no. 4
Numerical schemes for a three component Cahn-Hilliard model
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 4, pp. 697-738.

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.

DOI : 10.1051/m2an/2010072
Classification : 35K55, 65M60, 65M12, 76T30
Mots clés : finite element, Cahn-Hilliard model, numerical scheme, energy estimate 
@article{M2AN_2011__45_4_697_0,
     author = {Boyer, Franck and Minjeaud, Sebastian},
     title = {Numerical schemes for a three component {Cahn-Hilliard} model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {697--738},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {4},
     year = {2011},
     doi = {10.1051/m2an/2010072},
     mrnumber = {2804656},
     zbl = {1267.76127},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010072/}
}
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 
PY  - 2011
SP  - 697
EP  - 738
VL  - 45
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2010072/
DO  - 10.1051/m2an/2010072
LA  - en
ID  - M2AN_2011__45_4_697_0
ER  - 
%0 Journal Article
%A Boyer, Franck
%A Minjeaud, Sebastian
%T Numerical schemes for a three component Cahn-Hilliard model
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2011
%P 697-738
%V 45
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2010072/
%R 10.1051/m2an/2010072
%G en
%F M2AN_2011__45_4_697_0
Boyer, Franck; Minjeaud, Sebastian. Numerical schemes for a three component Cahn-Hilliard model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 4, pp. 697-738. doi : 10.1051/m2an/2010072. http://www.numdam.org/articles/10.1051/m2an/2010072/

[1] 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. | MR | Zbl

[2] 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. | MR | Zbl

[3] 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. | MR | Zbl

[4] J.W. Barrett, J.F. Blowey and H. Garcke, On fully practical finite element approximations of degenerate Cahn-Hilliard systems. ESAIM: M2AN 35 (2001) 713-748. | EuDML | Numdam | MR | Zbl

[5] 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. | MR | Zbl

[6] 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. | MR | Zbl

[7] F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31 (2002) 41-68. | Zbl

[8] F. Boyer and C. Lapuerta, Study of a three component Cahn-Hilliard flow model. ESAIM: M2AN 40 (2006) 653-687. | EuDML | Numdam | MR | Zbl

[9] 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. | MR | Zbl

[10] 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 Media 82 (2010) 463-483. | MR

[11] K. Deimling, Nonlinear functional analysis. Springer-Verlag (1985). | MR | Zbl

[12] Q. Du and R.A. Nicolaides, Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28 (1991) 1310-1322. | MR | Zbl

[13] 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. | MR | Zbl

[14] C.M. Elliott and H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix. Physica D 109 (1997) 242-256. | MR | Zbl

[15] 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).

[16] C.M. Elliott and A.M. Stuart, The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30 (1993) 1622-1663. | MR | Zbl

[17] A. Ern and J.-L. Guermond, Theory and Pratice of Finite Elements, Applied Mathematical Sciences 159. Springer (2004). | MR | Zbl

[18] 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. | MR

[19] 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. | MR

[20] 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. | MR | Zbl

[21] H. Garcke and B. Stinner, Second order phase field asymptotics for multi-component systems. Interface Free Boundaries 8 (2006) 131-157. | MR | Zbl

[22] 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. | MR | Zbl

[23] J. Kim and J. Lowengrub, Phase field modeling and simulation of three-phase flows. Interfaces Free Boundaries 7 (2005) 435-466. | MR | Zbl

[24] J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys. 193 (2004) 511-543. | MR | Zbl

[25] J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for ternary Cahn-Hilliard systems. Commun. Math. Sci. 2 (2004) 53-77. | MR | Zbl

[26] 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).

[27] H.G. Lee and J. Kim, A second-order accurate non-linear difference scheme for the n-component Cahn-Hilliard system. Physica A 387 (2008) 4787-4799. | MR

[28] B. Nestler, H. Garcke and B. Stinner, Multicomponent alloy solidification: Phase-field modeling and simulations. Phys. Rev. E 71 (2005) 041609.

[29] PELICANS, Collaborative Development environment, https://gforge.irsn.fr/gf/project/pelicans/.

[30] J.S. Rowlinson and B. Widom, Molecular theory of capillarity. Clarendon Press (1982).

[31] 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.

[32] J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | MR | Zbl

Cité par Sources :