High order regularization of nearly singular surface integrals

TL;DR

This work develops high-order regularization formulas for nearly singular surface integrals in boundary integral methods for harmonic and Stokes problems. By replacing singular kernels with erf-based smooth kernels and adding higher-order correction terms, it achieves $O(\delta^p)$ accuracy with $p\in\{3,5,7\}$ and attains overall $O(h^{pq})$ convergence when $\delta=\kappa h^q$ with $q<1$. The authors provide explicit near-surface and on-surface kernels for single/double layer potentials and Stokes operators, validate the theory with extensive numerical experiments on spheres, ellipsoids, and closely spaced surfaces, and show how to extend near-surface values to the entire grid via Mayo extrapolation. The results enable the use of standard quadrature with high accuracy for nearly singular integrals, with practical guidelines for choosing $\delta$ and a publicly available implementation.

Abstract

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 $O(δ^p)$ where $δ$ is the smoothing radius and $p = 3$, $5$, $7$. With spacing $h$ in the quadrature, we choose $δ= κh^q$ 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.

Figures (10)

• Figure 1: Errors for the harmonic single layer on a unit sphere, at grid points within distance $h$ to the sphere.
• Figure 2: Errors for the harmonic double layer on a unit sphere, at grid points within distance $h$ to the sphere.
• Figure 3: Errors for the harmonic solution on the molecular surface, at grid points randomly chosen within distance $h$ to the surface. Here $\kappa$ is chosen so that $\delta/h = \kappa_0$ when $h = 1/64$.
• Figure 4: Errors for the harmonic solution on an ellipsoid (1, .6, .4) (left) and ellipsoid (1, .4, .3) (right), at grid points randomly chosen within distance $h$ to the surface. Here $\kappa$ is chosen so that $\delta/h = \kappa_0$ when $h = 1/64$.
• Figure 5: Errors for the Stokes double layer on the spheroid (1, .5, .5), at grid points randomly chosen within distance $h$ to the surface. Here $\kappa$ is chosen so that $\delta/h = \kappa_0$ when $h = 1/64$.