Finite element approximations of a phase field model, based on the Cahn-Hilliard equation in the presence of an electric field and kinetics

We consider fully practical finite element approximations of the nonlinear parabolic Cahn-Hilliard system [Mathematical equation appears here. To view, please open pdf attachment] subject to an initial condition u0(.) ∈ [−1, 1] on the conserved order parameter u ∈ [−1, 1], and mixed boundary conditions. Here γ ∈ R>0 is the interfacial parameter, [Symbol appears here. To view, please open pdf attachment]∈ R>0 is a time-scaling parameter, α ∈ R≥0 is the field strength parameter, [Symbol appears here. To view, please open pdf attachment] is the obstacle potential, and c(x, u) and b(x, u) are the diffusion coefficients. Furthermore, w is the chemical potential, φ is the electro-static potential and {v, p} are the velocity and pressure. The system, in the limit γ → 0, models the evolution of an unstable interface between two dielectric media in the presence of an electric field, which is quasi-static, and a Stokes flow for the dielectric media. Our goal is to produce stable fully practical finite element approximations to the phase field model above. Additionally, we would like to reproduce the morphologies observed in studies by Buxton and Clarke in [28], and Kim and Lu in [53, 56]. The presence of the electric field and kinetics should drive the interface growth. Initially restricting ourselves to the case without kinetics, we consider coupled and decoupled finite element approximations of the Cahn-Hilliard system. A coupled system is a non-linear algebraic system where the constituent systems are solved simultaneously at each time-step. A decoupled system splits the constituent systems so that they are solved separately and sequentially. Existence, stability and convergence results are presented for a coupled scheme and numerical results are given in two space dimensions. To develop a computationally efficient approximation we present a decoupled scheme with conditional stability in two space dimensions. Numerical results demonstrate that it is a suitable approximation to the coupled scheme. Introducing kinetics to the system requires the careful consideration of both the boundary conditions and mass conservation of the system. A modified coupled scheme admits existence, stability and convergence results. We investigate the applicability of several fast solution methods for the Stokes system. We also present evidence that the MINI-element for the velocity space is more computationally efficient than the Taylor-Hood element. Using further optimisation techniques, such as solving the Stokes system on a coarser mesh, we are able compute results in three dimensions efficiently. Numerous numerical results are presented in two and three dimensions