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

¹ Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy.
pierluigi.colli@unipv.it; gianni.gilardi@unipv.it
² Institute of Mathematics, Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1, Czech Republic.
krejci@math.cas.cz
³ Accademia Nazionale dei Lincei and Department of Mathematics, University of Rome TorVergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy.
ppg@uniroma2.it
⁴ Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany.
sprekels@wias-berlin.de

Received: 16 April 2013
Revised: 5 November 2013

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.