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arXiv · 2605.01670

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

Carlos Pérez-Arancibia·Catalin Turc
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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.

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Carlos Pérez-Arancibia, Catalin Turc. 2026-05-03. Maxwell \`a la Helmholtz: Direct boundary integral equations for 3D scattering by perfect electric conductors via Helmholtz operators. https://arxiv.org/abs/2605.01670

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