The Wavelet Element Method part I. Construction and analysis

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 R(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. (C) 1999 Academic Press.