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.