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)(−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 nn-point quadratures, which are exact on both of the spaces P2n−2ℓ−1[−1,1]P2n−2ℓ−1[−1,1] (the space of algebraic polynomials of degree at most 2n−2ℓ−12n−2ℓ−1) and L2ℓ−1[−1,1]=span{xklog|x|}k=02ℓ−1 (the logarithmic space), where 1≤ℓ≤n1≤ℓ≤n, are presented. Regarding a direct connection of these 2m2m-point quadratures with mm-point quadratures of Gaussian type with respect to the weight function t↦t−1/2t↦t−1/2 over (0,1)(0,1), the method of construction is significantly simplified. Gaussian quadratures on (0,1)(0,1) are exact for integrands of the form t↦p(t)+q(t)logtt↦p(t)+q(t)logt, where pp and qq are algebraic polynomials of degree at most 2m−ℓ−12m−ℓ−1 and ℓ−1ℓ−1   (1≤ℓ≤2m)(1≤ℓ≤2m), respectively. The obtained quadratures can be used in a software implementation of the boundary element method.