Transforming smoothers for PDE constrained optimization problems

Transforming smoothers are known as a successful approach to the multigrid treatment of saddlepoint problems resulting from variational problems. In this paper we analyze similar multigrid methods in the context of the practically important class of optimization problems with partial differential equation constraint, which lead to a different kind of saddle point problems. We prove level independent convergence properties for the resulting multigrid methods and demonstrate this also in numerical investigations for a relevant model problem.