Multigrid preconditioning for discontinuous Galerkin discretizations of an elliptic optimal control problem with a convection-dominated state equation
<|reference_start|>Multigrid preconditioning for discontinuous Galerkin discretizations of an elliptic optimal control problem with a convection-dominated state equation: 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.<|reference_end|>
@article{liu2024multigrid,
    title={Multigrid preconditioning for discontinuous Galerkin discretizations of
  an elliptic optimal control problem with a convection-dominated state
  equation},
    author={Sijing Liu and Valeria Simoncini},
    journal={arXiv preprint arXiv:2406.09276},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.09276},
    primaryClass={math.NA cs.NA}
}