Spherical configurations and quadrature methods for integral equations of the second kind

TL;DR

This paper develops a new quadrature-based method for solving second-kind integral equations on the sphere, utilizing Marcinkiewicz--Zygmund properties to improve approximation accuracy and error analysis.

Contribution

It introduces a novel product integration approach on the sphere that broadens quadrature rule options and links geometric quadrature data with error estimates.

Findings

The method achieves accurate approximation of integral equations on the sphere.

Error bounds are derived based on best approximation and quadrature properties.

Numerical examples confirm the theoretical error estimates.

Abstract

In this paper, we propose and analyze a product integration method for the second-kind integral equation with weakly singular and continuous kernels on the unit sphere $\mathbb{S}^2$. We employ quadrature rules that satisfy the Marcinkiewicz--Zygmund property to construct hyperinterpolation for approximating the product of the continuous kernel and the solution, in terms of spherical harmonics. By leveraging this property, we significantly expand the family of candidate quadrature rules and establish a connection between the geometrical information of the quadrature points and the error analysis of the method. We then utilize product integral rules to evaluate the singular integral with the integrand being the product of the singular kernel and each spherical harmonic. We derive a practical $L^{\infty}$ error bound, which consists of two terms: one controlled by the best approximation of the product of the continuous kernel and the solution, and the other characterized by the Marcinkiewicz--Zygmund property and the best approximation polynomial of this product. Numerical examples validate our numerical analysis.