A Fully Discrete NonLinear Galerkin Method for the 3D Navier–Stokes Equations J.-L. Guermond, 1, * Serge Prudhomme 2 1 Department of Mathematics, Texas A&M University 3368 TAMU, College Station, Texas 77843-3368 2 ICES, The University of Texas at Austin, Austin, Texas 78712 Received 1 October 2006; accepted 12 June 2007 Published online 11 September 2007 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.20287 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 pro- vided 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 H 1 -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 peri- odicity or no-slip BC is enforced). © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 24: 759–775, 2008 Keywords: Navier–Stokes equations; nonlinear Galerkin method; suitable solutions; turbulence I. INTRODUCTION A dissipative evolution equation over an Hilbert space H is said to have an Inertial Manifold if the manifold in question contains the global attractor, is positively invariant under the flow, attracts all the orbits exponentially, and is given as the graph of a C 1 map over a finite-dimensional subspace of H . This class of object has been proved to exist for many equations, but for the Navier–Stokes equations, even in dimension two, the question of the existence of an Inertial Manifold is still open. To fill this gap, the concept of Approximate Inertial Manifold (AIM) has been introduced [1, 2]. An AIM is a sequence of finite-dimensional manifolds in H , of increasing dimension, which are constructed so that the global attractor lies in small neighborhoods of these manifolds and the width of which rapidly shrinks as the dimension of the manifolds goes to infinity. Approximate Inertial Manifold have been shown to exist for the Navier–Stokes equations in two space dimensions [1]. The Nonlinear Galerkin Method (NLGM) is an approximation tech- nique that aims at constructing Approximate Inertial Manifolds (AIM) of nonlinear PDE’s; see Correspondence to: Jean Luc Guermond, Department of Mathematics, Texas A&M University 3368 TAMU, College Station, TX 77843-3368 (e-mail: guermond@math.tamu.edu) *On leave from LIMSI (CNRS-UPR 3251), BP 133, 91403, Orsay, France. © 2007 Wiley Periodicals, Inc.