Adaptive wavelet methods for saddle point problems

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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.

Original languageEnglish
JournalMathematical Modelling and Numerical Analysis. Modélisation mathématique et analyse numérique
Issue number5
Pages (from-to)1003-1022
Number of pages20
Publication statusPublished - 01.09.2000
Externally publishedYes

Bibliographical note

© EDP Sciences, SMAI, 2000

The work of the first two authors has been supported by Deutsche Forschungsgemeinschaft (DFG) under Grants Da 117/13-1 and Ho 1846/1-1, respectively. Moreover, this work was supported by the European Commission within the TMR project (Training and Mobility for Researchers) Wavelets and Multiscale Methods in Numerical Analysis and Simulation, No. ERB FMRX CT98 018T4 and by the German Academic Exchange Service (DAAD) within the Vigoni–Project Multilevel–
Zerlegungsverfahren f ̈ur Partielle Differentialgleichungen. This paper was partially written when the third author was in residence at the Istituto di Analisi Numerica del C.N.R. in Pavia, Italy.

    Research areas

  • Mathematics
  • A posteriori error estimates, Adaptive schemes, Multiscale methods, Saddle point problems, Uzawa's algorithm, Wavelets