Optimal approximation of elliptic problems by linear and nonlinear mappings IV: Errors in L2 and other norms

  • Authors:

  • Stephan Dahlke;Erich Novak;Winfried Sickel

  • Affiliations:

  • Philipps-Universität Marburg, FB12 Mathematik und Informatik, Hans-Meerwein Straíe, 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
  • Year:
  • 2010

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Abstract

We study the optimal approximation of the solution of an operator equation A(u)=f by linear and different types of nonlinear mappings. In our earlier papers we only considered the error with respect to a certain H^s-norm where s was given by the operator since we assumed that A:H"0^s(@W)-H^-^s(@W) is an isomorphism. The most typical case here is s=1. It is well known that for certain regular problems the order of convergence is improved if one takes the L"2-norm. In this paper we study error bounds with respect to such a weaker norm, i.e., we assume that H"0^s(@W) is continuously embedded into a space X and we measure the error in the norm of X. A major example is X=L"2(@W) or X=H^r(@W) with r