Accurate quadrature of nearly singular line integrals in two and three dimensions by singularity swapping

TL;DR

This paper introduces a novel singularity swapping technique that improves the accuracy and efficiency of evaluating nearly singular line integrals in 2D and 3D, especially near curved geometries, by leveraging complex analysis and Newton iteration.

Contribution

It extends Helsing's 2D high-order quadrature method to 3D by swapping singularities, enabling accurate near-singular integral evaluation for smooth curves in three dimensions.

Findings

The 3D method is several times more efficient than adaptive integration.

The approach achieves high-order accuracy near curved geometries.

It successfully evaluates Stokes flow near curved filaments with improved speed.

Abstract

The method of Helsing and co-workers evaluates Laplace and related layer potentials generated by a panel (composite) quadrature on a curve, efficiently and with high-order accuracy for arbitrarily close targets. Since it exploits complex analysis, its use has been restricted to two dimensions (2D). We first explain its loss of accuracy as panels become curved, using a classical complex approximation result of Walsh that can be interpreted as "electrostatic shielding" of a Schwarz singularity. We then introduce a variant that swaps the target singularity for one at its complexified parameter preimage; in the latter space the panel is flat, hence the convergence rate can be much higher. The preimage is found robustly by Newton iteration. This idea also enables, for the first time, a near-singular quadrature for potentials generated by smooth curves in 3D, building on recurrences of Tornberg-Gustavsson. We apply this to accurate evaluation of the Stokes flow near to a curved filament in the slender body approximation. Our 3D method is several times more efficient (both in terms of kernel evaluations, and in speed in a C implementation) than the only existing alternative, namely, adaptive integration.