Hostname: page-component-cb9f654ff-pvkqz Total loading time: 0 Render date: 2025-08-10T17:20:14.826Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  09 September 2016

ROBERT NÜRNBERG  and
EDWARD J. W. TUCKER
ROBERT NÜRNBERG
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK emails: robert.nurnberg@imperial.ac.uk, e.tucker08@imperial.ac.uk
EDWARD J. W. TUCKER
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK emails: robert.nurnberg@imperial.ac.uk, e.tucker08@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a fully practical finite element approximation of the Cahn–Hilliard–Stokes system:

$$\begin{align*}\gamma \tfrac{\partial u}{\partial t} + \beta v \cdot \nabla u -\nabla \cdot \left(\nablaw\right) & = 0 \,, \quadw= -\gamma \Delta u + \gamma ^{-1} \Psi ' (u) - \tfrac12 \alpha c'(\cdot,u)| \nabla \phi |^2\,, \\\nabla \cdot (c(\cdot,u) \nabla \phi) & = 0\,,\quad\begin{cases}-\Delta v + \nabla p = \varsigma w \nabla u,\\\nabla \cdot v = 0, \end{cases}\end{align*}$$
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.

Keywords

Cahn–Hilliard–Stokes systemphase field modelfinite element approximationstability analysis

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 3 , June 2017 , pp. 470 - 498
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Barrett, J. W., Blowey, J. F. & Garcke, H. (1999) Finite element approximation of the Cahn–Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1), 286–318.CrossRefGoogle Scholar
[2] Barrett, J. W. & Nürnberg, R. (2004) Finite element approximation of a Stefan problem with degenerate Joule heating. M2AN Math. Model. Numer. Anal. 38 (4), 633–652.Google Scholar
[3] Barrett, J. W., Nürnberg, R. & Styles, V. (2004) Finite element approximation of a phase field model for void electromigration. SIAM J. Numer. Anal. 42 (2), 738–772.CrossRefGoogle Scholar
[4] Blowey, J. F. & Elliott, C. M. (1991) The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis. Eur. J. Appl. Math. 2 (3), 233–280.CrossRefGoogle Scholar
[5] Blowey, J. F. & Elliott, C. M. (1992) The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy. Part II: Numerical analysis. Eur. J. Appl. Math. 3 (2), 147–179.CrossRefGoogle Scholar
[6] Boffi, D. (1997) Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34 (2), 664–670.CrossRefGoogle Scholar
[7] Braess, D. (2007) Finite Elements. Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd ed., Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[8] Brezzi, F. & Fortin, M. (1991) Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Vol. 15, Springer-Verlag, New York.CrossRefGoogle Scholar
[9] Buxton, G. A. & Clarke, N. (2006) Predicting structure and property relations in polymeric photovoltaic devices. Phys. Rev. B 74 (8), 085207.CrossRefGoogle Scholar
[10] Ciarlet, P. G. (1988) Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge.Google Scholar
[11] Davis, T. A. (2005) Algorithm 849: A concise sparse Cholesky factorization package. ACM Trans. Math. Softw. 31 (4), 587–591.CrossRefGoogle Scholar
[12] Eck, C., Fontelos, M., Grün, G., Klingbeil, F. & Vantzos, O. (2009) On a phase-field model for electrowetting. Interfaces Free Bound. 11 (2), 259–290.CrossRefGoogle Scholar
[13] Feng, X. (2006) Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44 (3), 1049–1072.CrossRefGoogle Scholar
[14] Girault, V. & Raviart, P.-A. (1986) Finite element methods for Navier–Stokes equations, Springer Series in Computational Mathematics, Theory and algorithms, Vol. 5, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[15] Kay, D., Styles, V. & Süli, E. (2009) Discontinuous Galerkin finite element approximation of the Cahn–Hilliard equation with convection. SIAM J. Numer. Anal. 47 (4), 2660–2685.CrossRefGoogle Scholar
[16] Kay, D., Styles, V. & Welford, R. (2008) Finite element approximation of a Cahn–Hilliard–Navier–Stokes system. Interfaces Free Bound. 10 (1), 15–43.CrossRefGoogle Scholar
[17] Kim, D. & Lu, W. (2006) Interface instability and nanostructure patterning. Comp. Mat. Sci. 38 (2), 418–425.CrossRefGoogle Scholar
[18] Lu, W. & Kim, D. (2006) Thin-film structures induced by electrostatic field and substrate kinetic constraint. App. Phys. Lett. 88 (15), 153116.CrossRefGoogle Scholar
[19] Nürnberg, R. & Tucker, E. J. W. (2015) Finite element approximation of a phase field model arising in nanostructure patterning. Numer. Methods Partial Differ. Equ. 31 (6), 1890–1924.Google Scholar
[20] Renardy, M. & Rogers, R. C. (1992) An Introduction to Partial Differential Equations, Springer-Verlag, New York.Google Scholar
[21] Rossi, R. & Savaré, G. (2006) Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM Control Optim. Calc. Var. 12 (3), 564–614.CrossRefGoogle Scholar
[22] Schmidt, A. & Siebert, K. G. (2005) Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering, Vol. 42, Springer-Verlag, Berlin.Google Scholar
[23] Tucker, E. J. W. (2013) Finite Element Approximations of a Phase Field Model, based on the Cahn–Hilliard Equation in the Presence of an Electric Field and Kinetics, PhD thesis, Imperial College London, London, UK.Google Scholar