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
2 Institute of Mathematics, Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1, Czech Republic.
3 Accademia Nazionale dei Lincei and Department of Mathematics, University of Rome TorVergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy.
4 Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany.
Revised: 5 November 2013
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.
Mathematics Subject Classification: 35A40 / 35K55 / 35Q70 / 65M12 / 65M15
Key words: Cahn–Hilliard equation / phase field model / time discretization / convergence / error estimates
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