{
  "id": "1206.2686",
  "version": "v1",
  "published": "2012-06-12T22:38:17.000Z",
  "updated": "2012-06-12T22:38:17.000Z",
  "title": "Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations",
  "authors": [
    "Kassem Mustapha",
    "William McLean"
  ],
  "comment": "24 pages, 2 figures",
  "categories": [
    "math.NA"
  ],
  "abstract": "We consider an initial-boundary value problem for $\\partial_tu-\\partial_t^{-\\alpha}\\nabla^2u=f(t)$, that is, for a fractional diffusion ($-1<\\alpha<0$) or wave ($0<\\alpha<1$) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near $t=0$, but the spatial mesh is quasiuniform. Previously, we proved that the error, measured in the spatial $L_2$-norm, is of order $k^{2+\\alpha_-}+h^2\\ell(k)$, uniformly in $t$, where $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, $\\alpha_-=\\min(\\alpha,0)\\le0$ and $\\ell(k)=\\max(1,|\\log k|)$. Here, we generalize a known result for the classical heat equation (i.e., the case $\\alpha=0$) by showing that at each time level $t_n$ the solution is superconvergent with respect to $k$: the error is of order $(k^{3+2\\alpha_-}+h^2)\\ell(k)$. Moreover, a simple postprocessing step employing Lagrange interpolation yields a superconvergent approximation for any $t$. Numerical experiments indicate that our theoretical error bound is pessimistic if $\\alpha<0$. Ignoring logarithmic factors, we observe that the error in the DG solution at $t=t_n$, and after postprocessing at all $t$, is of order $k^{3+\\alpha_-}+h^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-12T22:38:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A33",
      "35R09",
      "45K05",
      "47G20",
      "65M12",
      "65M15",
      "65M60"
    ],
    "keywords": [
      "discontinuous galerkin method",
      "fractional diffusion",
      "wave equations",
      "postprocessing step employing lagrange",
      "step employing lagrange interpolation yields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.2686M"
    }
  }
}