SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 974-997, April 2024.
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 and Helmholtz equations show that high-order accuracy can be achieved for arbitrarily close field evaluation.

中文翻译：

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基于梯形法则的近奇异积分奇异性交换方法

SIAM 数值分析杂志，第 62 卷，第 2 期，第 974-997 页，2024 年 4 月
。摘要。近奇异积分的准确计算在许多基于边界积分方程的数值方法中起着重要作用。在本文中，我们提出了一种奇点交换方法的变体，以准确评估任意接近目标的层势。我们的方法基于全局梯形规则和三角插值，产生明确的求积公式。该方法实现了闭合解析曲线上近奇异积分的谱精度。为了从复数距离函数中提取奇异点，提出了一种基于轮廓积分的高效求根方法。通过变量的改变，我们还将求积法推广到分段解析曲线上的积分。拉普拉斯和亥姆霍兹方程的数值例子表明，任意近场评估可以实现高阶精度。