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Analysis of a time discretization scheme for a nonstandardviscous 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

Abstract

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In this paper we propose a time discretization of a system of two parabolic equationsdescribing diffusion-driven atom rearrangement in crystalline matter. The equationsexpress the balances of microforces and microenergy; the two phase fields are the orderparameter and the chemical potential. The initial and boundary-value problem for theevolutionary system is known to be well posed. Convergence of the discrete scheme to thesolution of the continuous problem is proved by a careful development of uniformestimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, forthe difference of discrete and continuous solutions we prove an error estimate of orderone with respect to the time step.

Keywords

Cahn–Hilliard equationphase field modeltime discretizationconvergenceerror estimates

Information

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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