Adaptive wavelet methods for saddle point problems

Stephan Dahlke; Reinhard Hochmuth; Karsten Urban

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 5, page 1003-1022
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Dahlke, Stephan, Hochmuth, Reinhard, and Urban, Karsten. "Adaptive wavelet methods for saddle point problems." ESAIM: Mathematical Modelling and Numerical Analysis 34.5 (2010): 1003-1022. <http://eudml.org/doc/197569>.

@article{Dahlke2010,
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. },
author = {Dahlke, Stephan, Hochmuth, Reinhard, Urban, Karsten},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Adaptive schemes; aposteriori error estimates; multiscale methods; wavelets; saddle point problems; Uzawa's algorithm.; adaptive schemes; Uzawa's algorithm; a posteriori error estimates; numerical examples},
language = {eng},
month = {3},
number = {5},
pages = {1003-1022},
publisher = {EDP Sciences},
title = {Adaptive wavelet methods for saddle point problems},
url = {http://eudml.org/doc/197569},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Dahlke, Stephan
AU - Hochmuth, Reinhard
AU - Urban, Karsten
TI - Adaptive wavelet methods for saddle point problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 5
SP - 1003
EP - 1022
AB - 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.
LA - eng
KW - Adaptive schemes; aposteriori error estimates; multiscale methods; wavelets; saddle point problems; Uzawa's algorithm.; adaptive schemes; Uzawa's algorithm; a posteriori error estimates; numerical examples
UR - http://eudml.org/doc/197569
ER -

References

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