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Analysis of a time discretization scheme for a nonstandard viscous Cahn–Hilliard system

Published online by Cambridge University Press:  30 June 2014

Pierluigi Colli ,
Gianni Gilardi ,
Pavel Krejčí ,
Paolo Podio-Guidugli  and
Jürgen Sprekels
Pierluigi Colli
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy.. pierluigi.colli@unipv.it; gianni.gilardi@unipv.it
Gianni Gilardi
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy.. pierluigi.colli@unipv.it; gianni.gilardi@unipv.it
Pavel Krejčí
Affiliation:
Institute of Mathematics, Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1, Czech Republic.; krejci@math.cas.cz
Paolo Podio-Guidugli
Affiliation:
Accademia Nazionale dei Lincei and Department of Mathematics, University of Rome TorVergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy.; ppg@uniroma2.it
Jürgen Sprekels
Affiliation:
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany. ; sprekels@wias-berlin.de
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Abstract

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.

Keywords

Cahn–Hilliard equationphase field modeltime discretizationconvergenceerror estimates
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1061 - 1087
Copyright
© EDP Sciences, SMAI 2014

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