A robust family of exponential attractors for a time semi-discretization of the Ginzburg-Landau equation

We consider a time semidiscretization of the Ginzburg-Landau equation by the backward Euler scheme. For each time step $\tau$, we build an exponential attractor of the dynamical system associated to the scheme. We prove that, as $\tau$ tends to $0$, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the dynamical system associated to the Allen-Cahn equation. We also prove that the fractal dimension of the exponential attractor and of the global attractor is bounded by a constant independent of $\tau$.