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

Abstract: We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem and we propose multigrid methods to solve the discretized system. We prove that the $W$-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains. Numerical results are shown for both $W$ -cycle and $V$-cycle algorithms.