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

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_m1 with the largest time step Δt ₁; subsequent approximations are generated on a succession of higher-dimensional spaces H_mj 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. © Springer 2006.