{
  "id": "1002.1859",
  "version": "v3",
  "published": "2010-02-09T13:55:49.000Z",
  "updated": "2012-05-23T15:29:11.000Z",
  "title": "Polynomial of best uniform approximation to $x^{-1}$ and smoothing in two-level methods",
  "authors": [
    "Johannes K. Kraus",
    "Panayot S. Vassilevski",
    "Ludmil T. Zikatanov"
  ],
  "comment": "23 pages 5 tables and 3 figures",
  "categories": [
    "math.NA"
  ],
  "abstract": "We derive a three-term recurrence relation for computing the polynomial of best approximation in the uniform norm to $x^{-1}$ on a finite interval with positive endpoints. As application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-23T15:29:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65F10",
      "15A06",
      "65F50",
      "65H10"
    ],
    "keywords": [
      "best uniform approximation",
      "two-level method",
      "polynomial",
      "scalar elliptic partial differential equation",
      "best approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.1859K"
    }
  }
}