A two-level method in space and time for the Navier-Stokes equations

A two-level method in space and time for the time-dependent Navier-Stokes equations is considered in this article. The approximate solution u(M) is an element of H-M is decomposed into the large eddy component v is an element of H-m(m < M) and the small eddy component w is an element of H-m(perpendicular to). We obtain the large eddy component v by solving a standard Galerkin equation in a coarse-level subspace H-m with a time step length k, whereas the small eddy component w is derived by solving a linear equation in an orthogonal complement subspace H-m(perpendicular to) with a time step length pk, where p is a positive integer. The analysis shows that our two-level scheme has long-time stability and can reach the same accuracy as the standard Galerkin method in fine-level subspace H-M for an appropriate configuration of p and m. Moreover, some numerical examples are provided to complement our theoretical analysis. (C) 2012 Wiley Periodicals, Inc.