# Numerical schemes for a three component Cahn-Hilliard model

Franck Boyer; Sebastian Minjeaud

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 45, Issue: 4, page 697-738
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBoyer, Franck, and Minjeaud, Sebastian. "Numerical schemes for a three component Cahn-Hilliard model." ESAIM: Mathematical Modelling and Numerical Analysis 45.4 (2011): 697-738. <http://eudml.org/doc/197597>.

@article{Boyer2011,

abstract = {
In this article, we investigate numerical schemes for solving
a three component Cahn-Hilliard model. The space discretization is
performed by using
a Galerkin formulation and the finite element method.
Concerning the time discretization,
the main difficulty is to write a scheme ensuring,
at the discrete level, the decrease of the free energy
and thus the stability of the method.
We study three different schemes and prove
existence and convergence theorems. Theoretical results are
illustrated by various numerical examples showing that the new semi-implicit
discretization that we propose seems to be a good compromise between robustness
and accuracy.
},

author = {Boyer, Franck, Minjeaud, Sebastian},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Finite element; Cahn-Hilliard model; numerical scheme;
energy estimate; finite element; energy estimate},

language = {eng},

month = {1},

number = {4},

pages = {697-738},

publisher = {EDP Sciences},

title = {Numerical schemes for a three component Cahn-Hilliard model},

url = {http://eudml.org/doc/197597},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Boyer, Franck

AU - Minjeaud, Sebastian

TI - Numerical schemes for a three component Cahn-Hilliard model

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2011/1//

PB - EDP Sciences

VL - 45

IS - 4

SP - 697

EP - 738

AB -
In this article, we investigate numerical schemes for solving
a three component Cahn-Hilliard model. The space discretization is
performed by using
a Galerkin formulation and the finite element method.
Concerning the time discretization,
the main difficulty is to write a scheme ensuring,
at the discrete level, the decrease of the free energy
and thus the stability of the method.
We study three different schemes and prove
existence and convergence theorems. Theoretical results are
illustrated by various numerical examples showing that the new semi-implicit
discretization that we propose seems to be a good compromise between robustness
and accuracy.

LA - eng

KW - Finite element; Cahn-Hilliard model; numerical scheme;
energy estimate; finite element; energy estimate

UR - http://eudml.org/doc/197597

ER -

## References

top- J.W. Barrett and J.F. Blowey, An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal.19 (1999) 147–168.
- J.W. Barrett and J.F. Blowey, Finite element approximation of an Allen-Cahn/Cahn-Hilliard system. IMA J. Numer. Anal.22 (2002) 11–71.
- J.W. Barrett, J.F. Blowey and H. Garcke, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal.37 (1999) 286–318.
- J.W. Barrett, J.F. Blowey and H. Garcke, On fully practical finite element approximations of degenerate Cahn-Hilliard systems. ESAIM: M2AN35 (2001) 713–748.
- J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. Eur. J. Appl. Math.3 (1992) 147–179.
- J.F. Blowey, M.I.M. Copetti and C.M. Elliott, Numerical analysis of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal.16 (1996) 111–139.
- F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids31 (2002) 41–68.
- F. Boyer and C. Lapuerta, Study of a three component Cahn-Hilliard flow model. ESAIM: M2AN40 (2006) 653–687.
- F. Boyer, C. Lapuerta, S. Minjeaud and B. Piar, A local adaptive refinement method with multigrid preconditioning illustrated by multiphase flows simulations, in CANUM 2008, ESAIM Proc.27, EDP Sciences, Les Ulis (2009) 15–53.
- F. Boyer, C. Lapuerta, S. Minjeaud, B. Piar and M. Quintard, Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows. Transp. Porous Media82 (2010) 463–483.
- K. Deimling, Nonlinear functional analysis. Springer-Verlag (1985).
- Q. Du and R.A. Nicolaides, Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal.28 (1991) 1310–1322.
- C.M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in Mathematical models for phase change problems, Óbidos, 1988, Internat. Ser. Numer. Math.88, Birkhäuser, Basel (1989) 35–73.
- C.M. Elliott and H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix. Physica D109 (1997) 242–256.
- C.M. Elliott and S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy. IMA Preprint Series # 887 (1991).
- C.M. Elliott and A.M. Stuart, The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal.30 (1993) 1622–1663.
- A. Ern and J.-L. Guermond, Theory and Pratice of Finite Elements, Applied Mathematical Sciences159. Springer (2004).
- D.J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, in Computational and mathematical models of microstructural evolution, San Francisco, CA, 1998, Mater. Res. Soc. Sympos. Proc.529, MRS, Warrendale, PA (1998) 39–46.
- X. Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal.44 (2006) 1049–1072.
- X. Feng, Y. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids. Math. Comp.76 (2007) 539–571.
- H. Garcke and B. Stinner, Second order phase field asymptotics for multi-component systems. Interface Free Boundaries8 (2006) 131–157.
- H. Garcke, B. Nestler and B. Stoth, A multiphase field concept: numerical simulations of moving phase boundaries and multiple junctions. SIAM J. Appl. Math.60 (2000) 295–315.
- J. Kim and J. Lowengrub, Phase field modeling and simulation of three-phase flows. Interfaces Free Boundaries7 (2005) 435–466.
- J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys.193 (2004) 511–543.
- J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for ternary Cahn-Hilliard systems. Commun. Math. Sci.2 (2004) 53–77.
- C. Lapuerta, Échanges de masse et de chaleur entre deux phases liquides stratifiées dans un écoulement à bulles. Mathématiques appliquées, Université de Provence, France (2006).
- H.G. Lee and J. Kim, A second-order accurate non-linear difference scheme for the n-component Cahn-Hilliard system. Physica A387 (2008) 4787–4799.
- B. Nestler, H. Garcke and B. Stinner, Multicomponent alloy solidification: Phase-field modeling and simulations. Phys. Rev. E71 (2005) 041609.
- PELICANS, Collaborative Development environment, https://gforge.irsn.fr/gf/project/pelicans/.
- J.S. Rowlinson and B. Widom, Molecular theory of capillarity. Clarendon Press (1982).
- J.M. Seiler and K. Froment, Material effects on multiphase phenomena in late phases of severe accidents of nuclear reactors. Multiph. Sci. Technol.12 (2000) 117–257.
- J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl.146 (1987) 65–96.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.