{
  "id": "1508.02807",
  "version": "v1",
  "published": "2015-08-12T03:29:24.000Z",
  "updated": "2015-08-12T03:29:24.000Z",
  "title": "A first-degree FEM for an optimal control problem of fractional operators: error analysis",
  "authors": [
    "Enrique Otarola"
  ],
  "categories": [
    "math.NA",
    "math.OC"
  ],
  "abstract": "We study a discretization technique for a linear-quadratic optimal control problem involving fractional diffusion of order $s \\in (0,1)$. Since fractional diffusion can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic equation, we recast our problem as a nonuniformly elliptic optimal control problem. The rapid decay of the solution suggests a truncation that is suitable for numerical approximation. We discretize the truncated problem with a fully discrete scheme based on a piecewise linear finite element approximation on quasi-uniform meshes for the optimal control. The state variable is approximated via first--degree tensor product finite elements on anisotropic meshes. Based on derived H\\\"older and Sobolev regularity results for the optimal control, we develop an a priori error analysis for $s \\in (0,1)$. Numerical experiments validate the derived error estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-12T03:29:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "error analysis",
      "linear finite element approximation",
      "elliptic optimal control problem",
      "fractional operators",
      "first-degree fem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150802807O"
    }
  }
}