{
  "id": "0902.0788",
  "version": "v1",
  "published": "2009-02-04T20:41:39.000Z",
  "updated": "2009-02-04T20:41:39.000Z",
  "title": "On the characterization of asymptotic cases of the diffusion equation with rough coefficients and applications to preconditioning",
  "authors": [
    "Burak Aksoylu",
    "Horst R. Beyer"
  ],
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We consider the diffusion equation in the setting of operator theory. In particular, we study the characterization of the limit of the diffusion operator for diffusivities approaching zero on a subdomain $\\Omega_1$ of the domain of integration of $\\Omega$. We generalize Lions' results to covering the case of diffusivities which are piecewise $C^1$ up to the boundary of $\\Omega_1$ and $\\Omega_2$, where $\\Omega_2 := \\Omega \\setminus \\overline{\\Omega}_1$ instead of piecewise constant coefficients. In addition, we extend both Lions' and our previous results by providing the strong convergence of $(A_{\\bar{p}_\\nu}^{-1})_{\\nu \\in \\mathbb{N}^\\ast},$ for a monotonically decreasing sequence of diffusivities $(\\bar{p}_\\nu )_{\\nu \\in \\mathbb{N}^\\ast}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-04T20:41:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "47F05",
      "65J10",
      "65N99"
    ],
    "keywords": [
      "diffusion equation",
      "asymptotic cases",
      "rough coefficients",
      "characterization",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.0788A"
    }
  }
}