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Adaptive frame methods for elliptic operator equations

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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.

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Authors and Affiliations

  1. FB 12 Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein Straße, Lahnberge, 35032, Marburg, Germany

    Stephan Dahlke & Thorsten Raasch

  2. Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università “La Sapienza” in Roma, Via Antonio Scarpa, 16/B, 00161, Roma, Italy

    Massimo Fornasier

Authors
  1. Stephan Dahlke
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  2. Massimo Fornasier
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  3. Thorsten Raasch
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Corresponding author

Correspondence to Stephan Dahlke.

Additional information

Communicated by Y. Xu

Mathematics subject classifications (2000)

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

The authors acknowledge the financial support provided through the European Union's Human Potential Programme, under contract HPRN-CT-2002-00285 (HASSIP). The work of the first author was also supported through DFG, Grants Da 360/4-1, Da 360/4-2. The second author also wants to thank the AG Numerik/Wavelet-Analysis Group, Philipps-Universität Marburg, and the ZeTeM, Universität Bremen, Germany, for the hospitality and the stimulating cooperations during the preparation of this work. Currently he is a Postdoc visitor (Individual Marie Curie Fellow, contract MEIF-CT-2004-501018) at Universität Wien, Fakultät für Mathematik, NuHAG, Nordbergstraß e 15, A-1090 Wien, Austria.

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Dahlke, S., Fornasier, M. & Raasch, T. Adaptive frame methods for elliptic operator equations. Adv Comput Math 27, 27–63 (2007). https://doi.org/10.1007/s10444-005-7501-6

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