Journal of Computational Mathematics, Vol.26, No.2, 2008, 209–226. FULL DISCRETE TWO-LEVEL CORRECTION SCHEME FOR NAVIER-STOKES EQUATIONS * Yanren Hou and Liquan Mei College of Science, Xi’an Jiaotong University, Xi’an 710049, China Email: yrhou@mail.xjtu.edu.cn, lqmei@mail.xjtu.edu.cn Abstract In this paper, a full discrete two-level scheme for the unsteady Navier-Stokes equations based on a time dependent projection approach is proposed. In the sense of the new projection and its related space splitting, non-linearity is treated only on the coarse level subspace at each time step by solving exactly the standard Galerkin equation while a linear equation has to be solved on the fine level subspace to get the final approximation at this time step. Thus, it is a two-level based correction scheme for the standard Galerkin approximation. Stability and error estimate for this scheme are investigated in the paper. Mathematics subject classification: 65M55, 65M70. Key words: Two-level method, Galerkin approximation, Correction, Navier-Stokes equa- tion. 1. Introduction We consider the two-dimensional Navier-Stokes equations du dt + νAu + B(u, u)= f, u(0) = u 0 , (1.1) in certain divergence-free Hilbert space H , where u 0 is the initial velocity field, A the Stokes operator, B the projection of the non-linearity on H , ν> 0 the kinetic viscosity and f the external force. To get efficient numerical schemes, the two-level (two-grid) strategy has been widely studied. In particular, a class of two-level method in connection with the approximate inertial manifolds (AIMs) initialized by Foias, Manley and Temam [5] has been extensively studied in the past decades, which is usually called the nonlinear Galerkin method (NLG). Let φ i be the ith eigenvector of the Stokes operator A corresponding to the associated eigenvalue λ i . For given m, M ∈ N (m<M ), let P m (P M ) denote the spectral projection from H onto the space spanned by the first m (M ) eigenvectors. And we also set Q m = I − P m (Q M = I − P M ). The semi-discrete NLG reads: solve (1.1) up to a given time t 0 by a standard Galerkin method (SGM) in the fine-level subspace, and for t>t 0 find v m ∈ P m H and ˆ w m ∈ (P M − P m )H such that dv m dt + νAv m + P m B(v m +ˆ w m ,v m +ˆ w m )= P m f, (1.2) ˆ w m = Φ(v m ). (1.3) * Received November 21, 2006 / Revised version received February 28, 2007 / Accepted April 16, 2007 /