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Wavelet bases in $\mathbf{H}( \mathrm{div})$ and $\mathbf{H}(\mathbf{curl})$


Author: Karsten Urban
Journal: Math. Comp. 70 (2001), 739-766
MSC (2000): Primary 65T60; Secondary 35Q60, 35Q30
Published electronically: May 19, 2000
MathSciNet review: 1710628
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Abstract:

Some years ago, compactly supported divergence-free wavelets were constructed which also gave rise to a stable (biorthogonal) wavelet splitting of $\mathbf{H}(\mathrm{div};\Omega)$. These bases have successfully been used both in the analysis and numerical treatment of the Stokes and Navier-Stokes equations. In this paper, we construct stable wavelet bases for the stream function spaces $\mathbf{H}(\mathbf{curl};\Omega)$. Moreover, $\mathbf{curl}$-free vector wavelets are constructed and analysed. The relationship between $ \mathbf{H}(\mathrm{div};\Omega)$ and $\mathbf{H}(\mathbf{curl};\Omega)$ are expressed in terms of these wavelets. We obtain discrete (orthogonal) Hodge decompositions.

Our construction works independently of the space dimension, but in terms of general assumptions on the underlying wavelet systems in $L^2(\Omega)$ that are used as building blocks. We give concrete examples of such bases for tensor product and certain more general domains $\Omega\subset\mathbb{R}^n$. As an application, we obtain wavelet multilevel preconditioners in $\mathbf{H}(\mathrm{div};\Omega)$ and $\mathbf{H}(\mathbf{curl};\Omega)$.


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Additional Information

Karsten Urban
Affiliation: RWTH Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany
Address at time of publication: Istituto di Analisi Numerica del C.N.R., via Abbiategrasso 209, 27100 Pavia, Italy
Email: urban@igpm.rwth-aachen.de

DOI: https://doi.org/10.1090/S0025-5718-00-01245-X
Keywords: $\mathbf{H}(\mathrm{div};\Omega)$, $\mathbf{H}(\mathbf{curl};\Omega)$, stream function spaces, wavelets
Received by editor(s): January 4, 1999
Received by editor(s) in revised form: May 24, 1999
Published electronically: May 19, 2000
Additional Notes: I am very grateful to Franco Brezzi and Claudio Canuto for fruitful and interesting discussions as well as helpful remarks. This paper was written when the author was in residence at the Istituto di Analisi Numerica del C.N.R. in Pavia, Italy.
This work was supported by the European Commission within the TMR project (Training and Mobilty for Researchers) Wavelets and Multiscale Methods in Numerical Analysis and Simulation, No. ERB FMRX CT98 0184 and by the German Academic Exchange Service (DAAD) within the Vigoni–Project Multilevel–Zerlegungsverfahren für Partielle Differentialgleichungen.
Article copyright: © Copyright 2000 American Mathematical Society