Evaluating singular and near-singular integrals on $C^2$ smooth surfaces with quadratic geometric approximation and closed form expressions

TL;DR

This paper develops efficient methods for accurately computing singular and near-singular integrals of Green's functions and their derivatives on smooth surfaces discretized by triangles, improving over existing techniques.

Contribution

It introduces a geometric approximation approach for integrals involving Green's functions and their derivatives on smooth surfaces, enhancing accuracy and efficiency.

Findings

Provides explicit algorithms for integral computation on triangles.

Introduces a geometric approximation method using surface information.

Achieves faster computation than adaptive refinement methods.

Abstract

Most Fredholm integral equations involve integrals with weakly singular kernels. Once the domain of integration is discretized into flat triangular elements, these weakly singular kernels become strongly singular or near-singular. Common methods to compute these integrals when the kernel is a Green's function include the Duffy transform, polar coordinates with closed analytic formulas, and singularity extraction. However, these methods do not generalize well to the normal derivatives of Green's functions due to the strongly singular behavior of these functions on triangular elements. We provide methods to integrate both the Green's function and its normal derivative on smooth surfaces discretized by triangular elements in three dimensions for many commonly encountered differential operators. For strongly singular integrals involving normal derivatives of Green's functions, we introduce a more refined approximation of the derivatives of the Green's function on flat triangles. This method uses geometric information of the true surface of integration to approximate the original integral on the true domain using push-forward maps. This is better than simply setting the singular integrals to zero, while being faster than adaptive refinement methods. We provide an algorithm for explicit computations on triangles, and present necessary analytic formulas that the algorithm requires in the appendix.