{
  "id": "1605.09104",
  "version": "v1",
  "published": "2016-05-30T05:06:20.000Z",
  "updated": "2016-05-30T05:06:20.000Z",
  "title": "A priori estimates of a finite element method for fractional diffusion problems by energy arguments",
  "authors": [
    "Samir Karaa",
    "Kassem Mustapha",
    "Amiya K. Pani"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "In this article, the Galerkin piecewise-linear finite element (FE) method is applied to approximate the solution of time-fractional diffusion equations with variable diffusivity on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, {\\it a priori} optimal error bounds in $L^2(\\Omega)$-, $H^1(\\Omega)$-, and quasi-optimal in $L^{\\infty}(\\Omega)$-norms are derived for the semidiscrete FE scheme for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a $t^m$ type of weights to take care of the singular behavior at $t=0.$ The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-30T05:06:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite element method",
      "fractional diffusion problems",
      "energy arguments",
      "priori estimates",
      "galerkin piecewise-linear finite element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}