A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions

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November 2010, 27(4): 1511-1533. doi: 10.3934/dcds.2010.27.1511

A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions

Laurence Cherfils ^1, , Madalina Petcu ^2, and Morgan Pierre ^2,

1.	Université de La Rochelle, Laboratoire de Mathématiques Images et Applications EA 3165, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1, France
2.	Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France

Received December 2009 Revised February 2010 Published March 2010

Abstract
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We consider a finite element space semi-discretization of the Cahn-Hilliard equation with dynamic boundary conditions. We prove optimal error estimates in energy norms and weaker norms, assuming enough regularity on the solution. When the solution is less regular, we prove a convergence result in some weak topology. We also prove the stability of a fully discrete problem based on the backward Euler scheme for the time discretization. Some numerical results show the applicability of the method.

Keywords: backward Euler scheme, semilinear parabolic problem, Finite element method, error estimates, Łojasiewicz inequality., linearized Crank-Nicolson scheme.

Mathematics Subject Classification: 65M60, 65M1.

Citation: Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511

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