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IMA Journal of Numerical Analysis Advance Access published online on February 9, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn055
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Uniform convergence for a finite-element discretization of a viscous diffusion equation

Morgan Pierre{dagger}

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

{dagger} Email: pierre{at}math.univ-poitiers.fr

Received on 25 October 2007. Revised on 29 July 2008.


  Abstract

We study a space semidiscretization of a viscous diffusion equation, obtained as a singular limit of the viscous Cahn–Hilliard equation by letting the interfacial energy tend to 0. The semidiscrete solution is shown to converge uniformly in time and space to the continuous solution, on finite time intervals, as the discretization parameter h tends to 0 (in space dimension one, two and three). We obtain an optimal error bound in space dimension one assuming only a piecewise Lipschitz regularity on the initial value. This approach allows us to obtain some counterexamples concerning lower and upper bounds of solutions to Cahn–Hilliard equations. Numerical simulations confirm the theoretical results.

Key Words: viscous Cahn–Hilliard equation; discrete maximum principle


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