Volume 20, Issue 5
Quadrature Weights on Tensor-Product Nodes for Accurate Integration of Hypersingular Functions over Some Simple 3-D Geometric Shapes

Commun. Comput. Phys., 20 (2016), pp. 1283-1312.

Published online: 2018-04

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In this paper, we present accurate and economic integration quadratures for hypersingular functions over three simple geometric shapes in $\mathbb{R}^3$ (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.

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@Article{CiCP-20-1283, author = {}, title = {Quadrature Weights on Tensor-Product Nodes for Accurate Integration of Hypersingular Functions over Some Simple 3-D Geometric Shapes}, journal = {Communications in Computational Physics}, year = {2018}, volume = {20}, number = {5}, pages = {1283--1312}, abstract = {

In this paper, we present accurate and economic integration quadratures for hypersingular functions over three simple geometric shapes in $\mathbb{R}^3$ (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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2015-0005}, url = {http://global-sci.org/intro/article_detail/cicp/11190.html} }
TY - JOUR T1 - Quadrature Weights on Tensor-Product Nodes for Accurate Integration of Hypersingular Functions over Some Simple 3-D Geometric Shapes JO - Communications in Computational Physics VL - 5 SP - 1283 EP - 1312 PY - 2018 DA - 2018/04 SN - 20 DO - http://doi.org/10.4208/cicp.OA-2015-0005 UR - https://global-sci.org/intro/article_detail/cicp/11190.html KW - AB -

In this paper, we present accurate and economic integration quadratures for hypersingular functions over three simple geometric shapes in $\mathbb{R}^3$ (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.

Brian Zinser, Wei Cai & Duan Chen. (2020). Quadrature Weights on Tensor-Product Nodes for Accurate Integration of Hypersingular Functions over Some Simple 3-D Geometric Shapes. Communications in Computational Physics. 20 (5). 1283-1312. doi:10.4208/cicp.OA-2015-0005
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