  In this work we construct multigrid preconditioners to accelerate the
solution process of a linear-quadratic optimal control problem constrained by
the Stokes system. The first order optimality conditions of the control problem
form a linear system (the KKT system) connecting the state, adjoint, and
control variables. Our approach is to eliminate the state and adjoint variables
by essentially solving two Stokes systems, and to construct efficient multigrid
preconditioners for the Schur-complement of the block associated with the state
and adjoint variables. These multigrid preconditioners are shown to be of
optimal order with respect to the convergence properties of the discrete
methods used to solve the Stokes system. In particular, the number of conjugate
gradient iterations is shown to decrease as the resolution increases, a feature
shared by similar multigrid preconditioners for elliptic constrained optimal
control problems.
