Efficient convolution with the Newton potential in d dimensions

Wolfgang Hackbusch

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Submission date: 04. Sep. 2007 (revised version: July 2008)
Pages: 30
published in: Numerische Mathematik, 110 (2008) 4, p. 449-489 
DOI number (of the published article): 10.1007/s00211-008-0171-9
Bibtex
Keywords and phrases: newton potential, convolution, coulomb potential
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Abstract:
The paper is concerned with the evaluation of the convolution integral in d dimensions (usually d=3), when f is given as piecewise polynomial of possibly large degree, i.e., f may be considered as an hp-finite element function. The underlying grid is locally refined using various levels of dyadically organised grids. The result of the convolution is approximated in the same kind of mesh. If f is given in tensor product form, the d-dimensional convolution can be reduced to one-dimensional convolutions.

Although the details are given for the kernel the basis techniques can be generalised to homogeneous kernels, e.g., the fundamential solution of the d-dimensional Poisson equation.