A second order accuracy for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme

We study a second-order two-grid scheme fully discrete in time and space for solving the Navier-Stokes equations. The two-grid strategy consists in discretizing, in the first step, the fully non-linear problem, in space on a coarse grid with mesh-size H and time step Delta t and, in the second step, in discretizing the linearized problem around the velocity u (H) computed in the first step, in space on a fine grid with mesh-size h and the same time step. The two-grid method has been applied for an analysis of a first order fully-discrete in time and space algorithm and we extend the method to the second order algorithm. This 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 (2) = H (3) = (Delta t)(2), then the global error of the two-grid algorithm is of the order of h (2), the same as would have been obtained if the non-linear problem had been solved directly on the fine grid.