An efficient quadrature rule for weakly and strongly singular integrals

In this paper, we consider the weakly and strongly singular integrals that arose from physical and engineering problems with corners. A fast and stable quadrature rule is designed for such integrals with nodes following a Clenshaw–Curtis distribution (i.e., extreme points of the Chebyshev polynomials). By a recurrence relation for the moments involved and Fast Fourier Transform (FFT), the presented quadrature rule can be implemented in O(nlogn) operations. Particular error estimates of the proposed algorithm are studied and verified by ample numerical illustrations. Finally, a specific Nyström method with the presented quadrature is applied to the two-dimensional scattering problem.