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Mathematics of Computation

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An approximate inertial manifolds approach
to postprocessing the Galerkin method
for the Navier-Stokes equations

Authors: Bosco García-Archilla, Julia Novo and Edriss S. Titi
Journal: Math. Comp. 68 (1999), 893-911
MSC (1991): Primary 65P25
Published electronically: February 19, 1999
MathSciNet review: 1627785
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Abstract: In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.

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Additional Information

Bosco García-Archilla
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Julia Novo
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain

Edriss S. Titi
Affiliation: Department of Mathematics and Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697-3875, USA
Email: etiti@math.uci-edu

Keywords: Dissipative equations, spectral methods, approximate inertial manifolds, nonlinear Galerkin methods
Received by editor(s): June 19, 1996
Received by editor(s) in revised form: February 9, 1998
Published electronically: February 19, 1999
Article copyright: © Copyright 1999 American Mathematical Society