A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme

We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity u(H) computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of u(H) to the error in the non-linear term, is measured in the L-2 norm in space and time, and thus has a higher-order than if it were measured in the H-1 norm in space. We present the following results: if h = H-2 = k, then the global error of the two-grid algorithm is of the order of h, the same as would have been obtained if the non-linear problem had been solved directly on the fine grid.