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

García-Archilla, Bosco; Novo, Julia; Titi, Edriss

doi:10.1090/S0025-5718-99-01057-1

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ISSN 1088-6842(online) ISSN 0025-5718(print)

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
Posted: February 19, 1999
MathSciNet review: 1627785
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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