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Quadrature Weights on Tensor-Product Nodes for Accurate Integration of Hypersingular Functions over Some Simple 3-D Geometric Shapes

Part of: Physiological, cellular and medical topics Higher-dimensional theory

Published online by Cambridge University Press:  02 November 2016

Brian Zinser ,
Wei Cai  and
Duan Chen
Brian Zinser*
Affiliation:
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
Wei Cai*
Affiliation:
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
Duan Chen*
Affiliation:
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
*
*Corresponding author. Email addresses:bzinser@sandia.gov (B. Zinser), wcai@uncc.edu (W. Cai), duan.chen@uncc.edu (D. Chen)
*Corresponding author. Email addresses:bzinser@sandia.gov (B. Zinser), wcai@uncc.edu (W. Cai), duan.chen@uncc.edu (D. Chen)
*Corresponding author. Email addresses:bzinser@sandia.gov (B. Zinser), wcai@uncc.edu (W. Cai), duan.chen@uncc.edu (D. Chen)
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Abstract

In this paper, we present accurate and economic integration quadratures for hypersingular functions over three simple geometric shapes in ℝ3 (spheres, cubes, and cylinders). The quadrature nodes are made of the tensor-product of 1-D Gauss nodes on [–1,1] for non-periodic variables or uniform nodes on [0,2π] or [0,π] for periodic ones. The quadrature weights are converted from a brute-force integration of the hypersingular function through interpolating the smooth component of the integrand. Numerical results are presented to validate the accuracy and efficiency of computing hypersingular integrals, as in the computations of Cauchy principal values, with a minimum number of quadrature nodes. The pre-calculated quadrature tables can be then readily used to implement Nyström collocation methods of hypersingular volume integral equations such as the one for Maxwell equations.

Keywords

Hypersingular integralCauchy principal valuetensor-product quadratures

MSC classification

Secondary: 31B05: Harmonic, subharmonic, superharmonic functions 92C05: Biophysics 65Z05: Applications to physics
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1283 - 1312
Copyright
Copyright © Global-Science Press 2016 

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References

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