An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier-Stokes equations

An optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two-dimensional steady incompressible Navier-Stokes equations. This method is based on two finite element spaces X-H and X-h for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h much less than H, respectively, and a finite element space M-h for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H-5. If we choose H = O(h(2/5)), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. (C) 2003 Wiley Periodicals, Inc.