Istituto di Matematica Applicata e Tecnologie Informatiche     
Canuto C., Tabacco A., Urban K. The wavelet element method. Part I: Construction and analysis. Preprint ercim.cnr.ian//1997-1038, 1997.
The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some $er^n$. The domain of interest is split into subdomains which are mapped to a simple reference domain, here $n$--dimensional cubes. Thus, one has to construct appropriate biorthogonal wavelets on the reference domain such that mapping them to each subdomain and matching along the interfaces leads to a wavelet system on the domain. In this paper we use adapted biorthogonal wavelet systems on the interval in such a way, that tensor products of these functions can be used for the construction of wavelet bases on the reference domain. We describe the matching procedure in any dimension $n$ in order to impose continuity and prove that it leads to a construction of a biorthogonal wavelet system on the domain. These wavelet systems characterize Sobolev spaces measuring both piecewise and global regularity. The construction is detailed for a bivariate example and an application to the numerical solution of second order partial differential equations is given.
Subject 65-XX

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