Convergence analysis of multigrid methods with collective point smoothers for optimal control problems

In this paper we consider multigrid methods for solving saddle point problems. The choice of an appropriate smoothing strategy is a key issue in this case. Here we focus on the widely used class of collective point smoothers. These methods are constructed by a point-wise grouping of the unknowns leading to, e.g., collective Richardson, Jacobi or Gauss-Seidel relaxation methods. Their smoothing properties are well-understood for scalar problems in the symmetric and positive definite case. In this work the analysis of these methods is extended to a special class of saddle point problems, namely to the optimality system of optimal control problems. For elliptic distributed control problems we show that the convergence rates of multigrid methods with collective point smoothers are bounded independent of the grid size and the regularization (or cost) parameter.