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The wavelet element method. I: Construction and analysis. (English) Zbl 0949.42024

Summary: The wavelet element method 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 \(\mathbb{R}^n\). 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.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A15 Spline approximation
65T60 Numerical methods for wavelets
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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