Stable discretizations of the Cahn-Hilliard-Gurtin equations

Sami Injrou; Morgan Pierre

doi:10.3934/dcds.2008.22.1065

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2008, Volume 22, Issue 4: 1065-1080. Doi: 10.3934/dcds.2008.22.1065

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Stable discretizations of the Cahn-Hilliard-Gurtin equations

Sami Injrou^1, and
Morgan Pierre^2,

1.
Université de Poitiers, Laboratoire de Mathématiques et Applications, Téléport 2 - BP 30179, SP2MI, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil
2.
Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil

Received: February 28, 2007

Revised: May 31, 2007

Published: September 2008

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Abstract

We study space and time discretizations of the Cahn-Hilliard-Gurtin equations with a polynomial nonlinearity. We first consider a space semi-discrete version of the equations, and we prove in particular that any solution converges to a steady state (as in the continuous case). Then, we study the numerical stability of the fully discrete scheme obtained by applying the Euler backward scheme to the space semi-discrete problem. In particular, we show that this fully discrete problem is unconditionally stable. Numerical simulations in one space dimension conclude the paper.

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Mathematics Subject Classification: 65M12, 65M60.

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