Numerische Mathematik

, Volume 139, Issue 1, pp 121–153 | Cite as

Convergence of exponential attractors for a time semi-discrete reaction-diffusion equation

Article
  • 72 Downloads

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.

Keywords

Allen–Cahn equation Backward Euler scheme Global attractor Exponential attractor 

Mathematics Subject Classification

37L30 65M10 

Notes

Acknowledgements

The author is thankful to Alain Miranville for helpful discussions. The author also thanks the two anonymous referees for their useful comments.

References

  1. 1.
    Aida, M., Yagi, A.: Global stability of approximation for exponential attractors. Funkcial. Ekvac. 47(2), 251–276 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Allen, S., Cahn, J.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsing. Acta. Metall. 27, 1084–1095 (1979)CrossRefGoogle Scholar
  3. 3.
    Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Studies in Mathematics and its Applications, vol. 25. North-Holland Publishing Co., Amsterdam (1992)MATHGoogle Scholar
  4. 4.
    Cazenave, T., Haraux, A.: Introduction aux problèmes d’évolution semi-linéaires, Mathématiques & Applications (Paris), vol. 1. Ellipses, Paris (1990)MATHGoogle Scholar
  5. 5.
    Chafee, N., Infante, E.: A bifurcation problem for a nonlinear partial differential equation of parabolic type. SIAM J. Appl. Anal. 4, 17–37 (1974)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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)MATHGoogle Scholar
  7. 7.
    Coti Zelati, M., Tone, F.: Multivalued attractors and their approximation: applications to the Navier–Stokes equations. Numer. Math. 122(3), 421–441 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Applied Mathematics, vol. 37. Wiley, Paris (1994)MATHGoogle Scholar
  9. 9.
    Efendiev, M., Miranville, A.: The dimension of the global attractor for dissipative reaction-diffusion systems. Appl. Math. Lett. 16(3), 351–355 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Efendiev, M., Miranville, A., Zelik, S.: Exponential attractors for a singularly perturbed Cahn–Hilliard system. Math. Nachr. 272, 11–31 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ezzoug, E., Goubet, O., Zahrouni, E.: Semi-discrete weakly damped nonlinear 2-D Schrödinger equation. Diff. Integr. Equ. 23(3–4), 237–252 (2010)MATHGoogle Scholar
  14. 14.
    Fabrie, P., Galusinski, C., Miranville, A.: Uniform inertial sets for damped wave equations. Discrete Contin. Dyn. Syst. 6(2), 393–418 (2000)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    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)MathSciNetMATHGoogle Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    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)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)MATHGoogle Scholar
  19. 19.
    Hale, J.K.: Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence, RI (1988)Google Scholar
  20. 20.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)CrossRefGoogle Scholar
  21. 21.
    Ladyzhenskaya, O.: Attractors for Semigroups and Evolution Equations. Lezioni Lincee. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  22. 22.
    Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris (1969)MATHGoogle Scholar
  23. 23.
    Marion, M.: Attractors for reaction–diffusion equations: existence and estimate of their dimension. Appl. Anal. 25(1–2), 101–147 (1987)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    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)Google Scholar
  25. 25.
    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)Google Scholar
  26. 26.
    Sell, G.R., You, Y.: Dynamics of Evolutionary Equations, Applied Mathematical Sciences, vol. 143. Springer, New York (2002)CrossRefMATHGoogle Scholar
  27. 27.
    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)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Shen, J.: Long time stability and convergence for fully discrete nonlinear Galerkin methods. Appl. Anal. 38(4), 201–229 (1990)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    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)Google Scholar
  30. 30.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (1997)CrossRefMATHGoogle Scholar
  31. 31.
    Wang, X.: Approximation of stationary statistical properties of dissipative dynamical systems: time discretization. Math. Comput. 79(269), 259–280 (2010)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Wang, X.: Numerical algorithms for stationary statistical properties of dissipative dynamical systems. Discrete Contin. Dyn. Syst. 36(8), 4599–4618 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Applications, CNRSUniversité de PoitiersChasseneuilFrance

Personalised recommendations