Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization

Abstract

A fully discrete multi-level spectral Galerkin method in space–time for the two-dimensional nonstationary Navier–Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier–Stokes problem is only solved on the lowest-dimensional space $H_{m_{1}}$ with the largest time step Δt 1; subsequent approximations are generated on a succession of higher-dimensional spaces $H_{m_{j}}$ with small time step Δt j by solving a linearized Navier–Stokes problem about the solution on the previous level. Some error estimates are also presented for the J-level spectral Galerkin method. The scaling relations of the dimensional numbers and time mesh widths that lead to optimal accuracy of the approximate solution in H 1-norm and L 2-norm are investigated, i.e., m j∼m j−1 3/2 , Δt j∼Δt j−1 3/2 , j=2,. . .,J. We demonstrate theoretically that a fully discrete J-level spectral Galerkin method is significantly more efficient than the standard one-level spectral Galerkin method.