{
  "id": "2012.13003",
  "version": "v1",
  "published": "2020-12-23T22:19:06.000Z",
  "updated": "2020-12-23T22:19:06.000Z",
  "title": "Robust preconditioning and error estimates for optimal control of the convection-diffusion-reaction equation with limited observation in Isogeometric analysis",
  "authors": [
    "Kent-Andre Mardal",
    "Jarle Sogn",
    "Stefan Takacs"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In this paper we analyze an optimization problem with limited observation governed by a convection--diffusion--reaction equation. Motivated by a Schur complement approach, we arrive at continuous norms that enable analysis of well-posedness and subsequent derivation of error analysis and a preconditioner that is robust with respect to the parameters of the problem. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection. We also provide a priori error estimates for this discretization. The preconditioner requires a fourth order problem to be solved. For this reason, we use Isogeometric Analysis as a method of discretization. To efficiently realize the preconditioner, we consider geometric multigrid with a standard Gauss-Seidel smoother as well as a new macro Gauss-Seidel smoother. The latter smoother provides good results with respect to both the geometry mapping and the polynomial degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-23T22:19:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isogeometric analysis",
      "error estimates",
      "convection-diffusion-reaction equation",
      "limited observation",
      "optimal control"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}