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