Multigrid preconditioning for discontinuous Galerkin discretizations of an elliptic optimal control problem with a convection-dominated state equation

TL;DR

This paper develops a multigrid preconditioning technique for discontinuous Galerkin discretizations of elliptic optimal control problems constrained by convection-dominated equations, achieving efficient solutions with proven convergence.

Contribution

It introduces a novel multilevel preconditioner based on downwind ordering that effectively handles convection dominance in DG discretizations of optimal control problems.

Findings

Proven global optimal convergence rates with explicit parameter tracking.

Preconditioner requires only two approximate solves using multigrid methods.

Numerical results confirm the efficiency and effectiveness of the proposed approach.

Abstract

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.