We consider a fully practical finite element approximation of the Cahn–Hilliard–
Stokes system
γ
∂u
∂t + βv · ∇ u − ∇ · (∇ w) = 0 , w = −γ∆u + γ
−1Ψ
′
(u) −
1
2
αc′
(·, u)|∇ φ|
2
,
∇ · (c(·, u)∇ φ) = 0 ,
(
−∆v + ∇ p = ςw∇ u,
∇ · v = 0,
subject to an initial condition u
0
(.) ∈ [−1, 1] on the conserved order parameter u ∈
[−1, 1], and mixed boundary conditions. Here γ ∈ R>0 is the interfacial parameter,
α ∈ R≥0 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.