Wavelet bases in 𝐇(𝐝𝐢𝐯) and 𝐇(𝐜𝐮𝐫𝐥)

Urban, Karsten

doi:10.1090/S0025-5718-00-01245-X

Wavelet bases in $\mathbf {H}( \mathrm {div})$ and $\mathbf {H}(\mathbf {curl})$
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Math. Comp. 70 (2001), 739-766 Request permission

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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