The method of steepest descent for estimating quadrature errors

This work presents an application of the method of steepest descent to estimate quadrature errors. The method is used to provide a unified approach to estimating the truncation errors which occur when Gauss–Legendre quadrature is used to evaluate the nearly singular integrals that arise as part of the two dimensional boundary element method. The integrals considered here are of the form ∫1–1     h(x) dx    ((x−a)²+b²)^α , where h(x) is a “well-behaved” function, α > 0 and −1 < a < 1. Since 0 < b « 1, the integral is "nearly singular", with a sharply peaked integrand.

The method of steepest descent is used to estimate the (generally large) truncation errors that occur when Gauss–Legendre quadrature is used to evaluate these integrals, as well as to estimate the (much lower) errors that occur when Gauss–Legendre quadrature is performed on such integrals after a “sinh” transformation has been applied. The new error estimates are highly accurate in the case of the transformed integral and are shown to be comparable to those found in previous work by Elliott and Johnston (2007). One advantage of the new estimates is that they are given by just one formula each for the un-transformed and the transformed integrals, rather than the much larger set of formulae in the previous work. Another advantage is that the new method applies over a much larger range of α values than the previous method.