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.

中文翻译：

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通过奇点交换对二维和三维近乎奇异的线积分进行精确求积

Helsing 和他的同事的数值方法评估了由曲线上的面板（复合）正交产生的拉普拉斯和相关层电位，对于任意接近的目标，有效且具有高阶精度。由于它利用复杂分析，因此其使用仅限于二维 (2D)。我们首先使用 Walsh 的经典复近似结果来解释它在面板弯曲时的精度损失，该结果可以解释为 Schwarz 奇点的“静电屏蔽”。然后我们引入了一种变体，它在其复杂的参数原像上将目标奇点交换为一个；在后一个空间中，面板是平坦的，因此收敛速度可以高得多。通过牛顿迭代稳健地找到原像。这个想法还第一次使 基于 Tornberg-Gustavsson 的递归，由 3D 中的平滑曲线生成的电位的近奇异正交。我们将此应用于细长体近似中弯曲细丝附近的斯托克斯流的准确评估。我们的 3D 方法比唯一现有的替代方法（即自适应集成）高效数倍（在内核评估和 C 实现的速度方面）。