CONVERGENCE OF EXPONENTIAL ATTRACTORS FOR A TIME SEMI-DISCRETE REACTION-DIFFUSION EQUATION

Morgan Pierre (1)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]

We consider a time semi-discretization of a generalized Allen-Cahn equation with time-step parameter τ. For every τ , we build an exponential attrac-tor Mτ of the discrete-in-time dynamical system. We prove that Mτ converges to an exponential attractor M0 of the continuous-in-time dynamical system for the symmetric Hausdorff distance as τ tends to 0. We also provide an explicit estimate of this distance and we prove that the fractal dimension of Mτ is bounded by a constant independent of τ. Our construction is based on the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semi-group. Their result has been applied in many situations, but here, for the first time, the perturbation is a discretization. Our method is applicable to a large class of dissipative problems.

Mots clés: Allen-Cahn equation; backward Euler scheme; global attractor; exponential attractor

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Morgan Pierre. CONVERGENCE OF EXPONENTIAL ATTRACTORS FOR A TIME SEMI-DISCRETE REACTION-DIFFUSION EQUATION. 2017. ⟨hal-01518790v1⟩