TY - JOUR

T1 - Adaptive wavelet methods for saddle point problems

AU - Dahlke, Stephan

AU - Hochmuth, Reinhard

AU - Urban, Karsten

N1 - © 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.

PY - 2000/9/1

Y1 - 2000/9/1

N2 - 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.

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.

KW - Mathematics

KW - A posteriori error estimates

KW - Adaptive schemes

KW - Multiscale methods

KW - Saddle point problems

KW - Uzawa's algorithm

KW - Wavelets

UR - http://www.scopus.com/inward/record.url?scp=0034356805&partnerID=8YFLogxK

U2 - 10.1051/m2an:2000113

DO - 10.1051/m2an:2000113

M3 - Journal articles

VL - 34

SP - 1003

EP - 1022

JO - Mathematical Modelling and Numerical Analysis. Modélisation mathématique et analyse numérique

JF - Mathematical Modelling and Numerical Analysis. Modélisation mathématique et analyse numérique

SN - 0764-583X

IS - 5

ER -