Error analysis for a sinh 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 + b2) d x where j2 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.