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.
Mots-clés : 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 },
pages = {1061--1087},
publisher = {EDP-Sciences},
volume = {48},
number = {4},
year = {2014},
doi = {10.1051/m2an/2014005},
mrnumber = {3264346},
language = {en},
url = {http://www.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 PY - 2014 SP - 1061 EP - 1087 VL - 48 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/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 %D 2014 %P 1061-1087 %V 48 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/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 , Tome 48 (2014) no. 4, pp. 1061-1087. doi : 10.1051/m2an/2014005. http://www.numdam.org/articles/10.1051/m2an/2014005/
[1] , , , and , The viscous Cahn-Hilliard equation. I. Computations. Nonlinearity 8 (1995) 131-160. | MR | Zbl
[2] and , 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] and , 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] , and , Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286-318. | MR | Zbl
[5] , and , On fully practical finite element approximations of degenerate Cahn-Hilliard systems. ESAIM: M2AN 35 (2001) 713-748. | Numdam | MR | Zbl
[6] and , A posteriori error controlled local resolution of evolving interfaces for generalized Cahn-Hilliard equations. Interfaces Free Bound. 12 (2010) 45-73. | MR
[7] and , 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] and , 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] , Global solvability of a dissipative Frémond model for shape memory alloys. II. Existence. Quart. Appl. Math. 62 (2004) 53-76. | MR | Zbl
[10] , 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] and , Numerical analysis of a relaxed variational model of hysteresis in two-phase solids. ESAIM: M2AN 35 (2001) 865-878. | Numdam | MR | Zbl
[12] , and , Error control and adaptivity for a phase relaxation model. ESAIM: M2AN 34 (2000) 775-797. | Numdam | MR | Zbl
[13] , and , A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete Contin. Dyn. Syst. 27 (2010) 1511-1533. | MR | Zbl
[14] , A dissipative model for hydrogen storage: existence and regularity results. Math. Methods Appl. Sci. 34 (2011) 642-669. | MR | Zbl
[15] , and , Global existence of a solution to a phase field model for supercooling. Nonlinear Anal. Real World Appl. 2 (2001) 523-539. | MR | Zbl
[16] , , and , Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system. SIAM J. Appl. Math. 71 (2011) 1849-1870. | MR
[17] , , and , Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity. J. Differ. Equ. 254 (2013) 4217-4244. | MR
[18] , and , 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] , and , Quasistatic isothermal evolution of shape memory alloys. Math. Models Methods Appl. Sci. 21 (2011) 2409-2432. | MR
[20] and , Time-discretization and global solution for a doubly nonlinear Volterra equation. J. Differ. Equ. 228 (2006) 707-736. | MR | Zbl
[21] and , Multigrid methods for obstacle problems. J. Comput. Math. 27 (2009) 1-44. | MR | Zbl
[22] and , Nonsmooth Newton methods for set-valued saddle point problems. SIAM J. Numer. Anal. 47 (2009) 1251-1273. | MR | Zbl
[23] , and , 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] , Approximation of nonlinear evolution systems, vol. 164 of Math. Sci. Eng. Academic Press Inc., Orlando, FL (1983). | MR | Zbl
[25] and , 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] , Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12 (2012) 613-661. | MR
[27] and , Well-posedness of a thermo-mechanical model for shape memory alloys under tension. ESAIM: M2AN 44 (2010) 1239-1253. | Numdam | MR
[28] , and , 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] , Models of phase segregation and diffusion of atomic species on a lattice. Ric. Mat. 55 (2006) 105-118. | MR | Zbl
[30] , Existence of weak solutions for the Mullins-Sekerka flow. SIAM J. Math. Anal. 37 (2005) 291-301. | MR | Zbl
[31] , 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] , Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | MR | Zbl
[33] , Error control of a nonlinear evolution problem related to phase transitions. Numer. Funct. Anal. Optim. 20 (1999) 585-608. | MR | Zbl
[34] , 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] , 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] , Rothe's method for an isothermal phase-field model of a binary alloy with convection. Mat. Contemp. 32 (2007) 221-251. | MR | Zbl
Cité par Sources :
