A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term

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2011, 2011(Special): 543-552. doi: 10.3934/proc.2011.2011.543

A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term

Maurizio Grasselli ^1, , Nicolas Lecoq ^2, and Morgan Pierre ^3,

1.	Dipartimento di Matematica, Politecnico di Milano, 20133 Milano
2.	Groupe de Physique de Matériaux UMR CNRS6634, Université de Rouen, 1, Avenue de l'Université, 76801 Saint-Etienne du Rouvray, France
3.	Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil

Received June 2010 Revised April 2011 Published October 2011

Abstract
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Under a Creative Commons license

We prove the Lyapunov stability of a time and space discretization of the Cahn-Hilliard equation with inertial term. The space discretization is a mixed (or "splitting") nite element method with numerical integration which includes a standard nite dierence approximation. The time discretization is the backward Euler scheme. The smallness assumption on the time step does not depend on the mesh step.

Keywords: discrete negative norms, numerical integration, Cahn-Hilliard equation, mixed nite elements, Second-order gradient-like ow, Lojasiewicz inequality.

Mathematics Subject Classification: Primary: 65M12, 82C26; Secondary: 65M60, 65P40.

Citation: Maurizio Grasselli, Nicolas Lecoq, Morgan Pierre. A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term. Conference Publications, 2011, 2011 (Special) : 543-552. doi: 10.3934/proc.2011.2011.543

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