Convergence to equilibrium for the backward Euler scheme and applications

Benoît Merlet; Morgan Pierre

doi:10.3934/cpaa.2010.9.685

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2010, Volume 9, Issue 3: 685-702. Doi: 10.3934/cpaa.2010.9.685

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Convergence to equilibrium for the backward Euler scheme and applications

Benoît Merlet^1, and
Morgan Pierre^2,

1.
Laboratoire Analyse, Géométrie et Applications, UMR 7539, Université Paris 13 - Institut Galilée, 99, avenue J.B. Clément, 93430 Villetaneuse
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: April 30, 2009

Revised: August 31, 2009

Published: April 2010

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Abstract

We prove that, under natural assumptions, the solution of the backward Euler scheme applied to a gradient flow converges to an equilibrium, as time goes to infinity. Optimal convergence rates are also obtained. As in the continuous case, the proof relies on the well known Lojasiewicz inequality. We extend these results to the $\theta$-scheme with $\theta\in [1/2, 1]$, and to the semilinear heat equation. Applications to semilinear parabolic equations, such as the Allen-Cahn or Cahn-Hilliard equation, are given

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Mathematics Subject Classification: 65L06, 65L20, 65P05.

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