Finite element approximation of coupled Cahn-Hilliard equations with a logarithmic potential and nondegenerate mobility

This research presents a numerical analysis conducted on a system of coupled Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. These equations are derived from a model describing phase separation in a thin film of binary liquid mixture. The study proposes semi-discrete and fully-discrete piecewise linear finite element approximations to the continuous problem. Existence, uniqueness, and various stability estimates for the approximate solutions are established. Fully-discrete error bounds are derived, and optimal time discretisation error is demonstrated. An iterative method is introduced for solving the resulting nonlinear algebraic system, and linear stability analysis in one space dimension is investigated. The research concludes with numerical experiments, providing illustrations of some of the theoretical findings, conducted in both one and two space dimensions.