LARGE EDDY SIMULATION BY THE NON-LINEAR GALERKIN METHOD JEAN-LUC GUERMOND AND SERGE PRUDHOMME Abstract. The purpose of this paper is to show that the Fourier-based Nonlinear Galerkin Method (NLGM) constructs suitable weak solutions to the periodic Navier–Stokes equations in three dimensions. We re-interpret NLGM as a Large-Eddy Simulation technique (LES) and we rigorously deduce a rela- tionship between the mesh size and the large-eddy scale. Key Words. Navier–Stokes equations, turbulence, large eddy simulation, nonlinear Galerkin method, suitable solutions. 1. Introduction Large eddy simulation methods for approximating turbulent flows are commonly viewed as techniques in which the governing equations are derived by applying a low-pass filter to the Navier–Stokes equations. These filtered equations are similar to the original equations but for the presence of the so-called subgrid scale stresses accounting for the influence of the small scales onto the large ones. Assuming that the behavior of the small scale structures is more or less universal, the objective of LES is then to find some models for the subgrid scale stresses, the so-called closure problem, and to compute the dynamics of the large scales by using the filtered equa- tions. Although this description of LES has been widely accepted over the years (see for example the books by Geurts [8], John [13], or Sagaut [17]), it nevertheless falls short of an unambiguous mathematical theory. Indeed, the filtering operators, which implicitly appear in the definition of the subgrid scale tensor, are often ig- nored while constructing the LES models. It is a common practice to work with filter length scales regardless of the actual filters being used. More importantly, it is now known that the closure problem actually yields a paradox; namely that it is possible to close exactly the LES equations, i.e., without invoking ad hoc hypothe- ses, by choosing a bijective filtering operator, see [7, 9]. In this case, there exists a one-to-one correspondence between the solution set of the Navier-Stokes equations and that of the filtered equations, which means that the same “number of degrees of freedom” should be used in both cases to represent any given solution. Another unjustified practice very often consists of assuming that the filtering length scale is equal to the mesh size of the approximation method that is used, regardless on the method in question. The above observations have led us to develop a research program aiming at constructing a mathematical framework for the large eddy simulation of turbu- lent flows. Our first step in this direction is to introduce the concept of suitable approximation (see Section 2.2 and [10]). The definition is essentially based on 2000 Mathematics Subject Classification. 35R35, 49J40, 60G40. Version: March 23, 2005. Submitted to M2AN. 1