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An operator preconditioned combined field integral equation for electromagnetic scattering

Van Chien Le, Kristof Cools

Abstract

This paper aims to address two issues of integral equations for the scattering of time-harmonic electromagnetic waves by a perfect electric conductor with Lipschitz continuous boundary: ill-conditioned {boundary element Galerkin matrices} on fine meshes and instability at spurious resonant frequencies. The remedy to ill-conditioned matrices is operator preconditioning, and resonant instability is eliminated by means of a combined field integral equation. Exterior traces of single and double layer potentials are complemented by their interior counterparts for a purely imaginary wave number. We derive the corresponding variational formulation in the natural trace space for electromagnetic fields and establish its well-posedness for all wave numbers. A Galerkin discretization scheme is employed using conforming edge boundary elements on dual meshes, which produces well-conditioned discrete linear systems of the variational formulation. Some numerical results are also provided to support the numerical analysis.

An operator preconditioned combined field integral equation for electromagnetic scattering

Abstract

This paper aims to address two issues of integral equations for the scattering of time-harmonic electromagnetic waves by a perfect electric conductor with Lipschitz continuous boundary: ill-conditioned {boundary element Galerkin matrices} on fine meshes and instability at spurious resonant frequencies. The remedy to ill-conditioned matrices is operator preconditioning, and resonant instability is eliminated by means of a combined field integral equation. Exterior traces of single and double layer potentials are complemented by their interior counterparts for a purely imaginary wave number. We derive the corresponding variational formulation in the natural trace space for electromagnetic fields and establish its well-posedness for all wave numbers. A Galerkin discretization scheme is employed using conforming edge boundary elements on dual meshes, which produces well-conditioned discrete linear systems of the variational formulation. Some numerical results are also provided to support the numerical analysis.
Paper Structure (19 sections, 14 theorems, 98 equations, 3 figures)

This paper contains 19 sections, 14 theorems, 98 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The scattering boundary value problem
  3. Challenges in boundary integral equations
  4. Previous and related works
  5. Novelty and contributions
  6. Outline
  7. Traces and spaces
  8. Potentials and integral operators
  9. The combined field integral equation
  10. Uniqueness
  11. Coercivity
  12. Galerkin discretization
  13. Mixed variational formulation
  14. Galerkin boundary element discretization
  15. Matrix representation
  16. ...and 4 more sections

Key Result

Theorem 2.1

\newlabelthm:duality0 The pairing $\langle{\cdot, \cdot}\rangle_{\times, \Gamma}$ can be extended to a continuous bilinear form on $\mathop{\mathrm{\mathbf H}}\nolimits^{-1/2}_{\times}(\textup{div}_{\Gamma}, \Gamma)$. Moreover, the space $\mathop{\mathrm{\mathbf H}}\nolimits^{-1/2}_{\times}(\textu

Figures (3)

  • Figure 1: Average pointwise error of the scattered electric field $\bm{e}$ with respect to meshwidth $h$ for different purely imaginary wave numbers $i\kappa^\prime$ (left), and spectral condition number of matrices arising from the Galerkin discretization of the proposed CFIE with respect to coupling parameter $\eta$ (right) for a sphere of radius $1\mathrm{m}$.
  • Figure 2: Condition number of matrices arising from the Galerkin discretization of the EFIE and the proposed CFIE, together with number of GMRES iterations required to solve the corresponding discrete systems for a sphere of radius $1\mathrm{m}$. Left: the wave number is fixed at $\kappa = \tfrac{2\pi}{3}$, and the meshwidth is varying. Right: the meshwidth is fixed at $h = 0.15\mathrm{m}$, and the wave number is varying.
  • Figure 3: Condition number of matrices arising from the Galerkin discretization of the EFIE and the proposed CFIE, together with number of GMRES iterations required to solve the corresponding discrete systems for a cube of edge $1\mathrm{m}$. Left: the wave number is fixed at $\kappa = \frac{\pi}{2}$, and the meshwidth is varying. Right: the meshwidth is fixed at $h = 0.1\mathrm{m}$, and the wave number is varying.

Theorems & Definitions (24)

  • Theorem 2.1: self-duality of the space $\mathop{\mathrm{\mathbf H}}\nolimits^{-1/2}_{\times}(\textup{div}_{\Gamma}, \Gamma)$
  • Theorem 2.2: integration by parts formula
  • Theorem 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Theorem 3.3
  • Corollary 3.4
  • Theorem 4.1: Uniqueness
  • Proof 3
  • ...and 14 more