Maxwell \`a la Helmholtz: Direct boundary integral equations for 3D scattering by perfect electric conductors via Helmholtz operators

TL;DR

This paper develops direct boundary integral equations for 3D electromagnetic scattering by PEC objects, ensuring unique solvability at all frequencies and addressing low-frequency issues with new regularizations.

Contribution

It introduces direct formulations based on physical surface traces, proves their unique solvability, and proposes regularizations to improve numerical stability, especially at low frequencies.

Findings

Formulations are uniquely solvable at all frequencies.

Regularizations make the equations of the Fredholm second kind.

Numerical experiments confirm accuracy and robustness.

Abstract

This paper is the direct-formulation companion to [Burbano-Gallegos, P\'erez-Arancibia, and Turc, ESAIM: M2AN, 60(1):273--315, 2026], which developed indirect combined-field-only boundary integral equations (BIEs) for time-harmonic electromagnetic scattering by smooth perfectly electrically conducting (PEC) obstacles, relying entirely on Helmholtz boundary integral operators. Here we exploit the same equivalence between the Maxwell PEC scattering problem and a pair of vector Helmholtz boundary value problems -- one for the electric field and one for the magnetic field -- to derive direct BIE formulations whose unknowns are the Dirichlet and Neumann traces of the total fields, decomposed into their normal and tangential surface components. These unknowns carry direct physical meaning: in particular, the magnetic-field formulation yields the surface electric currents as part of its solution. The mixed regularity of the two field-trace components requires introducing a tailored product H\"older space, a distinctive feature absent from the indirect approach. We prove that the resulting Direct Electric and Magnetic Combined-Field-Only Integral Equations (D-ECFOIE and D-MCFOIE) are uniquely solvable at all frequencies, and introduce Calder\'on-type regularizations (RD-ECFOIE and RD-MCFOIE) that render them of the Fredholm second kind. We further examine the low-frequency breakdown affecting the electric-field formulation and introduce a modified equation that enforces the physical charge-conservation constraints, which restores numerical accuracy and well-conditioned linear systems for frequencies arbitrarily close to zero. Numerical experiments, performed using a high-order Nystr\"om solver based on the Density Interpolation Method and implemented in the Julia package Inti.jl, validate the accuracy and robustness of the proposed formulations across a range of geometries and frequencies.