{
  "id": "1709.10223",
  "version": "v1",
  "published": "2017-09-29T03:02:37.000Z",
  "updated": "2017-09-29T03:02:37.000Z",
  "title": "Superconvergence Points For The Spectral Interpolation Of Riesz Fractional Derivatives",
  "authors": [
    "Beichuan Deng",
    "Zhimin Zhang",
    "Xuan Zhao"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "In this paper, superconvergence points are located for the approximation of the Riesz derivative of order $\\alpha$ using classical Lobatto-type polynomials when $\\alpha \\in (0,1)$ and generalized Jacobi functions (GJF) for arbitrary $\\alpha > 0$, respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre-Lobatto expansion. It is observed that the convergence rate for different $\\alpha$ at the superconvergence points is at least $O(N^{-2})$ better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of order $\\alpha > 1$ with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be $O(N^{-(\\alpha+3)/2})$ for $\\alpha \\in (0,1)$ and $O(N^{-2})$ for $\\alpha > 1$. Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-29T03:02:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N35",
      "65M15",
      "26A33",
      "41A05",
      "41A10"
    ],
    "keywords": [
      "superconvergence points",
      "riesz fractional derivative",
      "spectral interpolation",
      "optimal global convergence rate",
      "solving model fdes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}