Error analysis for a transformation used in evaluating nearly singular boundary element integrals

In the two-dimensional boundary element method, one often needs to evaluate numerically integrals of the form ∫ - 1 1 g ( x ) j ( x ) f ( ( x - a ) 2 + b 2 ) d x where j 2 is a quadratic, g is a polynomial and f is a rational, logarithmic or algebraic function with a singularity at zero. The constants a and b are such that - 1 ⩽ a ⩽ 1 and 0 < b ⪡ 1 so that the singularities of f will be close to the interval of integration. In this case the direct application of Gauss–Legendre quadrature can give large truncation errors. By making the transformation x = a + b sinh ( μ u - η ) , where the constants μ and η are chosen so that the interval of integration is again [ - 1 , 1 ] , it is found that the truncation errors arising, when the same Gauss–Legendre quadrature is applied to the transformed integral, are much reduced. The asymptotic error analysis for Gauss–Legendre quadrature, as given by Donaldson and Elliott [A unified approach to quadrature rules with asymptotic estimates of their remainders, SIAM J. Numer. Anal. 9 (1972) 573–602], is then used to explain this phenomenon and justify the transformation.