Stabilizing the singularity swap quadrature for near-singular line integrals

TL;DR

Problem: evaluating near-singular line integrals in 3D suffers catastrophic cancellation when kernel numerators vanish. Approach: introduce target-specific translated bases (open curves: monomials; closed curves: modified Fourier) and stabilize the constant-term evaluation with a stabilized adjoint weight framework, preserving SSQ efficiency. Findings: achieves near machine-precision accuracy on prototype integrals and up to ten orders of magnitude improvement at extremely close distances with minimal overhead. Significance: enhances robustness of SSQ for scalar and tensor kernels in Stokes flow, elasticity, and related boundary-integral solvers in 3D, with applicability to 2D as well as higher-fidelity simulations of filamentary and wire-like geometries.

Abstract

Singularity swap quadrature (SSQ) is an effective method for the evaluation at nearby targets of potentials due to densities on curves in three dimensions. While highly accurate in most settings, it is known to suffer from catastrophic cancellation when the kernel exhibits both near-vanishing numerators and strong singularities, as arises with scalar double layer potentials or tensorial kernels in Stokes flow or linear elasticity. This precision loss turns out to be tied to the interpolation basis, namely monomial (for open curves) or Fourier (for closed curves). We introduce a simple yet powerful remedy: target-specific translated monomial and Fourier bases that explicitly incorporate the near-vanishing behavior of the kernel numerator. We combine this with a stable evaluation of the constant term which now dominates the integral, significantly reducing cancellation. We show that our approach achieves close to machine precision for prototype integrals, and up to ten orders of magnitude lower error than standard SSQ at extremely close evaluation distances, without significant additional computational cost.

Figures (6)

• Figure 1: Sketch of the principal idea of this paper, in the open-arc case. (a) shows a line integral along the curve $\Gamma = \boldsymbol{\gamma}([-1,1]) \subset \mathbb{R}^3$, with a target point ${\mathbf{x}}$ close to its nearest curve point $\boldsymbol{\gamma}(a)$. (b) shows the standard SSQ expansion of the integrand nominator $F(t)$ in the monomial basis: the small value $F(a)$---which dominates the near-singular integral---is given by the sum of larger cancelling terms, making the method unstable. (c) shows the proposed expansion in a translated monomial basis, which we call TSSQ (translated SSQ): now $F(a)$ is captured directly by its small constant term, computed stably from the density, while the remaining terms each contribute much less to the line integral. Catastrophic cancellation is thereby avoided.
• Figure 1: Panel (a) shows the relative error in computing the integral \ref{['eq:ex1_int']} using standard (std) and modified (mod) monomial bases, as the imaginary part of the target point $t_0=a+ib$ decreases. For stronger singularities ($m=3,5$), the standard basis suffers from cancellation, while the modified basis maintains high accuracy. Solid black lines with dots indicate estimated cancellation errors from Remark \ref{['rem:cancellation_errest']}. Panel (b) shows the maximum norm of the quadrature vector compared to the magnitude of the true integral $I_m$. For the standard basis with $m>1$, the quadrature vector is orders of magnitude larger than $|I_m|$, providing direct evidence that the loss of accuracy in panel (a) is caused by cancellation. Here, the colors indicate the value of $m$ as in (a).
• Figure 1: Panel (a) and (b) compares the minimum, maximum, and mean relative error of the standard SSQ and translated SSQ (TSSQ) methods in evaluating the flow field \ref{['eq:slender_layer_potential']} for different tolerances $\epsilon$ at 1000 random target points sampled at each distance $d$ from the filament $\Gamma$ shown in panel (c), with dots illustrating the discretization nodes (when $\epsilon=10^{-6}$).
• Figure 2: Relative error in evaluating the integral \ref{['eq:ex1_int']} for fixed $t_0=0.23+10^{-4}i$ and varying $\delta$. The modified (mod) basis maintains high accuracy across all $\delta$, while the standard (std) basis suffers from cancellation when the numerator becomes too small, a behavior that is captured well by the approximate cancellation error from Remark \ref{['rem:cancellation_errest']}, shown as the solid black line with dots.
• Figure 2: Panel (a) compares the minimum, maximum, and mean relative error of the standard SSQ and translated (TSSQ) methods in evaluating \ref{['eq:slender_layer_potential']} at 5000 random points sampled at each distance $d$ from the globally discretized closed curve $\Gamma$ of panel (b). Filled markers indicate that the reference solution is accurate to within $10^{-10}$, and unfilled markers denote distances ($d<3\times 10^{-6}$) where it is not.

Theorems & Definitions (14)

• Remark 2.1: Numerical cancellation error
• Definition 3.1: Translated monomial basis
• Lemma 3.2: Modified monomial basis integrals
• Proof 1
• Remark 3.3
• Definition 3.4: Modified Fourier basis
• Remark 3.5
• Theorem 3.6: Modified Fourier coefficients
• Proof 2
• Lemma 3.7: Modified Fourier basis integrals