Iterative methods which keep some problem--dependent subspace invariant have been proven to be appropriate smoothers for multigrid methods. The Braess-Sarazin smoother for saddle point problems belongs to this class, where the invariant subspace is the kernel of the operator describing the constraints. Performing one step of this iterative method, however, requires the accurate solution of some global linear system for the dual variables. Another iterative method with invariant subspace has been proposed by Joachim Schoeberl for parameter dependent elliptic problems and the limiting saddle point problem. If this smoother is accompanied with some specific (kernel-preserving) prolongation robust multigrid convergence results were shown by Schoeberl. The smoother as well as the prolongation can be realized by solving a number of local problems. In this talk it will be shown that multigrid convergence can also be achieved for standard prolongation and local kernel--preserving smoothers for saddle point problems.