WIAS Preprint No. 1784, (2013)

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


  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Gilardi, Gianni
    ORCID: 0000-0002-0651-4307
  • Krejčí, Pavel
    ORCID: 0000-0002-7579-6002
  • Podio-Guidugli, Paolo
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2010 Mathematics Subject Classification

  • 35A40 35K55 35Q70 65M12 65M15


  • Cahn--Hilliard equation, phase field model, time discretization, convergence, error estimates




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.

Appeared in

  • ESAIM Math. Model. Numer. Anal., 48 (2014) pp. 1061--1087.

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