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

Abstract

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 P2n-2ℓ-1[-1,1] (the space of algebraic polynomials of degree at most 2n-2ℓ-1) and L2ℓ-1[-1,1]=span{xklog|x|}k=02ℓ-1 (the logarithmic space), where 1≤ℓ≤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→t-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)logt, where p and q are algebraic polynomials of degree at most 2m-ℓ-1 and ℓ-1 (1≤ℓ≤2m), respectively. The obtained quadratures can be used in a software implementation of the boundary element method.