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A class of finite element methods based on orthonormal, compactly supported wavelets. (English) Zbl 0830.65084

The purpose of this paper is to extend the class of problems that can be treated effectively using wavelet Galerkin methods by deriving tensor product wavelet-based finite elements. These elements can be extended using tensor products to represent a class of irregular domains in higher dimension.
Further contributions of this paper are the determination of quasi- optimal convergence rates in these elements and the derivation of associated quadratures to insure quasi-optimal convergence rates. The accuracy of quadratures for the corresponding element calculations is established by constructing a quasi-interpolation scheme associated with the elements.
Two examples to verify the approximation properties of the Daubechies wavelet-based elements and quasi-interpolation formula are presented.

MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D05 Numerical interpolation
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