Extrapolated regularization of nearly singular integrals on surfaces

TL;DR

This work tackles the challenge of nearly singular surface integrals for harmonic potentials and Stokes flow by regularizing the kernel with a length scale $\delta$ and using extrapolation across three (or four for the $O(\delta^7)$ harmonic variant) $\delta$ values to achieve high-order accuracy $O(\delta^5)$ (and $O(\delta^7)$ in the enhanced harmonic case) uniformly near the surface. The authors derive a local expansion showing the leading regularization error depends on explicit integrals $I_0(\lambda)$ and $I_2(\lambda)$ with $\lambda=b/\delta$, and similarly for the double layer and Stokes kernels; this enables a simple $3\times3$ linear system to recover the true integral using standard quadrature. They analyze discretization errors arising from surface quadrature, provide guidelines for choosing $\delta$ relative to the grid spacing $h$, and validate the approach with extensive numerical experiments on multiple geometries, demonstrating near-surface errors of $O(h^5)$ (or $O(h^4)$ under refined scaling). The method offers a straightforward, robust alternative to analytic corrections and is readily extensible to multi-surface and moving-interface problems, with potential integration into fast summation methods for large-scale computations.

Abstract

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 $δ$ 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 $δ$ we can solve for an extrapolated value that has regularization error reduced to $O(δ^5)$, uniformly for target points on or near the surface. In examples with $δ/h$ 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 $δ$ 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 $O(δ^7)$ regularization; it typically has smaller errors but the order of accuracy is less predictable.

Figures (13)

• Figure 1: The rotated (1,.8,.6) ellipsoid (left) and the (1,.5,.5) spheroid (right).
• Figure 2: Errors for the single layer potential on the unit sphere, (left) at grid points within distance $h$, computed with the 5th and 7th order regularization, and (right) evaluated at distance between $h$ and $2h$, without regularization and with the 5th and 7th order methods.
• Figure 3: $L^2$ errors in the single layer potential on the unit sphere, evaluated at distance between $2h$ and $3h$ (left) or $3h$ and $4h$ (right).
• Figure 4: Errors for the double layer potential on the unit sphere, (left) at grid points within distance $h$, computed with the 5th and 7th order regularization, and (right) evaluated at distance between $h$ and $2h$, without regularization and with the 5th and 7th order methods.
• Figure 5: $L^2$ errors in the double layer potential on the unit sphere, evaluated at distance between $2h$ and $3h$ (left) or $3h$ and $4h$ (right).