Error estimate based adaptive quadrature for layer potentials over axisymmetric surfaces

TL;DR

This paper introduces an adaptive quadrature method for accurately evaluating layer potentials on axisymmetric surfaces, especially near boundaries, using error estimation and semi-analytical techniques for complex geometries.

Contribution

The paper presents a novel adaptive quadrature approach with automatic parameter tuning for axisymmetric surfaces, accommodating complex geometries with multiple bodies.

Findings

Effective in achieving specified error tolerances

Handles complex geometries with multiple axisymmetric bodies

Demonstrates high accuracy through numerical examples

Abstract

Layer potentials represent solutions to partial differential equations in an integral equation formulation. When numerically evaluating layer potentials at evaluation points close to the domain boundary, specialized quadrature techniques are required for accuracy because of rapid variations in the integrand. To efficiently achieve a specified error tolerance, we introduce an adaptive quadrature method with automatic parameter adjustment for axisymmetric surfaces, facilitated by error estimation. Notably, while each surface must be axisymmetric, the integrand itself need not be, allowing for applications with complex geometries featuring multiple axisymmetric bodies. The proposed quadrature method utilizes so-called interpolatory semi-analytical quadrature in conjunction with a singularity swap technique in the azimuthal angle. In the polar angle, such a technique is used as needed, depending on the integral kernel, combined with an adaptive subdivision of the integration interval. The method is tied to a regular quadrature method that employs a trapezoidal rule in the azimuthal angle and a Gauss-Legendre quadrature rule in the polar angle, which will be used whenever deemed sufficiently accurate, as determined by a quadrature error estimate [C. Sorgentone and A.-K. Tornberg, Advances in Computational Mathematics, 49 (2023), p. 87]. Error estimates for both numerical integration and interpolation are derived using complex analysis, and are used to determine the adaptive panel subdivision given the evaluation point and desired accuracy. Numerical examples are presented to demonstrate the method's efficacy.