CONVERGENCE OF EXPONENTIAL ATTRACTORS FOR A FINITE ELEMENT APPROXIMATION OF THE ALLEN-CAHN EQUATION - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2018

CONVERGENCE OF EXPONENTIAL ATTRACTORS FOR A FINITE ELEMENT APPROXIMATION OF THE ALLEN-CAHN EQUATION

Abstract

We consider a space semidiscretization of the Allen-Cahn equation by conforming Lagrange finite elements. For every mesh parameter h, we build an exponential attractor M h of the dynamical system associated to the approximate equations. We prove that, as h tends to 0, M h converges for the symmetric Haus-dorff distance to an exponential attractor M0 of the dynamical system associated to the Allen-Cahn equation. We also provide an explicit estimate of this distance and we prove that the fractal dimension of M h and of the global attractor is bounded by a constant independent of h. Our proof is adapted from the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attrac-tors under perturbation of the underlying semigroup. Here, for the first time, the perturbation is a space discretization. The case of a time semidiscretization has been analyzed in a previous paper.
Fichier principal
Vignette du fichier
attracteur_expo_h.pdf (215.13 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-01725042 , version 1 (07-03-2018)

Identifiers

Cite

Morgan Pierre. CONVERGENCE OF EXPONENTIAL ATTRACTORS FOR A FINITE ELEMENT APPROXIMATION OF THE ALLEN-CAHN EQUATION. 2018. ⟨hal-01725042⟩
168 View
162 Download

Altmetric

Share

Gmail Facebook X LinkedIn More