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

We semi-discretize in space a time-dependent Navier-Stokes system on a thre e-dimensional polyhedron by finite-elements schemes defined on two grids. I n the first step, the fully non-linear problem is semi-discretized on a coa rse grid, with mesh-size H. In the second step, the problem is linearized b y 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 me sh-size h. This approach is motivated by the fact that, on a convex polyhed ron and under adequate assumptions on the data, the contribution of uH to t he 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.