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An all-frequency stable integral system for Maxwell's equations in 3-D penetrable media: continuous and discrete model analysis

Mahadevan Ganesh, Stuart C. Hawkins, Darko Volkov

TL;DR

The paper tackles all-frequency stability in 3-D Maxwell scattering for penetrable media by deriving a self-adjoint, constraints-free surface integral equation that remains well-posed for all frequencies $\omega>0$ and avoids low-frequency breakdown as $\omega\to 0$. It achieves this by embedding stabilization into a self-adjoint operator $[\boldsymbol{I}+\boldsymbol{M}^*\boldsymbol{M}+\boldsymbol{M}^*+\boldsymbol{M}+\boldsymbol{J}^*\boldsymbol{J}]$, yielding a numerically tractable formulation equivalent to the constrained approach but more robust. A fully discrete, spectrally accurate solver on curved interfaces is developed using vector spherical polynomials and a carefully designed projection, with a proven convergence rate that scales favorably with smoothness. Numerical experiments on spheres and curved geometries confirm high-order accuracy, reciprocity, and scalability across dielectric and absorbing regimes, highlighting the method's potential for accurate 3-D Maxwell simulations on nontrivial geometries. Overall, the work advances all-frequency SIE theory and provides a practical, high-accuracy framework for dielectric scattering in 3-D with curved interfaces.

Abstract

We introduce a new system of surface integral equations for Maxwell's transmission problem in three dimensions. This system has two remarkable features, both of which we prove. First, it is well-posed at all frequencies. Second, the underlying linear operator has a uniformly bounded inverse as the frequency approaches zero, ensuring that there is no low-frequency breakdown. The system is derived from a formulation we introduced in our previous work, which required additional integral constraints to ensure well -posedness across all frequencies. In this study, we eliminate those constraints and demonstrate that our new self adjoint, constraints-free linear system expressed in the desirable form of an identity plus a compact weakly-singular operator is stable for all frequencies. Furthermore, we propose and analyze a fully discrete numerical method for these systems and provide a proof of spectrally accurate convergence for the computational method. We also computationally demonstrate the high-order accuracy of the algorithm using benchmark scatterers with curved surfaces.

An all-frequency stable integral system for Maxwell's equations in 3-D penetrable media: continuous and discrete model analysis

TL;DR

The paper tackles all-frequency stability in 3-D Maxwell scattering for penetrable media by deriving a self-adjoint, constraints-free surface integral equation that remains well-posed for all frequencies and avoids low-frequency breakdown as . It achieves this by embedding stabilization into a self-adjoint operator , yielding a numerically tractable formulation equivalent to the constrained approach but more robust. A fully discrete, spectrally accurate solver on curved interfaces is developed using vector spherical polynomials and a carefully designed projection, with a proven convergence rate that scales favorably with smoothness. Numerical experiments on spheres and curved geometries confirm high-order accuracy, reciprocity, and scalability across dielectric and absorbing regimes, highlighting the method's potential for accurate 3-D Maxwell simulations on nontrivial geometries. Overall, the work advances all-frequency SIE theory and provides a practical, high-accuracy framework for dielectric scattering in 3-D with curved interfaces.

Abstract

We introduce a new system of surface integral equations for Maxwell's transmission problem in three dimensions. This system has two remarkable features, both of which we prove. First, it is well-posed at all frequencies. Second, the underlying linear operator has a uniformly bounded inverse as the frequency approaches zero, ensuring that there is no low-frequency breakdown. The system is derived from a formulation we introduced in our previous work, which required additional integral constraints to ensure well -posedness across all frequencies. In this study, we eliminate those constraints and demonstrate that our new self adjoint, constraints-free linear system expressed in the desirable form of an identity plus a compact weakly-singular operator is stable for all frequencies. Furthermore, we propose and analyze a fully discrete numerical method for these systems and provide a proof of spectrally accurate convergence for the computational method. We also computationally demonstrate the high-order accuracy of the algorithm using benchmark scatterers with curved surfaces.
Paper Structure (8 sections, 9 theorems, 136 equations, 11 figures, 12 tables)

This paper contains 8 sections, 9 theorems, 136 equations, 11 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Surface integral operators and properties
  3. A stability constrained well-posed SIE system
  4. A self-adjoint SIE system without constraints: well-posedness and low frequency stability
  5. Numerical algorithm and convergence analysis
  6. Vector spherical polynomials
  7. Numerical experiments
  8. Singularity of standard coupling approach

Key Result

Theorem 1

Let $[\boldsymbol{f}, \boldsymbol{g}]^T$ be the trace on $\partial D$ of some incident field $[\boldsymbol{E}_\mathrm{inc}, \boldsymbol{H}_\mathrm{inc}]$. Assume that $\epsilon^\pm$ and $\mu^\pm$ are as specified in Section sec:int. Let $[\boldsymbol{E}, \boldsymbol{H}]$ be the unique solution of th In addition, if $\partial D$ is of class $C^{m,1}$ with $m\geq 1$, then $(\boldsymbol{e},\boldsymbo

Figures (11)

  • Figure 1: Schematic showing vertical (V) and horizontal (H) polarization with respect to the scattering plane for an incident wave with direction $\boldsymbol{d}$ and a receiver in the direction $\widehat{\boldsymbol{x}}(\theta)$.
  • Figure 2: Condition number $\kappa$ of the matrix in systems \ref{['eq:strawman']} (without stabilization) and \ref{['eq:SIE']} (with stabilization) for a dielectric unit sphere with refractive index $\nu = 1.303324617806156$ and frequency $\omega$.
  • Figure 3: Visualization of the radar cross section (HH-polarization) computed with $n={80}$ of a Chebyshev particle with refractive index $\nu = {1.584}$ and electromagnetic size $s = {8}$ in original orientation (blue) and $\pi/2$-rotated orientation (red). Coincidence of the curves at $\theta = \pi/2$ is marked by a $\circ$. The right figure shows a close-up of the region indicated by the box in the left figure.
  • Figure 4: Visualization of the radar cross section (HH-polarization) computed with $n={80}$ of a Chebyshev particle with refractive index $\nu = {1.5 + 0.02i}$ and electromagnetic size $s = {8}$ in original orientation (blue) and $\pi/2$-rotated orientation (red). Coincidence of the curves at $\theta = \pi/2$ is marked by a $\circ$. The right figure shows a close-up of the region indicated by the box in the left figure.
  • Figure 5: Visualization of the radar cross section (HH-polarization) computed with $n={80}$ of a Chebyshev particle with refractive index $\nu = {1.0925 + 0.248i}$ and electromagnetic size $s = {8}$ in original orientation (blue) and $\pi/2$-rotated orientation (red). Coincidence of the curves at $\theta = \pi/2$ is marked by a $\circ$. The right figure shows a close-up of the region indicated by the box in the left figure.
  • ...and 6 more figures

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • ...and 10 more