Singularity swapping method for nearly singular integrals based on trapezoidal rule

TL;DR

This paper introduces a novel singularity swapping technique utilizing the trapezoidal rule and trigonometric interpolation to accurately evaluate nearly singular boundary integrals with spectral accuracy.

Contribution

It presents a new explicit quadrature formula based on global trapezoidal rule and trigonometric interpolation, extending to piecewise analytic curves, with an efficient root finding method.

Findings

Achieves spectral accuracy for nearly singular integrals on closed curves

Extends quadrature to piecewise analytic curves

Demonstrates high order accuracy for Laplace and Helmholtz equations

Abstract

Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer potentials for arbitrarily close targets. Our method is based on the global trapezoidal rule and trigonometric interpolation, resulting in an explicit quadrature formula. The method achieves spectral accuracy for nearly singular integrals on closed analytic curves. In order to extract the singularity from the complexified distance function, an efficient root finding method is proposed based on contour integration. Through the change of variables, we also extend the quadrature method to integrals on the piecewise analytic curves. Numerical examples for Laplace's and Helmholtz equations show that high order accuracy can be achieved for arbitrarily close field evaluation.