Fast Convergent Quadrature Method for Evaluating the RWG- and SWG-Related Convolutional Integrals

The Rao–Wilton–Glisson (RWG) basis function (BF)-related convolutional surface integrals are intrinsically 2-D. The Schaubert–Wilton–Glisson (SWG) BF-related convolutional volume integrals are 3-D in nature. It has been shown that such integrals can be reduced to a summation of several 1-D integrals by repeatedly applying the divergence theorem. To the best of our knowledge, there are no known analytic formulas for the 1-D integrals, and consequently, one must choose a quadrature rule to get the final results. The 1-D integrals are prone to numerical errors when the observation point is close to the source. We propose that sinh-related transformations can be used to improve the accuracy, which has been shown to have exponential convergence with respect to the number of Gauss–Legendre quadrature points in the numerical examples. We can reach ten or more significant digits in the convolutional integrals pertinent to RWG or SWG functions with a small number of quadrature points.