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

Abstract 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 XH and Xh for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ≪ H, respectively, and a finite element space Mh 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 H5. If we choose H = O(h2/5), 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. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.