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An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations
Author(s):
Bosco
García-Archilla;
Julia
Novo;
Edriss
S.
Titi.
Journal:
Math. Comp.
68
(1999),
893-911.
MSC (1991):
Primary 65P25
Posted:
February 19, 1999
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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
Email:
bosco.garcia@uam.es
Julia
Novo
Affiliation:
Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain
Email:
jnovo@mac.mac.cie.uva.es
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
DOI:
10.1090/S0025-5718-99-01057-1
PII:
S 0025-5718(99)01057-1
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
Posted:
February 19, 1999
Copyright of article:
Copyright
1999,
American Mathematical Society
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