TY - JOUR
T1 - Quadrature Weights on Tensor-Product Nodes for Accurate Integration of Hypersingular Functions over Some Simple 3-D Geometric Shapes
AU - Brian Zinser, Wei Cai & Duan Chen
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 - <p style="text-align: justify;">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.</p>