{
  "id": "1903.10409",
  "version": "v1",
  "published": "2019-03-25T15:48:14.000Z",
  "updated": "2019-03-25T15:48:14.000Z",
  "title": "Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian",
  "authors": [
    "Markus Faustmann",
    "Jens Markus Melenk",
    "Dirk Praetorius"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "For the discretization of the integral fractional Laplacian $(-\\Delta)^s$, $0 < s < 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for a lack of $L^2$-regularity of the residual in the regime $3/4 < s < 1$, this weighted residual error estimator includes as an additional weight a power of the distance from the mesh skeleton. We prove optimal convergence rates for an $h$-adaptive algorithm driven by this error estimator. Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-25T15:48:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integral fractional laplacian",
      "quasi-optimal convergence rate",
      "adaptive method",
      "weighted residual",
      "optimal convergence rates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}