High-order accurate Nystrom discretization of integral equations with weakly singular kernels on smooth curves in the plane

TL;DR

This paper develops and compares four high-order quadrature schemes for discretizing weakly singular integral equations on smooth planar curves, improving accuracy and efficiency in solving Laplace and Helmholtz boundary problems.

Contribution

It introduces four novel quadrature methods for weakly singular kernels and compares their convergence and performance in various boundary integral problem settings.

Findings

All four schemes achieve high-order convergence.

Performance varies significantly in iterative solvers.

Some methods extend easily to problems with corners.

Abstract

Boundary integral equations and Nystrom discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weakly-singular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarithmically-singular kernels. Only smooth boundaries are considered, but some of the techniques extend straightforwardly to the case of corners. Three are modifications of the global periodic trapezoid rule, due to Kapur-Rokhlin, to Alpert, and to Kress. The fourth is a modification to a quadrature based on Gauss-Legendre panels due to Kolm-Rokhlin; this formulation allows adaptivity. We compare in numerical experiments the convergence of the four schemes in various settings, including low- and high-frequency planar Helmholtz problems, and 3D axisymmetric Laplace problems. We also find striking differences in performance in an iterative setting. We summarize the relative advantages of the schemes.