Numerical analysis of a coupled pair of Cahn-Hilliard equations

Abstract

A mathematical analysis has been carried out for a coupled pair of Cahn-Hilliard equations, which appear in modelling a phase separation on a thin film of binary liquid mixture coating substrate, which is wet by one component. Existence and uniqueness are proved for a weak formulation of the problem, which possesses a Lyapunov functional. Regularity results are presented for the weak formulation. A fully practical piecewise linear finite element approximation is proposed where existence and uniqueness of the numerical solution, and its convergence to the solution of the continuous problem are proven. An error bound between the discrete and continuous solutions is given in three space dimensions. A practical algorithm for solving the resulting algebraic problem at each time step is suggested and its convergence is proven. Finally, linear stability analysis for one space dimension is presented, and some numerical simulations in one and two spaces dimension are exhibited.