A fully discrete nonLinear Galerkin method for the 3D Navier-Stokes equations

The purpose of this paper is twofold: (i) We show that the Fourier-based Nonlinear Galerkin Method (NLGM) constructs suitable weak solutions to the periodic Navier-Stokes equations in three space dimensions provided the large scale/small scale cutoff is appropriately chosen, (ii) If smoothness is assumed, NLGM always outperforms the Galerkin method by a factor equal to 1 in the convergence order of the H¹-norm for the velocity and the L ²-norm for the pressure. This is a purely linear superconvergence effect resulting from standard elliptic regularity and holds independently of the nature of the boundary conditions (whether periodicity or no-slip BC is enforced).