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

Ludvig Af Klinteberg,Alex H Barnett

doi:10.1007/s10543-020-00820-5

Ludvig Af Klinteberg, Alex H Barnett

Open Access

https://doi.org/10.1007/s10543-020-00820-5

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Abstract

The numerical 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.

Highlights

Integral equation methods enable efficient numerical solutions to piecewise-constant coefficient elliptic boundary value problems, by representing the solution as a layer potential generated by a so-called “density” function defined on a lower-dimensional source geometry [4,31]
For each point-panel interaction, we evaluate the singularity swap quadrature (SSQ) using data upsampled to 32 Gauss–Legendre points, for all target points within the Bernstein radius ρ = 3 [corresponding via (41) to full accuracy]
The error is measured against a reference computed using the adaptive quadrature and a discretization with 18-point panels, created using = 5 × 10−14

Summary

Introduction

Integral equation methods enable efficient numerical solutions to piecewise-constant coefficient elliptic boundary value problems, by representing the solution as a layer potential generated by a so-called “density” function defined on a lower-dimensional source geometry [4,31]. This work is concerned with accurate evaluation of such layer potentials close to their source, when this source is a one-dimensional curve embedded in 2D or 3D space. In 3D, such line integrals represent the fluid velocity in non-local slender-body theory (SBT) for filaments in a viscous (Stokes) flow [17,26,28], and may represent the solution fields in the electrostatic [42] or elecromagnetic [11] response of thin wire conductors. In this work we focus on non-oscillatory kernels arising from Laplace (electrostatic) and Stokes applications, we expect that by singularity splitting In this work we focus on non-oscillatory kernels arising from Laplace (electrostatic) and Stokes applications, we expect that by singularity splitting (e.g. [20]) the methods we present could be adapted for oscillatory or Yukawa kernels

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