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We consider discontinuous Galerkin methods for an elliptic distributedoptimal control problem and we propose multigrid methods to solve thediscretized system. We prove that the $W$-cycle algorithm is uniformlyconvergent in the energy norm and is robust with respect to a regularizationparameter on convex domains. Numerical results are shown for both $W$ -cycleand $V$-cycle algorithms.

Robust Multigrid Methods for Discontinuous Galerkin Discretizations of an Elliptic Optimal Control Problem