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.
Similar content being viewed by others
References
Aida, M., Yagi, A.: Global stability of approximation for exponential attractors. Funkcial. Ekvac. 47(2), 251–276 (2004)
Allen, S., Cahn, J.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsing. Acta. Metall. 27, 1084–1095 (1979)
Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Studies in Mathematics and its Applications, vol. 25. North-Holland Publishing Co., Amsterdam (1992)
Cazenave, T., Haraux, A.: Introduction aux problèmes d’évolution semi-linéaires, Mathématiques & Applications (Paris), vol. 1. Ellipses, Paris (1990)
Chafee, N., Infante, E.: A bifurcation problem for a nonlinear partial differential equation of parabolic type. SIAM J. Appl. Anal. 4, 17–37 (1974)
Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. American Mathematical Society Colloquium Publications, vol. 49. American Mathematical Society, Providence, RI (2002)
Coti Zelati, M., Tone, F.: Multivalued attractors and their approximation: applications to the Navier–Stokes equations. Numer. Math. 122(3), 421–441 (2012)
Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Applied Mathematics, vol. 37. Wiley, Paris (1994)
Efendiev, M., Miranville, A.: The dimension of the global attractor for dissipative reaction-diffusion systems. Appl. Math. Lett. 16(3), 351–355 (2003)
Efendiev, M., Miranville, A., Zelik, S.: Exponential attractors for a nonlinear reaction-diffusion system in \({{R}}^3\). C. R. Acad. Sci. Paris Sér. I Math. 330(8), 713–718 (2000)
Efendiev, M., Miranville, A., Zelik, S.: Exponential attractors for a singularly perturbed Cahn–Hilliard system. Math. Nachr. 272, 11–31 (2004)
Efendiev, M., Yagi, A.: Continuous dependence on a parameter of exponential attractors for chemotaxis-growth system. J. Math. Soc. Jpn. 57(1), 167–181 (2005)
Ezzoug, E., Goubet, O., Zahrouni, E.: Semi-discrete weakly damped nonlinear 2-D Schrödinger equation. Diff. Integr. Equ. 23(3–4), 237–252 (2010)
Fabrie, P., Galusinski, C., Miranville, A.: Uniform inertial sets for damped wave equations. Discrete Contin. Dyn. Syst. 6(2), 393–418 (2000)
Fabrie, P., Galusinski, C., Miranville, A., Zelik, S.: Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete Contin. Dyn. Syst. 10(1–2), 211–238 (2004)
Galusinski, C.: Perturbations singulières de problèmes dissipatifs: étude dynamique via l’existence et la continuité d’attracteurs exponentiels. Ph.D. thesis, Université de Bordeaux (1996)
Gatti, S., Grasselli, M., Miranville, A., Pata, V.: A construction of a robust family of exponential attractors. Proc. Am. Math. Soc. 134(1), 117–127 (2006). (electronic)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)
Hale, J.K.: Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence, RI (1988)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)
Ladyzhenskaya, O.: Attractors for Semigroups and Evolution Equations. Lezioni Lincee. Cambridge University Press, Cambridge (1991)
Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris (1969)
Marion, M.: Attractors for reaction–diffusion equations: existence and estimate of their dimension. Appl. Anal. 25(1–2), 101–147 (1987)
Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. In: Dafermos, C.M., Pokorný, M. (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. IV, pp. 103–200. Elsevier, Amsterdam (2008)
Raugel, G.: Global attractors in partial differential equations. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, vol. 2, pp. 885–982. North-Holland, Amsterdam (2002)
Sell, G.R., You, Y.: Dynamics of Evolutionary Equations, Applied Mathematical Sciences, vol. 143. Springer, New York (2002)
Shen, J.: Convergence of approximate attractors for a fully discrete system for reaction–diffusion equations. Numer. Funct. Anal. Optim. 10(11–12), 1213–1234 (1989). (1990)
Shen, J.: Long time stability and convergence for fully discrete nonlinear Galerkin methods. Appl. Anal. 38(4), 201–229 (1990)
Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis, Cambridge Monographs on Applied and Computational Mathematics, vol. 2. Cambridge University Press, Cambridge (1996)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (1997)
Wang, X.: Approximation of stationary statistical properties of dissipative dynamical systems: time discretization. Math. Comput. 79(269), 259–280 (2010)
Wang, X.: Numerical algorithms for stationary statistical properties of dissipative dynamical systems. Discrete Contin. Dyn. Syst. 36(8), 4599–4618 (2016)
Acknowledgements
The author is thankful to Alain Miranville for helpful discussions. The author also thanks the two anonymous referees for their useful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-017-0937-z