Divergence-Free Multiwavelets

In this paper we construct IR n-valued biorthogonal, compactly supported multiwavelet families such that one of the biorthogo-nal pairs consists of divergence-free vector wavelets. The construction is based largely on Lemari e's idea of multiresolution analyses intertwined by diierentiation. We show that this technique extends nontrivially to multiwavelets via Strela's two-scale transform. An example based on the Donovan-Geronimo-Hardin-Massopust (DGHM) multiwavelets is given. x1. Introduction The study of divergence-free wavelets originated in the early 1990s with two completely diierent constructions: one due to Battle and Federbush 1], the other due to Lemari e 7]. The Battle-Federbush machine built interscale orthogonal wavelets. Later reenements gave orthogonal wavelets in dimensions 1, 2, 4, 8. None of these wavelet families have compact support. The Lemari e wavelets, on the other hand, are biorthogonal and compactly supported. Only one of the two biorthogonal families is divergence-free. His construction hinged on an important realization that multiresolution analyses (MRA's) could be related to one-another by diierentiation, which is why biorthogonality is required (cf. 6]). Federbush 3] used divergence-free wavelets to study uniqueness for the Navier-Stokes equations (NSE). Numerical analysis of NSE using variations of Lemari e wavelets has been implemented by K. Urban, et al. 9]. Other wavelet-Galerkin approaches to NSE do not use divergence-free wavelets, cf. Glowinski 5]. It is our thesis that, if wavelets are to have an impact on the numerical analysis of NSE, the wavelets should be divergence-free; in addition, they should have the best possible tradeoo between localization and approximation. This is a hypothesis at this stage: testing it will be the subject of future eeorts. The burden of the present paper is to show how to construct adequate wavelets. The procedure is simple enough. We follow Lemari e's method for the most part, but we do so using multiwavelets. The tool for relating biorthogonal All rights of reproduction in any form reserved.