Multi-level spectral galerkin method for the navier-stokes problem I : spatial discretization

Abstract

A multi-level spectral Galerkin method for the two-dimensional non-stationary Navier-Stokes equations is presented. The method proposed here is a multiscale method in which the fully nonlinear Navier-Stokes equations are solved only on a low-dimensional space ** subsequent approximations are generated on a succession of higher-dimensional spaces ** j=2, . . . ,J, by solving a linearized Navier-Stokes problem around the solution on the previous level. Error estimates depending on the kinematic viscosity 0<ν<1 are also presented for the J-level spectral Galerkin method. The optimal accuracy is achieved when ** We demonstrate theoretically that the J-level spectral Galerkin method is much more efficient than the standard one-level spectral Galerkin method on the highest-dimensional space **.