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

Abstract

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. © 1998 John Wiley & Sons, Ltd.