Quadrature Weights on Tensor-Product Nodes for Accurate Integration of Hypersingular Functions over Some Simple 3-D Geometric Shapes (Communications in Computational Physics, 2020)
In this paper, we present accurate and economic integration quadratures forhypersingular functions over three simple geometric shapes in $\mathbb{R}^3$ (spheres, cubes, andcylinders). The quadrature nodes are made of the tensor-product of 1-D Gauss nodeson [−1,1] for non-periodic variables or uniform nodes on [0,2π] or [0,π] for periodicones. The quadrature weights are converted from a brute-force integration of the hypersingularfunction through interpolating the smooth component of the integrand.Numerical results are presented to validate the accuracy and efficiency of computinghypersingular integrals, as in the computations of Cauchy principal values, with aminimum number of quadrature nodes. The pre-calculated quadrature tables can bethen readily used to implement Nyström collocation methods of hypersingular volumeintegral equations such as the one for Maxwell equations.