We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem constrained by a convection-dominated problem. We prove global optimal convergence rates using an inf-sup condition, with the diffusion parameter $\varepsilon$ and regularization parameter $\beta$ explicitly tracked. We then propose a multilevel preconditioner based on downwind ordering to solve the discretized system. The preconditioner only requires two approximate solves of single convection-dominated equations using multigrid methods. Moreover, for the strongly convection-dominated case, only two sweeps of block Gauss-Seidel iterations are needed. We also derive a simple bound indicating the role played by the multigrid preconditioner. Numerical results are shown to support our findings.

翻译：本文研究对流占优问题约束的椭圆分布最优控制问题的间断Galerkin方法。利用inf-sup条件，我们证明了全局最优收敛率，并显式追踪了扩散参数$\varepsilon$和正则化参数$\beta$。随后，我们提出了一种基于下风排序的多层预处理器来求解离散化系统。该预处理器仅需使用多重网格方法对单个对流占优方程进行两次近似求解。此外，对于强对流占优情形，仅需两次块Gauss-Seidel迭代扫描。我们还推导了一个简单界，用以说明多重网格预处理器所起的作用。数值结果验证了我们的结论。