We present a multigrid solution concept for the optimal distributed control of
the time-dependent Navier--Stokes equation. This problem is described by a fully
coupled KKT system in space and time involving primal and dual variables for
velocity and pressure.
In this talk we present basic concepts and ingredients which are necessary for
setting up a hierarchical solver for such systems. The underlying KKT system is
discretised in a monolithic way on the whole space-time domain using finite
elements in space and a one-step-$/theta$-schemes in time. A global Newton
solver is applied to solve for the nonlinearity, while a space-time multigrid
solver is used for the linear subproblems. We obtain a robust solver whose
convergence behaviour is quite independent of the number of unknowns of the
discrete problem and robust with respect to the considered flow configuration. A
set of numerical examples demonstrates the feasibility of this approach.