We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter $\delta$ in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of $\delta$ we can solve for an extrapolated value that has regularization error reduced to $O(\delta^5)$. In examples with $\delta/h$ constant and moderate resolution we observe total error about $O(h^5)$. For convergence as $h \to 0$ we can choose $\delta$ proportional to $h^q$ with $q < 1$ to ensure the discretization error is dominated by the regularization error. With $q = 4/5$ we find errors about $O(h^4)$. For harmonic potentials we extend the approach to a version with $O(\delta^7)$ regularization; it typically has smaller errors but the order of accuracy is less predictable.

翻译：针对调和势或斯托克斯流中单层或双层面积分在邻近点求值时的近奇异积分计算问题，本文提出一种方法。此类求值需求可能源于求解两个相邻曲面间的积分方程，或获取网格点上的函数值。为控制离散误差，我们引入含长度参数$\delta$的正则化核替代奇异核。基于奇点附近的误差分析，正则化误差表达式可展开为已知量乘以未知系数的项。通过选取三种不同$\delta$值计算积分，可解出正则化误差降低至$O(\delta^5)$的外推值。在$\delta/h$恒定且中等分辨率算例中，总误差约为$O(h^5)$。为保证$h\to 0$时收敛，可选择$\delta$与$h^q$成比例（$q<1$），使离散误差由正则化误差主导。取$q=4/5$时误差约为$O(h^4)$。针对调和势，我们进一步将方法扩展为$O(\delta^7)$正则化版本，该版本通常误差更小但精度阶数稳定性较低。