We consider a fully practical finite element approximation of the Cahn–Hilliard–Stokes system:
$$\begin{align*}
\gamma \tfrac{\partial u}{\partial t} + \beta v \cdot \nabla u -
\nabla \cdot \left(
\nabla w \right) & = 0 \,, \quad
w= -\gamma \Delta u + \gamma ^{-1} \Psi ' (u) - \tfrac12 \alpha c'(\cdot,u)
| \nabla \phi |^2\,, \\
\nabla \cdot (c(\cdot,u) \nabla \phi) & = 0\,,\quad
\begin{cases}
-\Delta v + \nabla p = \varsigma w \nabla u, \\
\nabla \cdot v = 0, \end{cases}
\end{align*}$$
subject to an initial condition
u
0(.) ∈ [−1, 1] on the conserved order parameter
u ∈ [−1, 1], and mixed boundary conditions. Here, γ ∈
$\mathbb{R}_{>0}$
is the interfacial parameter, α ∈
$\mathbb{R}_{\geq0}$
is the field strength parameter, Ψ is the obstacle potential,
c(⋅,
u) is the diffusion coefficient, and
c′(⋅,
u) denotes differentiation with respect to the second argument. Furthermore,
w is the chemical potential, φ is the electro-static potential, and (
v
,
p) are the velocity and pressure. The system has been proposed to model the manipulation of morphologies in organic solar cells with the help of an applied electric field and kinetics.