On a multilevel approach for the two dimensional Navier-Stokes equations with finite elements

We study if the multilevel algorithm introduced in Debussche et al. (Theor: Comput. Fluid Dynam., 7, 279-315 (1995)) and Dubois et al. (J. Sci. Comp., 8, 167-194 (1993)) for the 2D Navier-Stokes equations with periodic boundary conditions and spectral discretization can be generalized to more general boundary conditions and to finite elements. We first show that a direct generalization, as in Calgaro et al. (Appl. Numer. Math., 21, 1-40 (1997)), for the Burgers equation, would not be very efficient. We then propose a new approach where the domain of integration is decomposed in subdomains. This enables us to define localized small-scale components and we show that, in this context, there is a good separation of scales. We conclude that all the ingredients necessary for the implementation of the multilevel algorithm are present. (C) 1998 John Wiley & Sons, Ltd.