
65Z05 Applications to the sciences
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
76T99 Multiphase and multicomponent flows
35Q58 Other completely integrable PDE (MSC2000)
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References:
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