In this paper, we study in two and three space dimensions, the a posteriori error estimates for the large eddy simulation (LES) applied to the Navier-Stokes system. We begin by introducing the Navier-Stokes and the corresponding LES equations. Then we introduce the corresponding discrete problem based on the finite element method. We establish a posteriori error estimation with three types of error indicators related to the filter of the LES method, to the discretization and to the linearization. Finally, numerical investigations are shown and discussed.

Abstract. We study the two dimensional time dependent Large Eddy Simulation method applied to the incompressible Navier-Stokes system with Smagorinsky’s eddy viscosity model and a filter width that depends on the local mesh size. The discrete model is based on the implicit Euler scheme and a conforming finite element method for the time and space discretizations, respectively. We establish a reliable and efficient a posteriori error estimation between the numerical LES solution and the exact solution of the original Navier-Stokes system, which involves three types of error indicators respectively related to the filter and to the discretizations in time and space. Numerical results show the effectiveness of adaptive simulations.