  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.
