  In this paper we present an all-at-once multigrid method for a distributed
Stokes control problem (velocity tracking problem). For solving such a problem,
we use the fact that the solution is characterized by the optimality system
(Karush-Kuhn-Tucker-system). The discretized optimality system is a large-scale
linear system whose condition number depends on the grid size and on the choice
of the regularization parameter forming a part of the problem. Recently,
block-diagonal preconditioners have been proposed, which allow to solve the
problem using a Krylov space method with convergence rates that are robust in
both, the grid size and the regularization parameter or cost parameter. In the
present paper, we develop an all-at-once multigrid method for a Stokes control
problem and show robust convergence, more precisely, we show that the method
converges with rates which are bounded away from one by a constant which is
independent of the grid size and the choice of the regularization or cost
parameter.
