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 $O(\delta^5)$, uniformly for target points on or near the surface. In examples with $\delta/h$ constant and moderate resolution we observe total error about $O(h^5)$ close to the surface. 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：本文提出了一种计算近奇异积分的方法，该积分出现在调和势或斯托克斯流的单层或双层面积分在邻近点求值时。此类数值在求解一个曲面接近另一个曲面的积分方程时可能需要，或用于获取网格点处的数值。我们用一个具有长度参数$\delta$的正则化版本来替代奇异核，以控制离散化误差。对奇点附近的分析导出了正则化引起的误差表达式，其中包含未知系数乘以已知量的项。通过使用三种不同的$\delta$值计算积分，我们可以求解出一个外推值，其正则化误差被降至$O(\delta^5)$，且对于曲面上或曲面附近的目标点具有一致性。在$\delta/h$为常数且具有中等分辨率的示例中，我们观察到接近曲面处的总误差约为$O(h^5)$。当$h \to 0$时，为确保离散化误差由正则化误差主导，我们可以选择$\delta$与$h^q$成比例，其中$q < 1$。当$q = 4/5$时，我们发现误差约为$O(h^4)$。对于调和势，我们将该方法扩展至具有$O(\delta^7)$正则化的版本；该版本通常具有更小的误差，但精度阶数的可预测性较低。