Generalized quadrature rules of Gaussian type for numerical evaluation of singular integrals

An efficient method for constructing a class of generalized quadrature formulae of Gaussian type on (-1, 1) for integrands having logarithmic singularities is developed. That kind of singular integrals are very common in the boundary element method. Several special cases for n-point quadratures, which are exact on both of the spaces rfo - 2n-2E-1 [-1, 1] (the space of algebraic polynomials of degree at most 2n 2t 1) and 2e-1[-1, 1] spanfxk log IxN_e-01 (the logarithmic space), where 1 < t < n, are presented. Regarding a direct connection of these 2m-point quadratures with m-point quadratures of Gaussian type with respect to the weight function t r-1/2 over (0, 1), the method of construction is significantly simplified. Gaussian quadratures on (0, 1) are exact for integrands of the form t p(t) q(t) log t, where p and q are algebraic polynomials of degree at most 2m 1 and t 1 (1 < e < 2m), respectively. The obtained quadratures can be used in a software implementation of the boundary element method. (C) 2014 Elsevier B.V. All rights reserved.