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Coupled Boundary and Volume Integral Equations for Electromagnetic Scattering

Ignacio Labarca-Figueroa, Ralf Hiptmair

TL;DR

This work introduces a coupled boundary–volume integral equation (STF–VIE) formulation for frequency-domain Maxwell scattering by bounded, inhomogeneous obstacles. Building on Stratton–Chu representations, it fuses boundary Calderón operators with compact volume potentials to yield a first-kind boundary–volume system that reduces to STF for piecewise-constant interiors. The authors establish well-posedness via a variational framework and generalized Gårding (T-coercivity) arguments, and develop a Galerkin discretization using Curl-conforming and Div-conforming spaces, with numerical experiments indicating favorable convergence and potential reduced pollution at high frequency. The approach, complemented by H-matrix compression and careful quadrature, offers a promising alternative to purely boundary or volume methods, especially when inhomogeneities are localized, while leaving open questions on uniqueness in general settings and discrete duality stability.

Abstract

We study frequency domain electromagnetic scattering at a bounded, penetrable, and inhomogeneous obstacle $ Ω\subset \mathbb{R}^3 $. From the Stratton-Chu integral representation, we derive a new representation formula when constant reference coefficients are given for the interior domain. The resulting integral representation contains the usual layer potentials, but also volume potentials on $Ω$. Then it is possible to follow a single-trace approach to obtain boundary integral equations perturbed by traces of compact volume integral operators with weakly singular kernels. The coupled boundary and volume integral equations are discretized with a Galerkin approach with usual Curl-conforming and Div-conforming finite elements on the boundary and in the volume. Compression techniques and special quadrature rules for singular integrands are required for an efficient and accurate method. Numerical experiments provide evidence that our new formulation enjoys promising properties.

Coupled Boundary and Volume Integral Equations for Electromagnetic Scattering

TL;DR

This work introduces a coupled boundary–volume integral equation (STF–VIE) formulation for frequency-domain Maxwell scattering by bounded, inhomogeneous obstacles. Building on Stratton–Chu representations, it fuses boundary Calderón operators with compact volume potentials to yield a first-kind boundary–volume system that reduces to STF for piecewise-constant interiors. The authors establish well-posedness via a variational framework and generalized Gårding (T-coercivity) arguments, and develop a Galerkin discretization using Curl-conforming and Div-conforming spaces, with numerical experiments indicating favorable convergence and potential reduced pollution at high frequency. The approach, complemented by H-matrix compression and careful quadrature, offers a promising alternative to purely boundary or volume methods, especially when inhomogeneities are localized, while leaving open questions on uniqueness in general settings and discrete duality stability.

Abstract

We study frequency domain electromagnetic scattering at a bounded, penetrable, and inhomogeneous obstacle . From the Stratton-Chu integral representation, we derive a new representation formula when constant reference coefficients are given for the interior domain. The resulting integral representation contains the usual layer potentials, but also volume potentials on . Then it is possible to follow a single-trace approach to obtain boundary integral equations perturbed by traces of compact volume integral operators with weakly singular kernels. The coupled boundary and volume integral equations are discretized with a Galerkin approach with usual Curl-conforming and Div-conforming finite elements on the boundary and in the volume. Compression techniques and special quadrature rules for singular integrands are required for an efficient and accurate method. Numerical experiments provide evidence that our new formulation enjoys promising properties.
Paper Structure (35 sections, 25 theorems, 218 equations, 12 figures, 3 tables)

This paper contains 35 sections, 25 theorems, 218 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Maxwell Transmission Problem
  3. VIEs for electromagnetic scattering
  4. BIEs for piecewise-constant coefficients
  5. FEM-BEM coupling
  6. STF-VIEs
  7. Outline and main results
  8. Derivation of VIEs
  9. Preliminaries: Function spaces and trace operators
  10. Fundamental Solutions and Newton Potential
  11. Stratton-Chu Representation Formula
  12. Boundary integral operators
  13. Calderón Identities
  14. Boundary Integral Formulations for Transmission Problems
  15. Boundary-Volume Integral Representation
  16. ...and 20 more sections

Key Result

Theorem 2.1

The Newton potential defines a solution operator for the Helmholtz equation on $\mathbb{R}^3$, i.e. for $f \in L^2_{\mathrm{comp}}(\mathbb{R}^3)$ compactly supported in $\Omega_i$, $u \coloneqq \mathsf{N}_{\kappa}f$ satisfies and the Sommerfeld radiation conditions.

Figures (12)

  • Figure 1: Geometric setting. Inhomogeneous material.
  • Figure 1: Meshes used in Section \ref{['sec:test_cube']}, generated by uniform regular refinement
  • Figure 2: Domains for transmission problems with spatially varying coefficients.
  • Figure 2: Meshes used in Section \ref{['sec:test_tetra']}, generated by uniform regular refinement.
  • Figure 3: Meshes used in Section \ref{['sec:test_fichera']}, generated by uniform regular refinement
  • ...and 7 more figures

Theorems & Definitions (44)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • proof
  • Theorem 2.5: Stratton-Chu Integral Representation
  • Proposition 2.6: Maxwell Layer Potentials buffa2003galerkin
  • Lemma 2.7: buffa2003galerkin
  • Proposition 2.8: Jump relations buffa2003galerkin
  • ...and 34 more