Optimum mixed finite element nonlinear Galerkin method for the Navier-Stokes equations; I: Error estimates for spatial discretization

In this article we present an optimum nonlinear Galerkin method using mixed finite elements for the two-dimensional incompressible Navier-Stokes equations.The scheme is based on two finite element spaces XH and Xh for the approximation of the velocity , defined respectively on one coarse grid with grid size H and one fine grid with grid size h << H and finite element space Mh for the approximation of the pressure.Nonlinearity is treated on the coarse grid and linearity is treated on the fine grid .We prove that the difference between the optimum nonlinear Galerkin method and the standard Galerkin method is of the order of H 3 , both in velocity (H 1 (n)2 norm) and pressure (L2(n) norm) .