{
  "id": "1405.2567",
  "version": "v2",
  "published": "2014-05-11T18:54:24.000Z",
  "updated": "2015-03-28T23:27:37.000Z",
  "title": "A Spectral Method for Nonlinear Elliptic Equations",
  "authors": [
    "Kendall Atkinson",
    "David Chien",
    "Olaf Hansen"
  ],
  "comment": "26 pages. arXiv admin note: text overlap with arXiv:0909.3607",
  "categories": [
    "math.NA"
  ],
  "abstract": "Let $\\Omega$ be an open, simply connected, and bounded region in $\\mathbb{R}^{d}$, $d\\geq2$, and assume its boundary $\\partial\\Omega$ is smooth. Consider solving an elliptic partial differential equation $Lu=f$ over $\\Omega$ with zero Dirichlet boundary value. The function $f$ is a nonlinear function of the solution $u$. The problem is converted to an equivalent\\ elliptic problem over the open unit ball $\\mathbb{B}^{d}$ in $\\mathbb{R}^{d}$, say $\\widetilde{L}\\widetilde{u}=\\widetilde{f}$. Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials $\\widetilde{u}_{n}$ of degree $\\leq n$ that is convergent to $\\widetilde{u}$. The transformation from $\\Omega$ to $\\mathbb{B}^{d}$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\\in C^{\\infty}\\left( \\overline{\\Omega }\\right) $ and assuming $\\partial\\Omega$ is a $C^{\\infty}$ boundary, the convergence of $\\left\\Vert \\widetilde{u}-\\widetilde{u}_{n}\\right\\Vert _{H^{1}% }$ \\ to zero is faster than any power of $1/n$. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to $-\\Delta u+\\gamma u=f$ with a zero Neumann boundary condition is also presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-11T18:54:24.000Z",
      "abstract": "Let $\\Omega$ be an open, simply connected, and bounded region in $\\mathbb{R}^{d}$, $d\\geq2$, and assume its boundary $\\partial\\Omega$ is smooth. Consider solving an elliptic partial differential equation $Lu=f$ over $\\Omega$ with zero Dirichlet boundary value. The function $f$ is a nonlinear function of the solution $u$. The problem is converted to an equivalent\\ elliptic problem over the open unit ball $\\mathbb{B}^{d}$ in $\\mathbb{R}^{d}$, say $\\widetilde{L}\\widetilde{u}=\\widetilde{f}$. Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials $\\widetilde{u}_{n}$ of degree $\\leq n$ that is convergent to $\\widetilde{u}$. The transformation from $\\Omega$ to $\\mathbb{B}^{d}$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\\in C^{\\infty}\\left(\\overline{\\Omega}\\right) $ and assuming $\\partial\\Omega$ is a $C^{\\infty}$ boundary, the convergence of $\\left\\Vert \\widetilde{u}-\\widetilde{u}_{n}\\right\\Vert _{H^{1}%}$ \\ to zero is faster than any power of $1/n$. Numerical examples illustrate experimentally an exponential rate of convergence.",
      "comment": "16 pages. arXiv admin note: text overlap with arXiv:0909.3607",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-28T23:27:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N35"
    ],
    "keywords": [
      "nonlinear elliptic equations",
      "spectral method",
      "elliptic partial differential equation",
      "zero dirichlet boundary value",
      "convergent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.2567A"
    }
  }
}