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Stable finite element approximation of a Cahn–Hilliard–Stokes system coupled to an electric field

  • ROBERT NÜRNBERG (a1) and EDWARD J. W. TUCKER (a1)

Abstract

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.

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Stable finite element approximation of a Cahn–Hilliard–Stokes system coupled to an electric field

  • ROBERT NÜRNBERG (a1) and EDWARD J. W. TUCKER (a1)

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