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

TL;DR

This work tackles a second-kind integral equation on the sphere $\mathbb{S}^2$ with a weakly singular kernel by marrying product integration with hyperinterpolation. The authors replace the kernel–solution product by its spherical-harmonic hyperinterpolant and construct quadrature-based weights $W_j(x)$ via the Funk–Hecke framework, allowing a Nyström-type discretization that accommodates quadrature rules satisfying the Marcinkiewicz–Zygmund property. A central result is an $L^{\infty}$ error bound that decomposes into the best-approximation error of $K(x_0,\cdot)\varphi$ and a geometry-dependent MZ term, with convergence guaranteed as the hyperinterpolation degree $n\to\infty$ and the MZ-constant $\eta\to 0$. Numerical experiments across non-singular and singular kernels, and various point configurations, validate the theory and demonstrate robustness when quadrature points do not form exact polynomial rules. The approach broadens the choice of quadrature points and provides a practical, geometrically informed error control for spherical integral equations with weakly singular kernels.

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.

Figures (8)

• Figure 1: Non-singular $h=1$ and oscillatory $K(\bm{x},\bm{y})=\sin(10|\bm{x}-\bm{y}|)$: Numerical solutions ($n=20$ and $m = (2n+1)^2$) of the equation \ref{['equ:equation']} with exact solution being constant 1 and their absolute errors.
• Figure 2: Non-singular $h=1$ and oscillatory $K(\bm{x},\bm{y})=\sin(10|\bm{x}-\bm{y}|)$: Uniform errors with different $n$ and $m$.
• Figure 3: Singular $h(\bm{x},\bm{y})= |\bm{x}-\bm{y}|^{-0.5}$ and oscillatory $K(\bm{x},\bm{y})=\cos(10|\bm{x}-\bm{y}|)$: Numerical solutions ($n=20$ and $m = (2n+1)^2$) of the equation \ref{['equ:equation']} with exact solution being constant 1 and their absolute errors.
• Figure 4: Singular $h(\bm{x},\bm{y})= |\bm{x}-\bm{y}|^{-0.5}$ and oscillatory $K(\bm{x},\bm{y})=\cos(10|\bm{x}-\bm{y}|)$: Uniform errors with different $n$ and $m$.
• Figure 5: Singular $h(\bm{x},\bm{y})= \log|\bm{x}-\bm{y}|$ and non-oscillatory $K=1$: Numerical solutions ($n=5$ and $m = (2n+1)^2$) of the equation \ref{['equ:equation']} with exact solution being constant 1 and their absolute errors.

Theorems & Definitions (8)

• theorem 1: Main theorem
• Remark 3.1
• Lemma 3.1
• Remark 3.2
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5