In this talk, we deal with a special solver methodology for the numerical treatment of distributed optimal control problems based on the nonstationary Navier--Stokes problem. This problem is formulated as a KKT system involving a primal equation forward in time and a dual equation backward in time. A simple analysis shows that this space-time boundary value problem is of elliptic type -- a fact which indicates that multigrid techniques may work well when being applied to the whole space-time system.

We discretise the KKT system using LBB-stable finite elements in space and a one-step-$/theta$-scheme in time. For solving the discrete problem, we combine a space-time Newton solver with a space-time Multigrid solver and simultaneously solve for primal and dual variables in all timesteps.
Local subproblems in space are treated by a monolithic Multigrid solver.
The high efficiency of this combination is demonstrated on a number of numerical examples of driven-cavity and flow-around-cylinder type.
The solver is robust with regard to the considered flow confguration, converges in very few nonlinear steps and shows convergence rates which are independent on the number of unknowns.