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) [1] and Approximate Inertial Manifolds (AIMs) [2]. 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 the 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 $r\geq 0$.