{
  "id": "2406.09276",
  "version": "v1",
  "published": "2024-06-13T16:14:34.000Z",
  "updated": "2024-06-13T16:14:34.000Z",
  "title": "Multigrid preconditioning for discontinuous Galerkin discretizations of an elliptic optimal control problem with a convection-dominated state equation",
  "authors": [
    "Sijing Liu",
    "Valeria Simoncini"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-13T16:14:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J20",
      "49M41",
      "65N30",
      "65N55"
    ],
    "keywords": [
      "elliptic optimal control problem",
      "discontinuous galerkin discretizations",
      "convection-dominated state equation",
      "multigrid preconditioning",
      "elliptic distributed optimal control problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}