Advances in Computational Mathematics

, Volume 27, Issue 1, pp 27–63

Adaptive frame methods for elliptic operator equations

  • Stephan Dahlke
  • Massimo Fornasier
  • Thorsten Raasch
Article

DOI: 10.1007/s10444-005-7501-6

Cite this article as:
Dahlke, S., Fornasier, M. & Raasch, T. Adv Comput Math (2007) 27: 27. doi:10.1007/s10444-005-7501-6

Abstract

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are especially interested in discretization schemes based on frames. The central objective is to derive an adaptive frame algorithm which is guaranteed to converge for a wide range of cases. As a core ingredient we use the concept of Gelfand frames which induces equivalences between smoothness norms and weighted sequence norms of frame coefficients. It turns out that this Gelfand characteristic of frames is closely related to their localization properties. We also give constructive examples of Gelfand wavelet frames on bounded domains. Finally, an application to the efficient adaptive computation of canonical dual frames is presented.

Keywords

operator equations multiscale methods adaptive algorithms domain decomposition sparse matrices overdetermined systems Banach frames norm equivalences Banach spaces 

Mathematics subject classifications (2000)

41A25 41A46 42C15 42C40 46E35 65F10 65F20 65F50 65N12 65N55 65T60 

Copyright information

© Springer 2006

Authors and Affiliations

  • Stephan Dahlke
    • 1
  • Massimo Fornasier
    • 2
  • Thorsten Raasch
    • 1
  1. 1.FB 12 Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany
  2. 2.Dipartimento di Metodi e Modelli Matematici per le Scienze ApplicateUniversità “La Sapienza” in RomaRomaItaly

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