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Adaptive wavelet methods for saddle point problems

Published online by Cambridge University Press:  15 April 2002

Stephan Dahlke ,
Reinhard Hochmuth  and
Karsten Urban
Stephan Dahlke
Affiliation:
RWTH Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany. (dahlke@igpm.rwth-aachen.de)
Reinhard Hochmuth
Affiliation:
FU Berlin, FB Mathematik, Arnimallee 2-6, 14195 Berlin, Germany. (hochmuth@math.fu-berlin.de)
Karsten Urban
Affiliation:
RWTH Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany. (urban@igpm.rwth-aachen.de)
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Abstract

Recently, adaptive wavelet strategies for symmetric, positive definite operators have been introduced that were proven to converge. This paper is devoted to the generalization to saddle point problems which are also symmetric, but indefinite. Firstly, we investigate a posteriori error estimates and generalize the known adaptive wavelet strategy to saddle point problems. The convergence of this strategy for elliptic operators essentially relies on the positive definite character of the operator. As an alternative, we introduce an adaptive variant of Uzawa's algorithm and prove its convergence. Secondly, we derive explicit criteria for adaptively refined wavelet spaces in order to fulfill the Ladyshenskaja-Babuška-Brezzi (LBB) condition and to be fully equilibrated.

Keywords

Adaptive schemesaposteriori error estimatesmultiscale methodswaveletssaddle point problemsUzawa's algorithm.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 5 , September 2000 , pp. 1003 - 1022
Copyright
© EDP Sciences, SMAI, 2000

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