Two-grid finite-element schemes for the transient Navier-Stokes problem

We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size H. In the second step, the problem is linearized by substituting into the non-linear term, the velocity u(H) computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size h. This approach is motivated by the fact that, on a convex polyhedron and under adequate assumptions on the data, the contribution of uH to the error analysis is measured in the L-2 norm in space and time, and thus, for the lowest-degree elements, is of the order of H-2. Hence, an error of the order of h can be recovered at the second step, provided h = H-2.