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IMA Journal of Numerical Analysis Advance Access originally published online on February 16, 2007
IMA Journal of Numerical Analysis 2007 27(4):717-740; doi:10.1093/imanum/drl035
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Adaptive frame methods for elliptic operator equations: the steepest descent approach

Stephan Dahlke{dagger}, Thorsten Raasch{ddagger} and Manuel Werner§

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

Massimo Fornasier

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università "La Sapienza" in Roma, Via Antonio Scarpa, 16/B, I-00161 Roma, Italy

Rob Stevenson||

Department of Mathematics, Utrecht University, PO Box 80.010, NL-3508 TA Utrecht, The Netherlands

{dagger} Email: dahlke{at}mathematik.uni-marburg.de

{ddagger} Email: raasch{at}mathematik.uni-marburg.de

§ Corresponding author. Email: werner{at}mathematik.uni-marburg.de

Email: mfornasi{at}math.unipd.it

|| Email: stevenson{at}math.uu.nl

Received on 20 February 2006. Revised on 6 October 2006.


  Abstract

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are particularly interested in discretization schemes based on wavelet frames. We show that by using three basic subroutines an implementable, convergent scheme can be derived, which, moreover, has optimal computational complexity. The scheme is based on adaptive steepest descent iterations. We illustrate our findings by numerical results for the computation of solutions of the Poisson equation with limited Sobolev smoothness on intervals in 1D and L-shaped domains in 2D.

Key Words: operator equations; multiscale methods; adaptive algorithms; sparse matrices; Banach frames; norm equivalences


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