Maxwell \`a la Helmholtz: Electromagnetic scattering by 3D perfect electric conductors via Helmholtz integral operators

TL;DR

This paper develops a new class of boundary integral equations for 3D electromagnetic scattering by perfect electric conductors, using Helmholtz operators to ensure well-posedness and eliminate spurious resonances, with demonstrated robustness and accuracy.

Contribution

It introduces Helmholtz-based boundary integral formulations for Maxwell scattering that are provably well-posed, resonance-free, and extend the equivalence between Maxwell and Helmholtz problems.

Findings

Formulations are frequency-robust and free from spurious resonances.

Numerical examples show high accuracy and robustness.

Implementation in open-source Julia package demonstrates practical effectiveness.

Abstract

This paper introduces a novel class of indirect boundary integral equation (BIE) formulations for the solution of electromagnetic scattering problems involving smooth perfectly electric conductors (PECs) in three-dimensions. These combined-field-type BIE formulations rely exclusively on classical Helmholtz boundary operators, resulting in provably well-posed, frequency-robust, Fredholm second-kind BIEs. Notably, we prove that the proposed formulations are free from spurious resonances, while retaining the versatility of Helmholtz integral operators. The approach is based on the equivalence between the Maxwell PEC scattering problem and two independent vector Helmholtz boundary value problems for the electric and magnetic fields, with boundary conditions defined in terms of the Dirichlet and Neumann traces of the corresponding vector Helmholtz solutions. While certain aspects of this equivalence (for the electric field) have been previously exploited in the so-called field-only BIE formulations, we here rigorously establish and generalize the equivalence between Maxwell and Helmholtz problems for both fields. Finally, a variety of numerical examples highlights the robustness and accuracy of the proposed approach when combined with Density Interpolation-based Nystr\"om methods and fast linear algebra solvers, implemented in the open-source Julia package Inti$.$jl.