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 \(h \rightarrow 0\), 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.

我们提出了一种计算近奇异积分的方法，当在附近的点评估谐波势或斯托克斯流的单层或双层表面积分时，会发生这种近似奇异积分。当一个表面靠近另一个表面时求解积分方程或获取网格点处的值时可能需要这些值。我们用具有长度参数 \(\delta \) 的正则化版本替换奇异内核，以控制离散化误差。奇点附近的分析导致了由于正则化而产生的误差的表达式，该表达式具有未知系数乘以已知量的项。通过使用三种选择的 \(\delta \) 计算积分，我们可以求解出一个外推值，该值的正则化误差减少到 \(O(\delta ^5)\)，对于表面上或表面附近的目标点一致。在具有 \(\delta /h\) 恒定且中等分辨率的示例中，我们观察到接近表面的总误差约为 \(O(h^5)\)。对于收敛为 \(h \rightarrow 0\) 的情况，我们可以选择与 \(h^q\) 成正比且 \(q < 1\) 的 \(\delta \)，以确保离散化误差由正则化误差主导。对于 \(q = 4/5\)，我们发现关于 \(O(h^4)\) 的错误。对于谐波势，我们将方法扩展到具有 \(O(\delta ^7)\) 正则化的版本；它通常具有较小的误差，但准确度的顺序不太可预测。