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/
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