A complete error analysis for the evaluation of a two-dimensional nearly singular boundary element integral

An important aspect of numerical integration is to have some knowledge of the truncation error for a given number of integration points. In this paper we determine estimates for these errors in the application of Gauss–Legendre quadrature to evaluate numerically two dimensional integrals which arise in the boundary element method. Expressions for the truncation errors developed here require the approximate evaluation of two integrals in the complex plane. The second integral, which has been termed the “remainder of the remainder”, was assumed small by the authors in a previous attempt in developing error estimates. However, here this integral is included and it is evaluated using a novel approach for the choice of contour. We consider examples where ignoring the “remainder of the remainder” was a reasonable assumption and also consider cases where this remainder dominates the error. Finally, it is shown, for each of the integrals considered, that these new error estimates agree very closely with the actual quadrature error.