Optimal approximation of elliptic problems by linear and nonlinear mappings I

  • Authors:

  • Stephan Dahlke;Erich Novak;Winfried Sickel

  • Affiliations:

  • Philipps-Universität Marburg, FB12 Mathematik und Informatik, Hans-Meerwein Straße, Lahnberge, 35032 Marburg, Germany;Friedrich-Schiller-Universität Jena, Mathematisches Institut, Ernst-Abbe-Platz 2, 07743 Jena, Germany;Friedrich-Schiller-Universität Jena, Mathematisches Institut, Ernst-Abbe-Platz 2, 07743 Jena, Germany

  • Venue:
  • Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
  • Year:
  • 2006

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Abstract

We study the optimal approximation of the solution of an operator equation A(u)=f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A:H"0^s(@W)-H^-^s(@W), where s0 and @W is an arbitrary bounded Lipschitz domain in R^d. We prove that approximation by linear mappings is as good as the best n-term approximation with respect to an optimal Riesz basis. We discuss why nonlinear approximation still is important for the approximation of elliptic problems.