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Integral Equations and Operator Theory

, Volume 84, Issue 1, pp 33–68 | Cite as

Integral Equations for Electromagnetic Scattering at Multi-Screens

  • X. ClaeysEmail author
  • R. Hiptmair
Article

Abstract

In (X. Claeys and R. Hiptmair, Integral equations on multi-screens. Integral Equ Oper Theory, 77(2):167–197, 2013) we developed a framework for the analysis of boundary integral equations for acoustic scattering at so-called multi-screens, which are arbitrary arrangements of thin panels made of impenetrable material. In this article we extend these considerations to boundary integral equations for electromagnetic scattering. We view tangential multi-traces of vector fields from the perspective of quotient spaces and introduce the notion of single-traces and spaces of jumps. We also derive representation formulas and establish key properties of the involved potentials and related boundary operators. Their coercivity will be proved using a splitting of jump fields. Another new aspect emerges in the form of surface differential operators linking various trace spaces.

Keywords

Screen integral equation scattering wave propagation Helmholtz junction points 

List of Symbols

Γ

Multi-screen with boundary Γ

Ωj

Finite collection of Lipschitz domains adjacent to Γ, see Definition 2.3

\({{\rm H}^{1}(\mathbb{R}^{3}\backslash \bar\Gamma)}\)

Sobolev space of functions \({\mathbb{R}^{3} \backslash\bar\Gamma \to \mathbb{C}}\), see (3.1)

\({{\bf H}({\rm div},\mathbb{R}^{3}\backslash\bar\Gamma)}\)

Sobolev space of vector fields \({\mathbb{R}^{3}\backslash \bar\Gamma \to \mathbb{C}^{3}}\) with square integrable divergence

\({\mathbb{H}^{\pm \frac{1}{2}}(\Gamma)}\)

Scalar values multi-trace spaces, see Definition 3.1

\({\pi_{\rm {\sc{D}}}}\)

Dirichlet trace (point trace) \({{\rm H}^{1}(\mathbb{R}^{3}\backslash \bar\Gamma)\to \mathbb{H}^{1/2}(\Gamma)}\)

\({\pi_{\rm {\sc{N}}}}\)

Normal component trace \({{\bf H}({\rm div},\mathbb{R}^{3}\backslash\bar\Gamma) \to \mathbb{H}^{-1/2}(\Gamma)}\)

\({\langle\langle{\cdot},{\cdot}\rangle\rangle}\)

Bilinear duality pairing for scalar functions on Γ, see (3.3)

\({{\rm H}^{\pm \frac{1}{2}}([\Gamma])}\)

Scalar-valued single traces spaces, see Definition 3.2

\({{\bf H}({\bf curl}, \mathbb{R}^{3}\backslash \bar\Gamma)}\)

Sobolev space of vector fields \({\mathbb{R}^{3}\backslash \bar\Gamma \to \mathbb{C}^{3}}\) with square integrable curl

\({\gamma_{\rm \sc{T}}}\)

Standard tangential trace operator, see (4.2), (7.1)

\({\mathbb{H}^{-\frac{1}{2}}({\rm curl_\Gamma},\Gamma)}\)

Tangential multi-trace space, see Definition 4.4

\({\pi_{\rm \sc{T}}}\)

Tangential multi-trace operator; canonical projection onto \({\mathbb{H}^{-\frac{1}{2}}({\rm curl}_\Gamma,\Gamma)}\), see (4.5)

\({\langle\langle{\cdot},{\cdot}\rangle\rangle_\times}\)

Skew-symmetric duality pairing in \({\mathbb{H}^{-\frac{1}{2}}({\rm curl}_\Gamma,\Gamma)}\), see (4.6), (4.8)

\({{\bf H}^{-\frac{1}{2}}({\rm curl_\Gamma},[\Gamma])}\)

Tangential single-trace space, see Definition 4.12

\({\widetilde{\bf H}\,^{-1/2}({\rm curl_\Gamma},[\Gamma])}\)

Tangential jump space, see Definition 4.13

\({[\;\;]}\)

Jump operator, see (3.5) and Definitions 4.8

\({\nabla_{\Gamma}}\)

Surface gradient, see (5.2)

curlγ

Surface rotation, see (5.4)

\({\gamma_{\rm {\sc{R}}}}\)

Tangential trace of curl, see (7.1)

\({\mathcal{G}_{\kappa}}\)

Helmholtz fundamental solution with wave number κ

DLκ, SLκ

Vector single and double layer potentials, see (7.6)

\({{\bf \imath}}\)

Imaginary unit

\({{\bf H}^{\frac{1}{2}}_{\times}(\Gamma)}\)

Tangential trace space for \({(\mathbb{H}^{1}(\mathbb{R}^{3}))^{3}}\), see (8.5)

\({\mathcal{E}_{\rm \sc{T}}}\)

Dirichlet harmonic vector fields, see (9.1)

Mathematics Subject Classification

Primary 45A05 Secondary 65R20 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUPMC Univ. Paris 6, CNRS UMR 7598ParisFrance
  2. 2.INRIA-Paris-Rocquencourt, EPC AlpinesLe Chesnay CedexFrance
  3. 3.Seminar of Applied MathematicsETH ZürichZurichSwitzerland

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