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's and Helmholtz equations show that high order accuracy can be achieved for arbitrarily close field evaluation.

翻译：近奇异积分的精确计算在许多基于边界积分方程的数值方法中扮演着重要角色。本文提出一种奇异点交换方法的变体，用于精确评估任意接近目标的层势。该方法基于全局梯形法则和三角插值，推导出显式求积公式。对于闭合解析曲线上的近奇异积分，该方法可实现谱精度。为从复化距离函数中提取奇异点，提出一种基于轮廓积分的高效寻根方法。通过变量替换，我们还将求积方法推广至分段解析曲线上的积分。针对拉普拉斯方程和亥姆霍兹方程的数值算例表明，该方法可在任意接近场点评估中实现高阶精度。