Integral equation methods for elastance and mobility problems in two dimensions

TL;DR

This paper introduces new integral equation methods for elastance and mobility problems in 2D, resulting in well-conditioned systems that enable efficient large-scale simulations in electrostatics and fluid dynamics.

Contribution

The paper develops resonance-free, second-kind integral representations for elastance and mobility problems, improving computational efficiency and stability over previous methods.

Findings

Resonance-free Fredholm integral equations of the second kind.

Well-conditioned linear systems upon discretization.

Efficient large-scale problem solving with high-order quadrature and fast multipole methods.

Abstract

We present new integral representations in two dimensions for the elastance problem in electrostatics and the mobility problem in Stokes flow. These representations lead to resonance-free Fredholm integral equations of the second kind and well conditioned linear systems upon discretization. By coupling our integral equations with high order quadrature and fast multipole acceleration, large-scale problems can be solved with only modest computing resources. We also discuss some applications of these boundary value problems in applied physics.