{
  "id": "0905.1645",
  "version": "v2",
  "published": "2009-05-11T15:50:06.000Z",
  "updated": "2009-06-05T11:45:58.000Z",
  "title": "Numerical analysis of nonlinear eigenvalue problems",
  "authors": [
    "Eric Cancès",
    "Rachida Chakir",
    "Yvon Maday"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form $-{div} (A\\nabla u) + Vu + f(u^2) u = \\lambda u$, $\\|u\\|_{L^2}=1$. We focus in particular on the Fourier spectral approximation (for periodic problems) and on the $\\P_1$ and $\\P_2$ finite-element discretizations. Denoting by $(u_\\delta,\\lambda_\\delta)$ a variational approximation of the ground state eigenpair $(u,\\lambda)$, we are interested in the convergence rates of $\\|u_\\delta-u\\|_{H^1}$, $\\|u_\\delta-u\\|_{L^2}$ and $|\\lambda_\\delta-\\lambda|$, when the discretization parameter $\\delta$ goes to zero. We prove that if $A$, $V$ and $f$ satisfy certain conditions, $|\\lambda_\\delta-\\lambda|$ goes to zero as $\\|u_\\delta-u\\|_{H^1}^2+\\|u_\\delta-u\\|_{L^2}$. We also show that under more restrictive assumptions on $A$, $V$ and $f$, $|\\lambda_\\delta-\\lambda|$ converges to zero as $\\|u_\\delta-u\\|_{H^1}^2$, thus recovering a standard result for {\\em linear} elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error $u_\\delta-u$ in negative Sobolev norms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-06-05T11:45:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N15",
      "65N25",
      "65N30",
      "65T99",
      "35J60",
      "35P30"
    ],
    "keywords": [
      "nonlinear eigenvalue problems",
      "numerical analysis",
      "variational approximation",
      "nonlinear elliptic eigenvalue problems",
      "ground state eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.1645C"
    }
  }
}