Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the surface. Here we derive formulas to regularize the integrals with high accuracy, using analysis from Beale and Tlupova (Adv. Comput. Math., 2024), so that a standard quadrature can be used without special care near the singularity. We treat single or double layer integrals for harmonic functions or for Stokes flow. The nearly singular case, evaluation at points close to the surface, can be needed when surfaces are close to each other, or to find values at grid points near a surface. We derive formulas for regularized kernels with error $O(\delta^p)$ where $\delta$ is the smoothing radius and $p = 3$, $5$, $7$. With spacing $h$ in the quadrature, we choose $\delta = \kappa h^q$ with $q<1$ so that the discretization error is controlled as $h \to 0$. We see the predicted order of convergence $O(h^{pq})$ in various examples. Values at all grid points can be obtained from those near the surface in an efficient manner suggested in A. Mayo (SIAM J. Statist. Comput., 1985). With this technique we obtain high order accurate grid values for a harmonic function determined by interfacial conditions and for the pressure and velocity in Stokes flow around a translating spheroid.

翻译：偏微分方程的解常可表示为具有奇异基本解相关核的面积分。在曲面上或曲面附近精确计算该积分需要特殊方法。本文基于Beale与Tlupova（Adv. Comput. Math., 2024）的分析推导出高精度正则化积分公式，使得标准数值积分方法在奇点附近无需特殊处理即可使用。我们处理调和函数或斯托克斯流中的单层或双层积分。当曲面相互靠近时，或需要计算曲面附近网格点上的数值时，近奇异情形（在接近曲面的点处求值）可能成为必要。我们推导出正则化核的误差为$O(\delta^p)$的公式，其中$\delta$为平滑半径，$p = 3$, $5$, $7$。在数值积分中取步长$h$，我们选择$\delta = \kappa h^q$（$q<1$），使得离散化误差在$h \to 0$时受控。我们在多个算例中观察到预期的收敛阶数$O(h^{pq})$。所有网格点上的数值可通过A. Mayo（SIAM J. Statist. Comput., 1985）提出的高效方法，由曲面附近的数值获得。利用该技术，我们获得了由界面条件确定的调和函数，以及平移椭球体周围斯托克斯流中压力与速度的高阶精确网格值。