Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations

A fully discrete two-level finite element method (the two-level method) is presented for solving the two-dimensional time-dependent Navier-Stokes problem. The method requires a Crank-Nicolson extrapolation solution (u(H,tau0), p(H,tau0)) on a spatial-time coarse grid J(H,tau0) and a backward Euler solution (u(h,tau), p(h,tau)) on a space-time fine grid J(h,tau). The error estimates of optimal order of the discrete solution for the two-level method are derived. Compared with the standard Crank-Nicolson extrapolation method (the one-level method) based on a space-time fine grid J(h,tau), the two-level method is of the error estimates of the same order as the one-level method in the H-1-norm for velocity and the L-2-norm for pressure. However, the two-level method involves much less work than the one-level method.