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.
The talk gives an introduction to a space-time discretisation using the 2nd
order accurate Crank-Nicolson discretisation in time. A special arrangement of
the primal and dual solutions in time is proposed, which necessitates changes
to the prolongation and restriction operators. The resulting scheme has a
numerical complexity similar to a standard discretisation with implicit Euler.