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.

翻译：我们考虑椭圆分布最优控制问题的间断伽辽金方法，并提出多重网格方法求解离散系统。我们证明，$W$循环算法在能量范数下一致收敛，且在凸区域上对正则化参数具有鲁棒性。文中同时给出了$W$循环与$V$循环算法的数值结果。