Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, pp. 1061-1087.

In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step.

DOI: 10.1051/m2an/2014005
Classification: 35A40,  35K55,  35Q70,  65M12,  65M15
Keywords: Cahn-Hilliard equation, phase field model, time discretization, convergence, error estimates
@article{M2AN_2014__48_4_1061_0,
     author = {Colli, Pierluigi and Gilardi, Gianni and Krej\v{c}{\'\i}, Pavel and Podio-Guidugli, Paolo and Sprekels, J\"urgen},
     title = {Analysis of a time discretization scheme for a nonstandard viscous {Cahn-Hilliard} system},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {1061--1087},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {4},
     year = {2014},
     doi = {10.1051/m2an/2014005},
     mrnumber = {3264346},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2014005/}
}
TY  - JOUR
AU  - Colli, Pierluigi
AU  - Gilardi, Gianni
AU  - Krejčí, Pavel
AU  - Podio-Guidugli, Paolo
AU  - Sprekels, Jürgen
TI  - Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system
JO  - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY  - 2014
DA  - 2014///
SP  - 1061
EP  - 1087
VL  - 48
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2014005/
UR  - https://www.ams.org/mathscinet-getitem?mr=3264346
UR  - https://doi.org/10.1051/m2an/2014005
DO  - 10.1051/m2an/2014005
LA  - en
ID  - M2AN_2014__48_4_1061_0
ER  - 
%0 Journal Article
%A Colli, Pierluigi
%A Gilardi, Gianni
%A Krejčí, Pavel
%A Podio-Guidugli, Paolo
%A Sprekels, Jürgen
%T Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system
%J ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
%D 2014
%P 1061-1087
%V 48
%N 4
%I EDP-Sciences
%U https://doi.org/10.1051/m2an/2014005
%R 10.1051/m2an/2014005
%G en
%F M2AN_2014__48_4_1061_0
Colli, Pierluigi; Gilardi, Gianni; Krejčí, Pavel; Podio-Guidugli, Paolo; Sprekels, Jürgen. Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, pp. 1061-1087. doi : 10.1051/m2an/2014005. http://archive.numdam.org/articles/10.1051/m2an/2014005/

[1] F. Bai, C.M. Elliott, A. Gardiner, A. Spence and A.M. Stuart, The viscous Cahn-Hilliard equation. I. Computations. Nonlinearity 8 (1995) 131-160. | MR | Zbl

[2] J.W. Barrett and J.F. Blowey, An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy. Numer. Math. 72 (1995) 1-20. | MR | Zbl

[3] 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 with a concentration dependent mobility matrix. Numer. Math. 88 (2001) 255-297. | MR | Zbl

[4] 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

[5] 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. | Numdam | MR | Zbl

[6] S. Bartels and R. Müller, A posteriori error controlled local resolution of evolving interfaces for generalized Cahn-Hilliard equations. Interfaces Free Bound. 12 (2010) 45-73. | MR

[7] S. Bartels and R. Müller, Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential. Numer. Math. 119 (2011) 409-435. | MR | Zbl

[8] 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

[9] E. Bonetti, Global solvability of a dissipative Frémond model for shape memory alloys. II. Existence. Quart. Appl. Math. 62 (2004) 53-76. | MR | Zbl

[10] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, no. 5. Notas de Matemática. North-Holland Publishing Co., Amsterdam (1973). | MR | Zbl

[11] C. Carstensen and P. Plecháč, Numerical analysis of a relaxed variational model of hysteresis in two-phase solids. ESAIM: M2AN 35 (2001) 865-878. | Numdam | MR | Zbl

[12] Z. Chen, R.H. Nochetto and A. Schmidt, Error control and adaptivity for a phase relaxation model. ESAIM: M2AN 34 (2000) 775-797. | Numdam | MR | Zbl

[13] L. Cherfils, M. Petcu and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete Contin. Dyn. Syst. 27 (2010) 1511-1533. | MR | Zbl

[14] E. Chiodaroli, A dissipative model for hydrogen storage: existence and regularity results. Math. Methods Appl. Sci. 34 (2011) 642-669. | MR | Zbl

[15] P. Colli, M. Frémond and O. Klein, Global existence of a solution to a phase field model for supercooling. Nonlinear Anal. Real World Appl. 2 (2001) 523-539. | MR | Zbl

[16] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system. SIAM J. Appl. Math. 71 (2011) 1849-1870. | MR

[17] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity. J. Differ. Equ. 254 (2013) 4217-4244. | MR

[18] C. Eck, B. Jadamba and P. Knabner, Error estimates for a finite element discretization of a phase field model for mixtures. SIAM J. Numer. Anal. 47 (2010) 4429-4445. | MR | Zbl

[19] S. Frigeri, P. Krejčí and U. Stefanelli, Quasistatic isothermal evolution of shape memory alloys. Math. Models Methods Appl. Sci. 21 (2011) 2409-2432. | MR

[20] G. Gilardi and U. Stefanelli, Time-discretization and global solution for a doubly nonlinear Volterra equation. J. Differ. Equ. 228 (2006) 707-736. | MR | Zbl

[21] C. Gräser and R. Kornhuber, Multigrid methods for obstacle problems. J. Comput. Math. 27 (2009) 1-44. | MR | Zbl

[22] C. Gräser and R. Kornhuber, Nonsmooth Newton methods for set-valued saddle point problems. SIAM J. Numer. Anal. 47 (2009) 1251-1273. | MR | Zbl

[23] C. Gräser, R. Kornhuber and U. Sack, Nonsmooth Schur-Newton methods for vector-valued Cahn-Hilliard equations. Freie Universität Berlin, Fachbereich Mathematik und Informatik, Serie A Preprint no. 01 (2013) 1-16.

[24] J.W. Jerome, Approximation of nonlinear evolution systems, vol. 164 of Math. Sci. Eng. Academic Press Inc., Orlando, FL (1983). | MR | Zbl

[25] D. Kessler and J.-F. Scheid, A priori error estimates of a finite-element method for an isothermal phase-field model related to the solidification process of a binary alloy. IMA J. Numer. Anal. 22 (2002) 281-305. | MR | Zbl

[26] J. Kim, Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12 (2012) 613-661. | MR

[27] P. Krejčí and U. Stefanelli, Well-posedness of a thermo-mechanical model for shape memory alloys under tension. ESAIM: M2AN 44 (2010) 1239-1253. | Numdam | MR

[28] A. Mielke, L. Paoli and A. Petrov, On existence and approximation for a 3D model of thermally induced phase transformations in shape-memory alloys. SIAM J. Math. Anal. 41 (2009) 1388-1414. | MR | Zbl

[29] P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice. Ric. Mat. 55 (2006) 105-118. | MR | Zbl

[30] M. Röger, Existence of weak solutions for the Mullins-Sekerka flow. SIAM J. Math. Anal. 37 (2005) 291-301. | MR | Zbl

[31] A. Segatti, Error estimates for a variable time-step discretization of a phase transition model with hyperbolic momentum. Numer. Funct. Anal. Optim. 25 (2004) 547-569. | MR | Zbl

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

[33] U. Stefanelli, Error control of a nonlinear evolution problem related to phase transitions. Numer. Funct. Anal. Optim. 20 (1999) 585-608. | MR | Zbl

[34] U. Stefanelli, Error control for a time-discretization of the full one-dimensional Frémond model for shape memory alloys. Adv. Math. Sci. Appl. 10 (2000) 917-936. | MR | Zbl

[35] U. Stefanelli, Analysis of a variable time-step discretization for a phase transition model with micro-movements. Commun. Pure Appl. Anal. 5 (2006) 657-671. | MR | Zbl

[36] C.L.D. Vaz, Rothe's method for an isothermal phase-field model of a binary alloy with convection. Mat. Contemp. 32 (2007) 221-251. | MR | Zbl

Cited by Sources: