Integral equation methods for electrostatics, acoustics and electromagnetics in smoothly varying, anisotropic media

TL;DR

This paper develops well-conditioned integral equation methods for solving electrostatic, acoustic, and electromagnetic scattering problems in smoothly varying, anisotropic media, providing a unified framework and demonstrating their efficiency with iterative and FFT-based techniques.

Contribution

It introduces a new vector PDE for electrostatic and acoustic problems and modifies classical formulations for electromagnetics, resulting in well-conditioned integral equations.

Findings

Methods are well-conditioned and efficient.

Unified framework for different wave problems.

Effective iterative solutions with FFT acceleration.

Abstract

We present a collection of well-conditioned integral equation methods for the solution of electrostatic, acoustic or electromagnetic scattering problems involving anisotropic, inhomogeneous media. In the electromagnetic case, our approach involves a minor modification of a classical formulation. In the electrostatic or acoustic setting, we introduce a new vector partial differential equation, from which the desired solution is easily obtained. It is the vector equation for which we derive a well-conditioned integral equation. In addition to providing a unified framework for these solvers, we illustrate their performance using iterative solution methods coupled with the FFT-based technique of [1] to discretize and apply the relevant integral operators.