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 uH 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 L2 norm in space and time, and thus, for the lowest-degree elements, is of the order of H2. Hence, an error of the order of h can be recovered at the second step, provided h = H2.