An Integral Equation Formulation of the $N$-Body Dielectric Spheres Problem. Part I: Numerical Analysis

TL;DR

This paper develops an N-independent stability and convergence analysis for an integral equation modeling the mutual polarization of N dielectric spheres, enabling scalable numerical solutions for large N.

Contribution

It introduces a novel a priori error analysis that proves N-independent stability and convergence rates for the integral equation formulation of dielectric sphere interactions.

Findings

Stability constants are independent of N.

Convergence rates do not depend on the number of spheres.

The analysis supports scalable numerical methods for large N.

Abstract

In this article, we analyse an integral equation of the second kind that represents the solution of $N$ interacting dielectric spherical particles undergoing mutual polarisation. A traditional analysis can not quantify the scaling of the stability constants -- and thus the approximation error -- with respect to the number $N$ of involved dielectric spheres. We develop a new a priori error analysis that demonstrates $N$-independent stability of the continuous and discrete formulations of the integral equation. Consequently, we obtain convergence rates that are independent of $N$.