Direct integration of the Newton potential over cubes including a program description

Wolfgang Hackbusch

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Submission date: 25. Sep. 2001
Pages: 29
published in: Computing, 68 (2002) 3, p. 193-216 
DOI number (of the published article): 10.1007/s00607-001-1443-8
Bibtex
with the following different title: Direct integration of the Newton potential over cubes
MSC-Numbers: 65R20, 65N38, 68W30, 35Q99
Keywords and phrases: newton potential, coulomb potential, direct integration, integral equations
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Abstract:
In boundary element methods, the evaluation of the weakly singular integrals can be performed either a) numerically, b) symbolically, i.e., by explicit expressions, or c) in a combined manner. The explicit integration is of particular interest, when the integrals contain the singularity or if the singularity is rather close to the integration domain.

We describe the explicit expressions for the sixfold volume integrals arising for the Newton potential, i.e., for a 1/r integrand. The volume elements are axi-parallel bricks. The sixfold integrals are typical for the Galerkin method. However, the threefold integral arising from collocation methods can be derived in the same way.

Furthermore, this report contains a description of the program together with examples for its use. You can download the source as Pascal program or as C program (both GNU-zip compressed, approx. 27 kB).