Abstract
We consider a time semi-discretization of a generalized Allen–Cahn equation with time-step parameter \(\tau \). For every \(\tau \), we build an exponential attractor \(\mathcal {M}_\tau \) of the discrete-in-time dynamical system. We prove that \(\mathcal {M}_\tau \) converges to an exponential attractor \(\mathcal {M}_0\) of the continuous-in-time dynamical system for the symmetric Hausdorff distance as \(\tau \) tends to 0. We also provide an explicit estimate of this distance and we prove that the fractal dimension of \(\mathcal {M}_\tau \) is bounded by a constant independent of \(\tau \). 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.
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Acknowledgements
The author is thankful to Alain Miranville for helpful discussions. The author also thanks the two anonymous referees for their useful comments.
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Pierre, M. Convergence of exponential attractors for a time semi-discrete reaction-diffusion equation. Numer. Math. 139, 121–153 (2018). https://doi.org/10.1007/s00211-017-0937-z
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DOI: https://doi.org/10.1007/s00211-017-0937-z