Estimating nearly singular quadrature errors in boundary integral methods and quadrature by expansion

Ludvig af Klinteberg, Anna-Karin Tornberg

Published 2016-03-28Version 1

In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is singular or nearly singular. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. This error has not previously been well understood. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. Using these results, it is possible to derive accurate estimates for the quadrature errors related to QBX, when applied to the harmonic single layer potential in two and three dimensions. Together with previous results for the truncation error, this provides a new way of understanding the convergence of QBX.