Wavelet bases in \(\mathbf{H}( \text{div})\) and \(\mathbf{H}(\text \textbf{curl})\)
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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