Blended kernel approximation in the -matrix techniques

Wolfgang Hackbusch and Boris N. Khoromskij

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Submission date: 05. Nov. 2000
Pages: 19
published in: Numerical linear algebra with applications, 9 (2002) 4, p. 281-304 
DOI number (of the published article): 10.1002/nla.273
Bibtex
MSC-Numbers: 65F05, 65F50, 65F30
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Abstract:
Several types of -matrices were shown to provide a data-sparse approximation of nonlocal (integral) operators in FEM and BEM applications. The general construction is applied to the operators with asymptotically smooth kernel function (which is not necessary given explicitly) provided that the Galerkin ansatz space has a hierarchical structure.

The new class of -matrices is based on so-called blended FE and polynomial approximations of the kernel function and leads to matrix blocks with a tensor-product of block-Toeplitz (block-circulant) and rank-k matrices. This implies the translation (rotation) invariance of the kernel combined with the corresponding tensor-product grids. The approach is devoted to the fast evaluation of volume/boundary integral operators with possibly non-smooth kernels defined on canonical domains/manifolds in the FEM/BEM applications. In particular, we provide the error and complexity analysis for blended expansions to the Helmholtz kernel.