Wavelet bases in \(\mathbf{H}( \text{div})\) and \(\mathbf{H}(\text \textbf{curl})\) (Q2701562)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Wavelet bases in \(\mathbf{H}( \text{div})\) and \(\mathbf{H}(\text \textbf{curl})\) |
scientific article |
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19 February 2001
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wavelets
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Navier-Stokes equations
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Stokes equations
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equations of electromagnetic theory
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stream function spaces
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condition numbers
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multilevel preconditioners
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Wavelet bases in \(\mathbf{H}( \text{div})\) and \(\mathbf{H}(\text \textbf{curl})\) (English)
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The author constructs stable (biorthogonal) wavelet bases for the stream function spaces \(\mathbf H (\mathbf{curl};\Omega)\). Moreover, \textbf{curl}-free vector wavelets are constructed and analysed. The relationship between \(\mathbf H(\text{div};\Omega)\) and \(\mathbf H(\mathbf {curl};\Omega)\) are expressed in terms of these wavelets. Discrete (orthogonal) Hodge decomposition of \(\mathbf L^2(\Omega)\) is obtained. The author gives examples of domains and wavelet bases fulfilling the general hypothesis and computes \textbf{curl}-free vector wavelets explicitely starting from biorthogonal spline wavelets. Using this decomposition, the author constructs wavelet multilevel preconditioners in \(\mathbf H(\text{div};\Omega)\) and \(\mathbf H(\mathbf {curl};\Omega)\) that give rise to uniformly bounded condition numbers. As an application, wavelet multilevel preconditioners in \(\mathbf H(\text{div};\Omega)\) and \(\mathbf H(\mathbf {curl};\Omega)\) are obtained.
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