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Fourier nonlinear Galerkin method for Navier-Stokes equations
Nonlinear Galerkin Methods (NGMs) are numerical schemes for evolutionary partial
differential equations based on the theory of Inertial Manifolds
(IMs)  and Approximate
Inertial Manifolds (AIMs) . In this paper,
we focus our attention on the 2-D Navier-
Stokes equations with periodic boundary conditions, and use Fourier methods
to study its nonlinear Galerkin approximation which we call Fourier Nonlinear
Galerkin Methods (FNGMs) here. The first part is contributed to the semidiscrete case.
In this part, we derive the well-posedness of the nonlinear Galerkin form and
distance between the nonlinear Galerkin approximation
and the genuine solution in Sobolev spaces of any orders. The second part
is concerned about the full discrete case, in which, for a given numerical scheme
based on NGMs, we investigate the stability and error estimate respectively.
We derive the stability conditions for the scheme in any fractional Sobolev spaces.
Finally, we give its error estimate in $H^r$ for any $r\geq 0$.