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Numerische Mathematik

, Volume 87, Issue 3, pp 523–554 | Cite as

Efficient methods using high accuracy approximate inertial manifolds

  • Julia Novo
  • Edriss S. Titi
  • Shannon Wynne
Original article

Summary. We extend the idea of the post-processing Galerkin method, in the context of dissipative evolution equations, to the nonlinear Galerkin, the filtered Galerkin, and the filtered nonlinear Galerkin methods. In general, the post-processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear Galerkin method (and Galerkin method), for approximating smooth, bounded solutions, is much smaller than the total error of the method. In each case, an improvement in accuracy is obtained by post-processing these more accurate lower modes with an appropriately chosen, highly accurate, approximate inertial manifold (AIM). We present numerical experiments that support the theoretical improvements in accuracy. Both the theory and computations are presented in the framework of a two dimensional reaction-diffusion system with polynomial nonlinearity. However, the algorithm is very general and can be implemented for other dissipative evolution systems. The computations clearly show the post-processed filtered Galerkin method to be the most efficient method.

Mathematics Subject Classification (1991): 65P25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Julia Novo
    • 1
  • Edriss S. Titi
    • 2
  • Shannon Wynne
    • 3
  1. 1.Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain; e-mail: jnovo@mac.cie.uva.es ES
  2. 2.Department of Mathematics, and Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697-3875, USA; e-mail: etiti@math.uci.edu US
  3. 3.Department of Mathematics, University of California, Irvine, CA 92697-3875, USA; e-mail: swynne@math.uci.edu US

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