Abstract: This paper presents a optimum nonlinear Galerkin algorithm for solving the two-dimensional penalized Navier-Stokes equations. The algorithm consists in solving a nonlinear subproblem on a coarse grid finite element space X_H⊂X_h(h<H) and solving a linear subproblem on a fine grid incremental finite element space W_h. If the linear finite elements are used and H=o(ε^(-1/4)h^1/2), then the algorithm is of the convergence rate of same order as the finite element Galerkin algorithm. However, this algorithm can save a large amount of computing time.

Key words: Penalized Navier-Stokes equations, Finite element, Nonlinear Galerkin algorithm