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Second-Kind Boundary Integral Equations for Electromagnetic - - PowerPoint PPT Presentation

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Second-Kind Boundary Integral Equations for Electromagnetic Scattering at Composite Objects X. Claeys 2 , R. Hiptmair 1 , C. E. Spindler 1 1 Seminar for Applied Mathematics, ETH Zrich 2 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis

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SLIDE 1

Second-Kind Boundary Integral Equations for Electromagnetic Scattering at Composite Objects

  • X. Claeys2, R. Hiptmair1, C. E. Spindler 1

1 Seminar for Applied Mathematics, ETH Zürich 2 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions

RICAM Workshop on Analysis and Numerics

  • f Acoustic and Electromagnetic Problems

Oct 17-21, 2016

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 1 / 29

slide-2
SLIDE 2

Second-Kind Boundary Integral Equations for Electromagnetic Scattering at Composite Objects

  • X. Claeys2, R. Hiptmair1, C. E. Spindler 1

1 Seminar for Applied Mathematics, ETH Zürich 2 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions

RICAM Workshop on Analysis and Numerics

  • f Acoustic and Electromagnetic Problems

Oct 17-21, 2016

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 1 / 29

slide-3
SLIDE 3

Second-Kind Boundary Integral Equations for Electromagnetic Scattering at Composite Objects

  • X. Claeys2, R. Hiptmair1, C. E. Spindler 1

1 Seminar for Applied Mathematics, ETH Zürich 2 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions

RICAM Workshop on Analysis and Numerics

  • f Acoustic and Electromagnetic Problems

Oct 17-21, 2016

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 1 / 29

slide-4
SLIDE 4

Electromagnetic Transmission Problem

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 2 / 29

slide-5
SLIDE 5

Electromagnetic Transmission Problem

Frequency domain EM scattering at composite object (piecewise homogeneous material)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 2 / 29

slide-6
SLIDE 6

Electromagnetic Transmission Problem

Ω0

Γ

Ω1 Ω2 Ω3

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Frequency domain EM scattering at composite object (piecewise homogeneous material) curl curl Ui − κ2

i Ui = 0

in Ωi ,

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 2 / 29

slide-7
SLIDE 7

Electromagnetic Transmission Problem

Ω0

Γ

Ω1 Ω2 Ω3

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Frequency domain EM scattering at composite object (piecewise homogeneous material) curl curl Ui − κ2

i Ui = 0

in Ωi , + transmission conditions across interface Γij := ∂Ωi ∩ ∂Ωj γi

EU−γj EU

= γi

MU+γj MU

=

  • n Γij ,
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 2 / 29

slide-8
SLIDE 8

Electromagnetic Transmission Problem

Ω0

Γ

Ω1 Ω2 Ω3

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Frequency domain EM scattering at composite object (piecewise homogeneous material) curl curl Ui − κ2

i Ui = 0

in Ωi , + transmission conditions across interface Γij := ∂Ωi ∩ ∂Ωj γi

EU−γj EU

= γi

MU+γj MU

=

  • n Γij ,

Tangential traces: γEU = γtU := n × (U × n)|∂Ω (“electric trace”) γMU = γ× curl U := curl U × n|∂Ω (“magnetic trace”)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 2 / 29

slide-9
SLIDE 9

Electromagnetic Transmission Problem

Ω0

Γ

Ω1 Ω2 Ω3

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Frequency domain EM scattering at composite object (piecewise homogeneous material) curl curl Ui − κ2

i Ui = 0

in Ωi , + transmission conditions across interface Γij := ∂Ωi ∩ ∂Ωj γi

EU−γj EU

= γi

MU+γj MU

=

  • n Γij ,

Tangential traces: γEU = γtU := n × (U × n)|∂Ω (“electric trace”) γMU = γ× curl U := curl U × n|∂Ω (“magnetic trace”)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 2 / 29

slide-10
SLIDE 10

Electromagnetic Transmission Problem

Ω0

Γ

Ω1 Ω2 Ω3

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Frequency domain EM scattering at composite object (piecewise homogeneous material) curl curl Ui − κ2

i Ui = 0

in Ωi , + transmission conditions across interface Γij := ∂Ωi ∩ ∂Ωj γi

EU−γj EU

= γi

MU+γj MU

=

  • n Γij ,

+ Silver-Müller r.c. at ∞. Tangential traces: γEU = γtU := n × (U × n)|∂Ω (“electric trace”) γMU = γ× curl U := curl U × n|∂Ω (“magnetic trace”)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 2 / 29

slide-11
SLIDE 11

Electromagnetic Transmission Problem

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Frequency domain EM scattering at composite object (piecewise homogeneous material) curl curl Ui − κ2

i Ui = 0

in Ωi , + transmission conditions across interface Γij := ∂Ωi ∩ ∂Ωj γi

EU−γj EU

= γi

MU+γj MU

=

  • n Γij ,

+ Silver-Müller r.c. at ∞. Excitation by incident (plane) wave Uinc Tangential traces: γEU = γtU := n × (U × n)|∂Ω (“electric trace”) γMU = γ× curl U := curl U × n|∂Ω (“magnetic trace”)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 2 / 29

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SLIDE 12

Electromagnetic Trace Spaces

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

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SLIDE 13

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

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SLIDE 14

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω) ✓ ✒ ✏ ✑ γE : H(curl, Ω) γM : H(curl2, Ω)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

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SLIDE 15

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω) ✓ ✒ ✏ ✑ γE : H(curl, Ω) → H− 1

2 (curlΓ, Γ) := HE(Γ)

γM : H(curl2, Ω) → H− 1

2 (divΓ, Γ) := HM(Γ)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

slide-16
SLIDE 16

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω) ✓ ✒ ✏ ✑ γE : H(curl, Ω) → H− 1

2 (curlΓ, Γ) := HE(Γ)

γM : H(curl2, Ω) → H− 1

2 (divΓ, Γ) := HM(Γ)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

slide-17
SLIDE 17

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω) ✓ ✒ ✏ ✑ γE : H(curl, Ω) → H− 1

2 (curlΓ, Γ) := HE(Γ)

γM : H(curl2, Ω) → H− 1

2 (divΓ, Γ) := HM(Γ)

continuous & surjective

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

slide-18
SLIDE 18

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω) ✓ ✒ ✏ ✑ γE : H(curl, Ω) → H− 1

2 (curlΓ, Γ) := HE(Γ)

γM : H(curl2, Ω) → H− 1

2 (divΓ, Γ) := HM(Γ)

continuous & surjective ✓ ✒ ✏ ✑ L2

t (Γ)-duality:

HE(Γ) := H− 1

2 (curlΓ, Γ) ↔ H− 1 2 (divΓ, Γ) := HM(Γ)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

slide-19
SLIDE 19

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω) ✓ ✒ ✏ ✑ γE : H(curl, Ω) → H− 1

2 (curlΓ, Γ) := HE(Γ)

γM : H(curl2, Ω) → H− 1

2 (divΓ, Γ) := HM(Γ)

continuous & surjective ✓ ✒ ✏ ✑ L2

t (Γ)-duality:

HE(Γ) := H− 1

2 (curlΓ, Γ) ↔ H− 1 2 (divΓ, Γ) := HM(Γ)

[ Pairing (v, η) → v, ηΓ :=

  • Γ v(y) · η(y) dS(y) ]
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

slide-20
SLIDE 20

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω) ✓ ✒ ✏ ✑ γE : H(curl, Ω) → H− 1

2 (curlΓ, Γ) := HE(Γ)

γM : H(curl2, Ω) → H− 1

2 (divΓ, Γ) := HM(Γ)

continuous & surjective ✓ ✒ ✏ ✑ L2

t (Γ)-duality:

HE(Γ) := H− 1

2 (curlΓ, Γ) ↔ H− 1 2 (divΓ, Γ) := HM(Γ)

[ Pairing (v, η) → v, ηΓ :=

  • Γ v(y) · η(y) dS(y) ]

EM compound trace operator γ := (γE, γM) = (γt, curl · × n),

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

slide-21
SLIDE 21

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω) ✓ ✒ ✏ ✑ γE : H(curl, Ω) → H− 1

2 (curlΓ, Γ) := HE(Γ)

γM : H(curl2, Ω) → H− 1

2 (divΓ, Γ) := HM(Γ)

continuous & surjective ✓ ✒ ✏ ✑ L2

t (Γ)-duality:

HE(Γ) := H− 1

2 (curlΓ, Γ) ↔ H− 1 2 (divΓ, Γ) := HM(Γ)

[ Pairing (v, η) → v, ηΓ :=

  • Γ v(y) · η(y) dS(y) ]

EM compound trace operator γ := (γE, γM) = (γt, curl · × n), EM compound trace space γ : H(curl2, Ω) → H(Γ) := HE(Γ) × HM(Γ).

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

slide-22
SLIDE 22

Electromagnetic Trace Spaces

(Ω ⊂ R3 bounded Lipschitz domain, Γ := ∂Ω) ✓ ✒ ✏ ✑ γE : H(curl, Ω) → H− 1

2 (curlΓ, Γ) := HE(Γ)

γM : H(curl2, Ω) → H− 1

2 (divΓ, Γ) := HM(Γ)

continuous & surjective ✓ ✒ ✏ ✑ L2

t (Γ)-duality:

HE(Γ) := H− 1

2 (curlΓ, Γ) ↔ H− 1 2 (divΓ, Γ) := HM(Γ)

[ Pairing (v, η) → v, ηΓ :=

  • Γ v(y) · η(y) dS(y) ]

EM compound trace operator γ := (γE, γM) = (γt, curl · × n), EM compound trace space γ : H(curl2, Ω) → H(Γ) := HE(Γ) × HM(Γ). ➣ H(Γ): self-dual w.r.t. pairing

  • u

µ

  • ,

v ν

  • Γ

:= u, νΓ − v, µΓ , u µ

  • ,

v ν

  • ∈ T (Γ) .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

slide-23
SLIDE 23

Stratton-Chu Representation Formula

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

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SLIDE 24

Stratton-Chu Representation Formula

Representation formula: Uℓ ∈ H(curl, Ωℓ), − curl curl Uℓ − κ2

ℓUℓ = 0 in Ωℓ:

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

slide-25
SLIDE 25

Stratton-Chu Representation Formula

Representation formula: Uℓ ∈ H(curl, Ωℓ), − curl curl Uℓ − κ2

ℓUℓ = 0 in Ωℓ:

Sℓ[κℓ] (γℓ

MUℓ) − Dℓ[κℓ] (γℓ EUℓ) =

  • Uℓ

in Ωℓ , in R3 \ Ωℓ .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

slide-26
SLIDE 26

Stratton-Chu Representation Formula

Representation formula: Uℓ ∈ H(curl, Ωℓ), − curl curl Uℓ − κ2

ℓUℓ = 0 in Ωℓ:

Sℓ[κℓ] (γℓ

MUℓ) − Dℓ[κℓ] (γℓ EUℓ) =

  • Uℓ

in Ωℓ , in R3 \ Ωℓ . (Maxwell) single layer potential

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

slide-27
SLIDE 27

Stratton-Chu Representation Formula

Representation formula: Uℓ ∈ H(curl, Ωℓ), − curl curl Uℓ − κ2

ℓUℓ = 0 in Ωℓ:

Sℓ[κℓ] (γℓ

MUℓ) − Dℓ[κℓ] (γℓ EUℓ) =

  • Uℓ

in Ωℓ , in R3 \ Ωℓ . (Maxwell) single layer potential Sℓ[κ](ν) (x) = Vℓ[κ](ν)(x) + ∇xVℓ[κ](divΓν)(x) , x ∈ Γ ,

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

slide-28
SLIDE 28

Stratton-Chu Representation Formula

Representation formula: Uℓ ∈ H(curl, Ωℓ), − curl curl Uℓ − κ2

ℓUℓ = 0 in Ωℓ:

Sℓ[κℓ] (γℓ

MUℓ) − Dℓ[κℓ] (γℓ EUℓ) =

  • Uℓ

in Ωℓ , in R3 \ Ωℓ . (Maxwell) single layer potential Vℓ[κ](ϕ)(x) =

  • Γℓ

Φ[κ](x, y)ϕ(y) dS(y) , Sℓ[κ](ν) (x) = Vℓ[κ](ν)(x) + ∇xVℓ[κ](divΓν)(x) , x ∈ Γ , Helmholtz fundamental solution: Φ[κ](x, y) = exp(ıκ|x − y|) 4π|x − y| , x = y

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

slide-29
SLIDE 29

Stratton-Chu Representation Formula

Representation formula: Uℓ ∈ H(curl, Ωℓ), − curl curl Uℓ − κ2

ℓUℓ = 0 in Ωℓ:

Sℓ[κℓ] (γℓ

MUℓ) − Dℓ[κℓ] (γℓ EUℓ) =

  • Uℓ

in Ωℓ , in R3 \ Ωℓ . (Maxwell) single layer potential "smoothing": V : H− 1

2 (∂Ωℓ) → H1

loc(R3)

Vℓ[κ](ϕ)(x) =

  • Γℓ

Φ[κ](x, y)ϕ(y) dS(y) , Sℓ[κ](ν) (x) = Vℓ[κ](ν)(x) + ∇xVℓ[κ](divΓν)(x) , x ∈ Γ , Helmholtz fundamental solution: Φ[κ](x, y) = exp(ıκ|x − y|) 4π|x − y| , x = y

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

slide-30
SLIDE 30

Stratton-Chu Representation Formula

Representation formula: Uℓ ∈ H(curl, Ωℓ), − curl curl Uℓ − κ2

ℓUℓ = 0 in Ωℓ:

Sℓ[κℓ] (γℓ

MUℓ) − Dℓ[κℓ] (γℓ EUℓ) =

  • Uℓ

in Ωℓ , in R3 \ Ωℓ . (Maxwell) single layer potential (Maxwell) double layer potential Vℓ[κ](ϕ)(x) =

  • Γℓ

Φ[κ](x, y)ϕ(y) dS(y) , Sℓ[κ](ν) (x) = Vℓ[κ](ν)(x) + ∇xVℓ[κ](divΓν)(x) , Dℓ[κ](v) (x) = curl Vℓ(v)(x) . x ∈ Γ , Helmholtz fundamental solution: Φ[κ](x, y) = exp(ıκ|x − y|) 4π|x − y| , x = y

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

slide-31
SLIDE 31

Stratton-Chu Representation Formula

Representation formula: Uℓ ∈ H(curl, Ωℓ), − curl curl Uℓ − κ2

ℓUℓ = 0 in Ωℓ:

Gℓ[κℓ](γℓUℓ) := Sℓ[κℓ] (γℓ

MUℓ) − Dℓ[κℓ] (γℓ EUℓ) =

  • Uℓ

in Ωℓ , in R3 \ Ωℓ . (Maxwell) single layer potential (Maxwell) double layer potential Vℓ[κ](ϕ)(x) =

  • Γℓ

Φ[κ](x, y)ϕ(y) dS(y) , Sℓ[κ](ν) (x) = Vℓ[κ](ν)(x) + ∇xVℓ[κ](divΓν)(x) , Dℓ[κ](v) (x) = curl Vℓ(v)(x) . x ∈ Γ , Helmholtz fundamental solution: Φ[κ](x, y) = exp(ıκ|x − y|) 4π|x − y| , x = y

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

slide-32
SLIDE 32

(Subdomain) Boundary Integral Operators

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-33
SLIDE 33

(Subdomain) Boundary Integral Operators

R.F.: Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-34
SLIDE 34

(Subdomain) Boundary Integral Operators

R.F.: Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-35
SLIDE 35

(Subdomain) Boundary Integral Operators

R.F.: Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-36
SLIDE 36

(Subdomain) Boundary Integral Operators

R.F.: Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-37
SLIDE 37

(Subdomain) Boundary Integral Operators

R.F.: Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-38
SLIDE 38

(Subdomain) Boundary Integral Operators

R.F.: Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-39
SLIDE 39

(Subdomain) Boundary Integral Operators

R.F.: Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-40
SLIDE 40

(Subdomain) Boundary Integral Operators

R.F.: Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • EFIE Op.
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-41
SLIDE 41

(Subdomain) Boundary Integral Operators

R.F.: Gℓ[κℓ](γℓUℓ) := Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • MFIE Op.
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-42
SLIDE 42

(Subdomain) Boundary Integral Operators

R.F.: Gℓ[κℓ](γℓUℓ) := Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • γℓGℓ[κℓ](γℓUℓ) =

1

2Id + Aℓ[κℓ]

  • (γℓUℓ) = γℓUℓ

“on Γ ” .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-43
SLIDE 43

(Subdomain) Boundary Integral Operators

R.F.: Gℓ[κℓ](γℓUℓ) := Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • γℓGℓ[κℓ](γℓUℓ) =

1

2Id + Aℓ[κℓ]

  • (γℓUℓ) = γℓUℓ

“on Γ ” .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-44
SLIDE 44

(Subdomain) Boundary Integral Operators

R.F.: Gℓ[κℓ](γℓUℓ) := Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • γℓGℓ[κℓ](γℓUℓ) =

1

2Id + Aℓ[κℓ]

  • (γℓUℓ) = γℓUℓ

“on Γ ” . ✤ ✣ ✜ ✢ u =

  • u

ϕ

  • ∈ H(Γℓ)

are Cauchy data ⇔ γℓGℓ[κℓ](u) = 1

2Id − Cℓ[κℓ]

Sℓ[κℓ] S′

ℓ[κℓ] 1 2Id + C′ ℓ[κℓ]

  • u

ϕ

  • = u .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-45
SLIDE 45

(Subdomain) Boundary Integral Operators

R.F.: Gℓ[κℓ](γℓUℓ) := Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • γℓGℓ[κℓ](γℓUℓ) =

1

2Id + Aℓ[κℓ]

  • (γℓUℓ) = γℓUℓ

“on Γ ” . ✤ ✣ ✜ ✢ u =

  • u

ϕ

  • ∈ H(Γℓ)

are Cauchy data ⇔ γℓGℓ[κℓ](u) = 1

2Id − Cℓ[κℓ]

Sℓ[κℓ] S′

ℓ[κℓ] 1 2Id + C′ ℓ[κℓ]

  • u

ϕ

  • = u .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-46
SLIDE 46

(Subdomain) Boundary Integral Operators

R.F.: Gℓ[κℓ](γℓUℓ) := Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • γℓGℓ[κℓ](γℓUℓ) =

1

2Id + Aℓ[κℓ]

  • (γℓUℓ) = γℓUℓ

“on Γ ” . ✤ ✣ ✜ ✢ u =

  • u

ϕ

  • ∈ H(Γℓ)

are Cauchy data Compound traces of U: curl curl U − κ2

ℓU = 0 in Ωℓ

⇔ γℓGℓ[κℓ](u) = 1

2Id − Cℓ[κℓ]

Sℓ[κℓ] S′

ℓ[κℓ] 1 2Id + C′ ℓ[κℓ]

  • u

ϕ

  • = u .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-47
SLIDE 47

(Subdomain) Boundary Integral Operators

R.F.: Gℓ[κℓ](γℓUℓ) := Sℓ[κℓ](γℓ

MUℓ) − Dℓ[κℓ](γℓ EUℓ) = Uℓ

in Ωℓ . γℓ

ESℓ[κℓ] (γℓ MUℓ)

− γℓ

EDℓ[κℓ] (γℓ EUℓ)

= γℓ

EUℓ ,

γℓ

MSℓ[κℓ] (γℓ MUℓ)

− γℓ

MDℓ[κℓ] (γℓ EUℓ)

= γℓ

MUℓ .

  • 1

2Id − Cℓ[κℓ]

  • (γℓ

EUℓ)

+ Sℓ[κℓ] (γℓ

MUℓ)

S′

ℓ[κℓ] (γℓ EUℓ)

+ 1

2Id + C′ ℓ[κℓ]

  • (γℓ

MUℓ)

  • =

γℓ

EUℓ

γℓ

MUℓ

  • γℓGℓ[κℓ](γℓUℓ) =

1

2Id + Aℓ[κℓ]

  • (γℓUℓ) = γℓUℓ

“on Γ ” . ✤ ✣ ✜ ✢ u =

  • u

ϕ

  • ∈ H(Γℓ)

are Cauchy data ⇔ γℓGℓ[κℓ](u) = 1

2Id − Cℓ[κℓ]

Sℓ[κℓ] S′

ℓ[κℓ] 1 2Id + C′ ℓ[κℓ]

  • (interior) Calderón projector Pℓ = 1

2Id + Aℓ[κℓ]

u ϕ

  • = u .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

slide-48
SLIDE 48

What Next ?

1

Scattering at Composite Objects

2

BIE: Single Subdomain Setting

3

BIE: Composite Scatterer

4

Numerical Experiments

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 6 / 29

slide-49
SLIDE 49

Single Subdomain Setting (SSS)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 7 / 29

slide-50
SLIDE 50

Single Subdomain Setting (SSS)

= Homogeneous scatterer:

Ω0 = Ω

c

Γ Ω1 Uinc

n n n0 n n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 7 / 29

slide-51
SLIDE 51

Single Subdomain Setting (SSS)

= Homogeneous scatterer: κ(x) =

  • κ0

for x ∈ Ω0 , κ1 for x ∈ Ω1 ,

Ω0 = Ω

c

Γ Ω1 Uinc

n n n0 n n0

Electromagnetic scattering: curl curl U − κ(x)2U = 0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 7 / 29

slide-52
SLIDE 52

Single Subdomain Setting (SSS)

= Homogeneous scatterer: κ(x) =

  • κ0

for x ∈ Ω0 , κ1 for x ∈ Ω1 ,

Ω0 = Ω

c

Γ Ω1 Uinc

n n n0 n n0

Electromagnetic scattering: curl curl U − κ(x)2U = 0 Acoustic scattering: −∆U − κ(x)2U = 0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 7 / 29

slide-53
SLIDE 53

SSS: Helmholtz Transmission Problem

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

slide-54
SLIDE 54

SSS: Helmholtz Transmission Problem

If U solves Helmholtz scattering trans- mission problem

Ω0 = Ω

c

Γ Ω uinc

n n n0 n n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

slide-55
SLIDE 55

SSS: Helmholtz Transmission Problem

If U solves Helmholtz scattering trans- mission problem ( 1

2Id + A1[κ1])γ1U = 0 ,

( 1

2Id + A0[κ0])γ0U = . . . .

Ω0 = Ω

c

Γ Ω uinc

n n n0 n n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

slide-56
SLIDE 56

SSS: Helmholtz Transmission Problem

If U solves Helmholtz scattering trans- mission problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω uinc

n n n0 n n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

slide-57
SLIDE 57

SSS: Helmholtz Transmission Problem

If U solves Helmholtz scattering trans- mission problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω uinc

n n n0 n n0

compact operator (L2(Γ))2 → (L2(Γ))2

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

slide-58
SLIDE 58

SSS: Helmholtz Transmission Problem

If U solves Helmholtz scattering trans- mission problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω uinc

n n n0 n n0

compact operator (L2(Γ))2 → (L2(Γ))2 A1[κ0] − A1[κ0] = −δK1 δV1 δW1 δK′

1

  • [difference BI-Ops]
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

slide-59
SLIDE 59

SSS: Helmholtz Transmission Problem

If U solves Helmholtz scattering trans- mission problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω uinc

n n n0 n n0

compact operator (L2(Γ))2 → (L2(Γ))2 A1[κ0] − A1[κ0] = −δK1 δV1 δW1 δK′

1

  • [difference BI-Ops]

δV1(ϕ)(x) =

  • Γ

exp(ıκ0|x − y|) − exp(ıκ1|x − y|) 4π|x − y|

  • C0-kernel

ϕ(y) dS(y) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

slide-60
SLIDE 60

SSS: Helmholtz Transmission Problem

If U solves Helmholtz scattering trans- mission problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω uinc

n n n0 n n0

compact operator (L2(Γ))2 → (L2(Γ))2 ➤ 2nd-kind BIE A1[κ0] − A1[κ0] = −δK1 δV1 δW1 δK′

1

  • [difference BI-Ops]

δV1(ϕ)(x) =

  • Γ

exp(ıκ0|x − y|) − exp(ıκ1|x − y|) 4π|x − y|

  • C0-kernel

ϕ(y) dS(y) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

slide-61
SLIDE 61

SSS: Maxwell Transmission Problem

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

slide-62
SLIDE 62

SSS: Maxwell Transmission Problem

Copy Helmholtz approach: If U solves Maxwell scattering problem

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

slide-63
SLIDE 63

SSS: Maxwell Transmission Problem

Copy Helmholtz approach: If U solves Maxwell scattering problem ( 1

2Id + A1[κ1])γ1U = 0 ,

( 1

2Id + A0[κ0])γ0U = . . . .

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

slide-64
SLIDE 64

SSS: Maxwell Transmission Problem

Copy Helmholtz approach: If U solves Maxwell scattering problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

slide-65
SLIDE 65

SSS: Maxwell Transmission Problem

Copy Helmholtz approach: If U solves Maxwell scattering problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

compact operator (L2

t (Γ))2 → (L2 t (Γ))2?

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

slide-66
SLIDE 66

SSS: Maxwell Transmission Problem

Copy Helmholtz approach: If U solves Maxwell scattering problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

compact operator (L2

t (Γ))2 → (L2 t (Γ))2?

S1[κ1](ϕ) − S1[κ0](ϕ) =V1[κ0](ϕ) − V1[κ1](ϕ)+

1 κ2

0 ∇ΓV1[κ0](divΓϕ) − 1

κ2

1 ∇ΓV1[κ1](divΓϕ) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

slide-67
SLIDE 67

SSS: Maxwell Transmission Problem

Copy Helmholtz approach: If U solves Maxwell scattering problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

compact operator (L2

t (Γ))2 → (L2 t (Γ))2?

S1[κ1](ϕ) − S1[κ0](ϕ) = Cancellation of singularities V1[κ0](ϕ) − V1[κ1](ϕ)+

1 κ2

0 ∇ΓV1[κ0](divΓϕ) − 1

κ2

1 ∇ΓV1[κ1](divΓϕ) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

slide-68
SLIDE 68

SSS: Maxwell Transmission Problem

Copy Helmholtz approach: If U solves Maxwell scattering problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

compact operator (L2

t (Γ))2 → (L2 t (Γ))2?

S1[κ1](ϕ) − S1[κ0](ϕ) =V1[κ0](ϕ) − V1[κ1](ϕ)+ Remains hypersingular!

1 κ2

0 ∇ΓV1[κ0](divΓϕ) − 1

κ2

1 ∇ΓV1[κ1](divΓϕ) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

slide-69
SLIDE 69

SSS: Maxwell Transmission Problem

Copy Helmholtz approach: If U solves Maxwell scattering problem ( 1

2Id + A1[κ1])γ1U = 0 ,

+ ( 1

2Id + A0[κ0])γ0U = . . . .

(Id + (A1[κ0]−A1[κ1])) u = . . . .

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

compact operator (L2

t (Γ))2 → (L2 t (Γ))2?

NO!

S1[κ1](ϕ) − S1[κ0](ϕ) =V1[κ0](ϕ) − V1[κ1](ϕ)+

1 κ2

0 ∇ΓV1[κ0](divΓϕ) − 1

κ2

1 ∇ΓV1[κ1](divΓϕ) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

slide-70
SLIDE 70

Homogeneous Scatterer: Müller Formulation

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-71
SLIDE 71

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-72
SLIDE 72

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 ,

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-73
SLIDE 73

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 , Int ➁ : (−κ2

1V1[κ1] + curlΓV1[κ1] curlΓ)(γ1 EU) + ( 1 2Id − C′ 1[κ1])(γ1 MU)

= 0 ,

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-74
SLIDE 74

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 , Int ➁ : (−κ2

1V1[κ1] + curlΓV1[κ1] curlΓ)(γ1 EU) + ( 1 2Id − C′ 1[κ1])(γ1 MU)

= 0 ,

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-75
SLIDE 75

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 , Int ➁ : (−κ2

1V1[κ1] + curlΓV1[κ1] curlΓ)(γ1 EU) + ( 1 2Id − C′ 1[κ1])(γ1 MU)

= 0 , Ext ➂ : ( 1

2Id − C1[κ0])(γ1 EU) + (V1[κ0] + κ−2 0 ∇ΓV1[κ0]divΓ)(γ1 MU)

= . . .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-76
SLIDE 76

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 , Int ➁ : (−κ2

1V1[κ1] + curlΓV1[κ1] curlΓ)(γ1 EU) + ( 1 2Id − C′ 1[κ1])(γ1 MU)

= 0 , Ext ➂ : ( 1

2Id − C1[κ0])(γ1 EU) + (V1[κ0] + κ−2 0 ∇ΓV1[κ0]divΓ)(γ1 MU)

= . . . Ext ➃ : (κ2

0V1[κ0] − curlΓV1[κ0] curlΓ)(γ1 EU) + ( 1 2Id + C′ 1[κ0])(γ1 MU)

= . . .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-77
SLIDE 77

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 , Int ➁ : (−κ2

1V1[κ1] + curlΓV1[κ1] curlΓ)(γ1 EU) + ( 1 2Id − C′ 1[κ1])(γ1 MU)

= 0 , Ext ➂ : ( 1

2Id − C1[κ0])(γ1 EU) + (V1[κ0] + κ−2 0 ∇ΓV1[κ0]divΓ)(γ1 MU)

= . . . Ext ➃ : (κ2

0V1[κ0] − curlΓV1[κ0] curlΓ)(γ1 EU) + ( 1 2Id + C′ 1[κ0])(γ1 MU)

= . . .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-78
SLIDE 78

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 , Int ➁ : (−κ2

1V1[κ1] + curlΓV1[κ1] curlΓ)(γ1 EU) + ( 1 2Id − C′ 1[κ1])(γ1 MU)

= 0 , Ext ➂ : ( 1

2Id − C1[κ0])(γ1 EU) + (V1[κ0] + κ−2 0 ∇ΓV1[κ0]divΓ)(γ1 MU)

= . . . Ext ➃ : (κ2

0V1[κ0] − curlΓV1[κ0] curlΓ)(γ1 EU) + ( 1 2Id + C′ 1[κ0])(γ1 MU)

= . . . Idea: κ2

i -weighted sum of Calderón identities

κ2

1 · ➀ + κ2 0 · ➂

, ➁ + ➃.

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-79
SLIDE 79

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 , Int ➁ : (−κ2

1V1[κ1] + curlΓV1[κ1] curlΓ)(γ1 EU) + ( 1 2Id − C′ 1[κ1])(γ1 MU)

= 0 , Ext ➂ : ( 1

2Id − C1[κ0])(γ1 EU) + (V1[κ0] + κ−2 0 ∇ΓV1[κ0]divΓ)(γ1 MU)

= . . . Ext ➃ : (κ2

0V1[κ0] − curlΓV1[κ0] curlΓ)(γ1 EU) + ( 1 2Id + C′ 1[κ0])(γ1 MU)

= . . . Idea: κ2

i -weighted sum of Calderón identities

κ2

1 · ➀ + κ2 0 · ➂

, ➁ + ➃.

  • κ2

1+κ2

2

Id Id

  • +

κ2

1C1[κ1] − κ2 0C1[κ0]

−κ2

1V1[κ1] + κ2 0V1[κ0]

−κ2

1V1[κ1] + κ2 0V1[κ0]

−κ2

1C′ 1[κ1] + κ2 0C′ 1[κ0]

γ1

EU

γ1

MU

  • = . .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-80
SLIDE 80

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 , Int ➁ : (−κ2

1V1[κ1] + curlΓV1[κ1] curlΓ)(γ1 EU) + ( 1 2Id − C′ 1[κ1])(γ1 MU)

= 0 , Ext ➂ : ( 1

2Id − C1[κ0])(γ1 EU) + (V1[κ0] + κ−2 0 ∇ΓV1[κ0]divΓ)(γ1 MU)

= . . . Ext ➃ : (κ2

0V1[κ0] − curlΓV1[κ0] curlΓ)(γ1 EU) + ( 1 2Id + C′ 1[κ0])(γ1 MU)

= . . . Idea: κ2

i -weighted sum of Calderón identities

κ2

1 · ➀ + κ2 0 · ➂

, ➁ + ➃.

  • κ2

1+κ2

2

Id Id

  • +

κ2

1C1[κ1] − κ2 0C1[κ0]

−κ2

1V1[κ1] + κ2 0V1[κ0]

−κ2

1V1[κ1] + κ2 0V1[κ0]

−κ2

1C′ 1[κ1] + κ2 0C′ 1[κ0]

γ1

EU

γ1

MU

  • = . .

Cancellation of hypersingular operators!

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-81
SLIDE 81

Homogeneous Scatterer: Müller Formulation

Details: Maxwell Calderón identities (w.r.t Ω1) Int ➀ : ( 1

2Id + C1[κ1])(γ1 EU) − (V1[κ1] + κ−2 1 ∇ΓV1[κ1]divΓ)(γ1 MU)

= 0 , Int ➁ : (−κ2

1V1[κ1] + curlΓV1[κ1] curlΓ)(γ1 EU) + ( 1 2Id − C′ 1[κ1])(γ1 MU)

= 0 , Ext ➂ : ( 1

2Id − C1[κ0])(γ1 EU) + (V1[κ0] + κ−2 0 ∇ΓV1[κ0]divΓ)(γ1 MU)

= . . . Ext ➃ : (κ2

0V1[κ0] − curlΓV1[κ0] curlΓ)(γ1 EU) + ( 1 2Id + C′ 1[κ0])(γ1 MU)

= . . . Idea: κ2

i -weighted sum of Calderón identities

κ2

1 · ➀ + κ2 0 · ➂

, ➁ + ➃. No cancellation in kernels! Fredholm, index = 0, in (L2

t (Γ))2

  • D. MITREA, M. MITREA, AND J. PIPHER, Vector potential theory
  • n nonsmooth domains in R3 and applications to

electromagnetic scattering, J. Fourier Anal. Appl., 3 (1997),

  • pp. 131–192.
  • κ2

1+κ2

2

Id Id

  • +

κ2

1C1[κ1] − κ2 0C1[κ0]

−κ2

1V1[κ1] + κ2 0V1[κ0]

−κ2

1V1[κ1] + κ2 0V1[κ0]

−κ2

1C′ 1[κ1] + κ2 0C′ 1[κ0]

γ1

EU

γ1

MU

  • = . .

Cancellation of hypersingular operators!

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

slide-82
SLIDE 82

Composite Scattering Challenge

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 11 / 29

slide-83
SLIDE 83

Composite Scattering Challenge

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 11 / 29

slide-84
SLIDE 84

Composite Scattering Challenge

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Calderón identities set in same space: A0[κ0], A1[κ1] : H(γ) → H(Γ)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 11 / 29

slide-85
SLIDE 85

Composite Scattering Challenge

Ω0 = Ω

c

Γ Ω1 uinc

n n n0 n n0

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Calderón identities set in same space: A0[κ0], A1[κ1] : H(γ) → H(Γ) Local Calderón identities in local trace spaces H(Γk)

?

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 11 / 29

slide-86
SLIDE 86

Müller Formulation: Multi-Potential Approach

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

slide-87
SLIDE 87

Müller Formulation: Multi-Potential Approach

Multi-potential representation, U solves Maxwell TP: (1) U = − D0[κ0](γ0

EU) + S0[κ0](γ0 MU)

− D1[κ1](γ1

EU) + S1[κ1](γ1 MU)

+ . . . , (2)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

slide-88
SLIDE 88

Müller Formulation: Multi-Potential Approach

Multi-potential representation, U solves Maxwell TP: (κ2

1χ1 + κ2 0χ0)U = − κ2 0D0[κ0](γ0 EU) + κ2 0S0[κ0](γ0 MU)

− κ2

1D1[κ1](γ1 EU) + κ2 1S1[κ1](γ1 MU)

+ . . . , (1) U = − D0[κ0](γ0

EU) + S0[κ0](γ0 MU)

− D1[κ1](γ1

EU) + S1[κ1](γ1 MU)

+ . . . , (2)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

slide-89
SLIDE 89

Müller Formulation: Multi-Potential Approach

Multi-potential representation, U solves Maxwell TP: (κ2

1χ1 + κ2 0χ0)U = − κ2 0D0[κ0](γ0 EU) + κ2 0S0[κ0](γ0 MU)

− κ2

1D1[κ1](γ1 EU) + κ2 1S1[κ1](γ1 MU)

+ . . . , (1) U = − D0[κ0](γ0

EU) + S0[κ0](γ0 MU)

− D1[κ1](γ1

EU) + S1[κ1](γ1 MU)

+ . . . , (2) Apply traces γ1

E, γ0 E to (1)

& γ1

M, γ0 M on (2)

[signs adjusted]

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

slide-90
SLIDE 90

Müller Formulation: Multi-Potential Approach

Multi-potential representation, U solves Maxwell TP: (κ2

1χ1 + κ2 0χ0)U = − κ2 0D0[κ0](γ0 EU) + κ2 0S0[κ0](γ0 MU)

− κ2

1D1[κ1](γ1 EU) + κ2 1S1[κ1](γ1 MU)

+ . . . , (1) U = − D0[κ0](γ0

EU) + S0[κ0](γ0 MU)

− D1[κ1](γ1

EU) + S1[κ1](γ1 MU)

+ . . . , (2) Apply traces γ1

E, γ0 E to (1)

& γ1

M, γ0 M on (2)

[signs adjusted] 1

2(κ2 0 + κ2 1)Id + κ2 0C0[κ0] + κ2 1C1[κ1]

−κ2

0S0[κ0] + κ2 1S1[κ1]

S′

0[κ0] − S′ 1[κ1]

Id − C′

0[κ0] − C′ 1[κ1]

γ1

EU

γ1

MU

  • = . . .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

slide-91
SLIDE 91

Müller Formulation: Multi-Potential Approach

Multi-potential representation, U solves Maxwell TP: (κ2

1χ1 + κ2 0χ0)U = − κ2 0D0[κ0](γ0 EU) + κ2 0S0[κ0](γ0 MU)

− κ2

1D1[κ1](γ1 EU) + κ2 1S1[κ1](γ1 MU)

+ . . . , (1) U = − D0[κ0](γ0

EU) + S0[κ0](γ0 MU)

− D1[κ1](γ1

EU) + S1[κ1](γ1 MU)

+ . . . , (2) Apply traces γ1

E, γ0 E to (1)

& γ1

M, γ0 M on (2)

[signs adjusted] 1

2(κ2 0 + κ2 1)Id + κ2 0C0[κ0] + κ2 1C1[κ1]

−κ2

0S0[κ0] + κ2 1S1[κ1]

S′

0[κ0] − S′ 1[κ1]

Id − C′

0[κ0] − C′ 1[κ1]

γ1

EU

γ1

MU

  • = . . .
  • κ2

1+κ2

2

Id Id

  • +

κ2

1C1[κ1] − κ2 0C1[κ0]

−κ2

1V1[κ1] + κ2 0V1[κ0]

−κ2

1V1[κ1] + κ2 0V1[κ0]

−κ2

1C′ 1[κ1] + κ2 0C′ 1[κ0]

γ1

EU

γ1

MU

  • = . . .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

slide-92
SLIDE 92

What Next ?

1

Scattering at Composite Objects

2

BIE: Single Subdomain Setting

3

BIE: Composite Scatterer

4

Numerical Experiments

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 13 / 29

slide-93
SLIDE 93

Composite EM Scattering

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 14 / 29

slide-94
SLIDE 94

Composite EM Scattering

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

curl curl U − κ(x)2U = 0 , κ(x) = κi > 0 in Ωi .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 14 / 29

slide-95
SLIDE 95

Composite EM Scattering

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

curl curl U − κ(x)2U = 0 , κ(x) = κi > 0 in Ωi . Boundaries Γk := ∂Ωk Interfaces Γkℓ = ∂Ωk ∩ ∂Ωℓ Skeleton: Σ :=

  • k

∂Ωk =

  • k=ℓ

Γkℓ .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 14 / 29

slide-96
SLIDE 96

Skeleton Trace Spaces

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-97
SLIDE 97

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-98
SLIDE 98

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Multi-trace space (electric/magnetic): MT (Σ) := H(Γ0) × · · · × H(ΓN)∼ =MTE(Σ) × MTM(Σ) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-99
SLIDE 99

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Multi-trace space (electric/magnetic): MT (Σ) := H(Γl) = HE(Γℓ) × HM(Γℓ) H(Γ0) × · · · × H(ΓN)∼ =MTE(Σ) × MTM(Σ) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-100
SLIDE 100

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Multi-trace space (electric/magnetic): MT (Σ) := H(Γ0) × · · · × H(ΓN)∼ =MTE(Σ) × MTM(Σ) . Multi-trace operator: γΣ :

ℓ Hloc(curl2, Ωℓ) → MT (Σ),

γΣ := γ0 × · · · × γN ∼ = γΣ

E × γΣ M

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-101
SLIDE 101

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Multi-trace space (electric/magnetic): MT (Σ) := H(Γ0) × · · · × H(ΓN)∼ =MTE(Σ) × MTM(Σ) . Multi-trace operator: γΣ :

ℓ Hloc(curl2, Ωℓ) → MT (Σ),

γΣ := γ0 × · · · × γN ∼ = γΣ

E × γΣ M

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-102
SLIDE 102

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Multi-trace space (electric/magnetic): MT (Σ) := H(Γ0) × · · · × H(ΓN)∼ =MTE(Σ) × MTM(Σ) . Multi-trace operator: γΣ :

ℓ Hloc(curl2, Ωℓ) → MT (Σ),

γΣ := γ0 × · · · × γN ∼ = γΣ

E × γΣ M

Single-trace space (“traces satisfying transmission conditions”)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-103
SLIDE 103

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Multi-trace space (electric/magnetic): MT (Σ) := H(Γ0) × · · · × H(ΓN)∼ =MTE(Σ) × MTM(Σ) . Multi-trace operator: γΣ :

ℓ Hloc(curl2, Ωℓ) → MT (Σ),

γΣ := γ0 × · · · × γN ∼ = γΣ

E × γΣ M

Single-trace space (“traces satisfying transmission conditions”) ST (Σ) := u0 ϕ0

  • , . . . ,

uN ϕN

  • : ∃U ∈ Hloc(curl, Rd) : uℓ = γℓ

EU

∃V ∈ Hloc(curl2, Rd) : ϕℓ = γℓ

MV

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-104
SLIDE 104

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Multi-trace space (electric/magnetic): MT (Σ) := H(Γ0) × · · · × H(ΓN)∼ =MTE(Σ) × MTM(Σ) . Multi-trace operator: γΣ :

ℓ Hloc(curl2, Ωℓ) → MT (Σ),

γΣ := γ0 × · · · × γN ∼ = γΣ

E × γΣ M

Single-trace space (“traces satisfying transmission conditions”) ST (Σ) := u0 ϕ0

  • , . . . ,

uN ϕN

  • : ∃U ∈ Hloc(curl, Rd) : uℓ = γℓ

EU

∃V ∈ Hloc(curl2, Rd) : ϕℓ = γℓ

MV

  • ⊂MT (Σ)
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-105
SLIDE 105

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Multi-trace space (electric/magnetic): MT (Σ) := H(Γ0) × · · · × H(ΓN)∼ =MTE(Σ) × MTM(Σ) . Multi-trace operator: γΣ :

ℓ Hloc(curl2, Ωℓ) → MT (Σ),

γΣ := γ0 × · · · × γN ∼ = γΣ

E × γΣ M

Single-trace space (“traces satisfying transmission conditions”) ST (Σ) := u0 ϕ0

  • , . . . ,

uN ϕN

  • : ∃U ∈ Hloc(curl, Rd) : uℓ = γℓ

EU

∃V ∈ Hloc(curl2, Rd) : ϕℓ = γℓ

MV

  • ⊂MT (Σ)

Pairing:

  • u0

ϕ0

  • , . . . ,

uN ϕN

  • ,

v0 ψ0

  • , . . . ,

vN ψN

  • :=

N

  • ℓ=0

uℓ, ψℓℓ − vℓ, ϕℓℓ

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-106
SLIDE 106

Skeleton Trace Spaces

Ω0

Γ

Ω1 Ω2 Ω3 Uinc

n1 n3 n1 n0 n1 n0 n3 n2 n2 n0

Multi-trace space (electric/magnetic): MT (Σ) := H(Γ0) × · · · × H(ΓN)∼ =MTE(Σ) × MTM(Σ) . Multi-trace operator: γΣ :

ℓ Hloc(curl2, Ωℓ) → MT (Σ),

γΣ := γ0 × · · · × γN ∼ = γΣ

E × γΣ M

Single-trace space (“traces satisfying transmission conditions”) ST (Σ) := u0 ϕ0

  • , . . . ,

uN ϕN

  • : ∃U ∈ Hloc(curl, Rd) : uℓ = γℓ

EU

∃V ∈ Hloc(curl2, Rd) : ϕℓ = γℓ

MV

  • ⊂MT (Σ)

Pairing:

  • u0

ϕ0

  • , . . . ,

uN ϕN

  • ,

v0 ψ0

  • , . . . ,

vN ψN

  • :=

N

  • ℓ=0

uℓ, ψℓℓ − vℓ, ϕℓℓ ✎ ✍ ☞ ✌ Polarity:

  • u ∈ MT (Σ):
  • u ∈ ST (Σ)

⇐ ⇒

  • u,

v

  • = 0

∀ v ∈ ST (Σ) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

slide-107
SLIDE 107

Multi-Potential Representation

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-108
SLIDE 108

Multi-Potential Representation

U = solution of Maxwell transmission problem:

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-109
SLIDE 109

Multi-Potential Representation

U = solution of Maxwell transmission problem: U − χ0Uinc = N

k=0 Gk[κk]

  • γk(U − χ0Uinc)
  • .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-110
SLIDE 110

Multi-Potential Representation

U = solution of Maxwell transmission problem: N

k=0 κ2 kχkU − κ2 0χ0Uinc =

N

k=0 κ2 kGk[κk]

  • γk(U − χ0Uinc)
  • ,

U − χ0Uinc = N

k=0 Gk[κk]

  • γk(U − χ0Uinc)
  • .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-111
SLIDE 111

Multi-Potential Representation

U = solution of Maxwell transmission problem: N

k=0 κ2 kχkU − κ2 0χ0Uinc =

N

k=0 κ2 kGk[κk]

  • γk(U − χ0Uinc)
  • ,

U − χ0Uinc = N

k=0 Gk[κk]

  • γk(U − χ0Uinc)
  • .

Superposition of multiple potentials everywhere in R3 \ Σ

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-112
SLIDE 112

Multi-Potential Representation

U = solution of Maxwell transmission problem: N

k=0 κ2 kχkU − κ2 0χ0Uinc =

N

k=0 κ2 kGk[κk]

  • γk(U − χ0Uinc)
  • ,

U − χ0Uinc = N

k=0 Gk[κk]

  • γk(U − χ0Uinc)
  • .

Skeleton compound integral operator M : MT (Σ) → MT (Σ)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-113
SLIDE 113

Multi-Potential Representation

U = solution of Maxwell transmission problem: N

k=0 κ2 kχkU − κ2 0χ0Uinc =

N

k=0 κ2 kGk[κk]

  • γk(U − χ0Uinc)
  • ,

U − χ0Uinc = N

k=0 Gk[κk]

  • γk(U − χ0Uinc)
  • .

Skeleton compound integral operator M : MT (Σ) → MT (Σ) Mu :=  γΣ

E

N

k=0 κ2 kGk[κk](uk)

γΣ

M

N

k=0 Gk[κk](uk)

  , u = (u0, . . . , uN) ∈ MT (Σ) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-114
SLIDE 114

Multi-Potential Representation

U = solution of Maxwell transmission problem: N

k=0 κ2 kχkU − κ2 0χ0Uinc =

N

k=0 κ2 kGk[κk]

  • γk(U − χ0Uinc)
  • ,

U − χ0Uinc = N

k=0 Gk[κk]

  • γk(U − χ0Uinc)
  • .

Skeleton compound integral operator M : MT (Σ) → MT (Σ) Mu :=  γΣ

E

N

k=0 κ2 kGk[κk](uk)

γΣ

M

N

k=0 Gk[κk](uk)

  , u = (u0, . . . , uN) ∈ MT (Σ) .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-115
SLIDE 115

Multi-Potential Representation

U = solution of Maxwell transmission problem: N

k=0 κ2 kχkU − κ2 0χ0Uinc =

N

k=0 κ2 kGk[κk]

  • γk(U − χ0Uinc)
  • ,

U − χ0Uinc = N

k=0 Gk[κk]

  • γk(U − χ0Uinc)
  • .

Skeleton compound integral operator M : MT (Σ) → MT (Σ) Mu :=  γΣ

E

N

k=0 κ2 kGk[κk](uk)

γΣ

M

N

k=0 Gk[κk](uk)

  , u = (u0, . . . , uN) ∈ MT (Σ) . ✬ ✫ ✩ ✪ γΣU solves skeleton boundary integral equation: (Dκ − M) γΣU = (Dκ − M) (γ0Uinc, 0, . . . , 0) = Dκ0γΣUinc .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-116
SLIDE 116

Multi-Potential Representation

U = solution of Maxwell transmission problem: N

k=0 κ2 kχkU − κ2 0χ0Uinc =

N

k=0 κ2 kGk[κk]

  • γk(U − χ0Uinc)
  • ,

U − χ0Uinc = N

k=0 Gk[κk]

  • γk(U − χ0Uinc)
  • .

Skeleton compound integral operator M : MT (Σ) → MT (Σ) Mu :=  γΣ

E

N

k=0 κ2 kGk[κk](uk)

γΣ

M

N

k=0 Gk[κk](uk)

  , u = (u0, . . . , uN) ∈ MT (Σ) . ✬ ✫ ✩ ✪ γΣU solves skeleton boundary integral equation: (Dκ − M) γΣU = (Dκ − M) (γ0Uinc, 0, . . . , 0) = Dκ0γΣUinc .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-117
SLIDE 117

Multi-Potential Representation

U = solution of Maxwell transmission problem: N

k=0 κ2 kχkU − κ2 0χ0Uinc =

N

k=0 κ2 kGk[κk]

  • γk(U − χ0Uinc)
  • ,

U − χ0Uinc = N

k=0 Gk[κk]

  • γk(U − χ0Uinc)
  • .

Skeleton compound integral operator M : MT (Σ) → MT (Σ) Mu :=  γΣ

E

N

k=0 κ2 kGk[κk](uk)

γΣ

M

N

k=0 Gk[κk](uk)

  , u = (u0, . . . , uN) ∈ MT (Σ) . ✬ ✫ ✩ ✪ γΣU solves skeleton boundary integral equation: (Dκ − M) γΣU = (Dκ − M) (γ0Uinc, 0, . . . , 0) = Dκ0γΣUinc . [Dκ u0

µ0

  • , . . . ,

uN

µN

  • :=

κ2

0u0

µ0

  • , . . . ,

κ2

NuN

µN

  • ]
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 16 / 29

slide-118
SLIDE 118

Regularization of Skeleton BIE

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 17 / 29

slide-119
SLIDE 119

Regularization of Skeleton BIE

✗ ✖ ✔ ✕ Vanishing multi-potential: ϑ > 0 fixed ➣ N

k=0 Gk[ϑ](uk) = 0

∀u = (u0, . . . , uN) ∈ ST (Σ)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 17 / 29

slide-120
SLIDE 120

Regularization of Skeleton BIE

✗ ✖ ✔ ✕ Vanishing multi-potential: ϑ > 0 fixed ➣ N

k=0 Gk[ϑ](uk) = 0

∀u = (u0, . . . , uN) ∈ ST (Σ) M∗(u) =  γΣ

E

N

k=0 ϑ2Gk[ϑ](uk)

γΣ

E

N

k=0 Gk[ϑ](uk)

  = 0 ∀u ∈ ST (Σ) ,

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 17 / 29

slide-121
SLIDE 121

Regularization of Skeleton BIE

✗ ✖ ✔ ✕ Vanishing multi-potential: ϑ > 0 fixed ➣ N

k=0 Gk[ϑ](uk) = 0

∀u = (u0, . . . , uN) ∈ ST (Σ) M∗(u) =  γΣ

E

N

k=0 ϑ2Gk[ϑ](uk)

γΣ

E

N

k=0 Gk[ϑ](uk)

  = 0 ∀u ∈ ST (Σ) ,

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 17 / 29

slide-122
SLIDE 122

Regularization of Skeleton BIE

✗ ✖ ✔ ✕ Vanishing multi-potential: ϑ > 0 fixed ➣ N

k=0 Gk[ϑ](uk) = 0

∀u = (u0, . . . , uN) ∈ ST (Σ) M∗(u) =  γΣ

E

N

k=0 ϑ2Gk[ϑ](uk)

γΣ

E

N

k=0 Gk[ϑ](uk)

  = 0 ∀u ∈ ST (Σ) , (Dκ − (M − M∗)) γΣU = Dκ0γΣUinc .

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 17 / 29

slide-123
SLIDE 123

Regularization of Skeleton BIE

✗ ✖ ✔ ✕ Vanishing multi-potential: ϑ > 0 fixed ➣ N

k=0 Gk[ϑ](uk) = 0

∀u = (u0, . . . , uN) ∈ ST (Σ) M∗(u) =  γΣ

E

N

k=0 ϑ2Gk[ϑ](uk)

γΣ

E

N

k=0 Gk[ϑ](uk)

  = 0 ∀u ∈ ST (Σ) , (Dκ − (M − M∗)) γΣU = Dκ0γΣUinc . Variational skeleton BIE: seek u ∈ ST (Σ)

  • (Dκ − (M − M∗)) u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ MT (Σ) .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 17 / 29

slide-124
SLIDE 124

Regularization of Skeleton BIE

✗ ✖ ✔ ✕ Vanishing multi-potential: ϑ > 0 fixed ➣ N

k=0 Gk[ϑ](uk) = 0

∀u = (u0, . . . , uN) ∈ ST (Σ) M∗(u) =  γΣ

E

N

k=0 ϑ2Gk[ϑ](uk)

γΣ

E

N

k=0 Gk[ϑ](uk)

  = 0 ∀u ∈ ST (Σ) , (Dκ − (M − M∗)) γΣU = Dκ0γΣUinc . Variational skeleton BIE: seek u ∈ ST (Σ)

  • (Dκ − (M − M∗)) u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ MT (Σ) .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 17 / 29

slide-125
SLIDE 125

Regularization of Skeleton BIE

✗ ✖ ✔ ✕ Vanishing multi-potential: ϑ > 0 fixed ➣ N

k=0 Gk[ϑ](uk) = 0

∀u = (u0, . . . , uN) ∈ ST (Σ) M∗(u) =  γΣ

E

N

k=0 ϑ2Gk[ϑ](uk)

γΣ

E

N

k=0 Gk[ϑ](uk)

  = 0 ∀u ∈ ST (Σ) , (Dκ − (M − M∗)) γΣU = Dκ0γΣUinc . Variational skeleton BIE: seek u ∈ ST (Σ)

  • (Dκ − (M − M∗)) u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ MT (Σ) .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 17 / 29

slide-126
SLIDE 126

Regularized Skeleton BIE

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-127
SLIDE 127

Regularized Skeleton BIE

For u = u0

ν0

  • , . . . ,

uN

νN

  • ∈ ST (Σ)

(M − M∗)(u)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-128
SLIDE 128

Regularized Skeleton BIE

For u = u0

ν0

  • , . . . ,

uN

νN

  • ∈ ST (Σ)

(M − M∗)(u) =  γΣ

E

N

k=0

  • κ2

kGk[κk] − ϑ2Gk[ϑ]

  • (uk)

γΣ

M

N

k=0 (Gk[κk] − Gk[ϑ]) (uk)

 

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-129
SLIDE 129

Regularized Skeleton BIE

For u = u0

ν0

  • , . . . ,

uN

νN

  • ∈ ST (Σ)

(M − M∗)(u) =  γΣ

E

N

k=0

  • κ2

kGk[κk] − ϑ2Gk[ϑ]

  • (uk)

γΣ

M

N

k=0 (Gk[κk] − Gk[ϑ]) (uk)

 

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-130
SLIDE 130

Regularized Skeleton BIE

For u = u0

ν0

  • , . . . ,

uN

νN

  • ∈ ST (Σ)

(M − M∗)(u) =  γΣ

E

N

k=0

  • κ2

kGk[κk] − ϑ2Gk[ϑ]

  • (uk)

γΣ

M

N

k=0 (Gk[κk] − Gk[ϑ]) (uk)

  =              γj

E

N

  • k=0

( κ2

kDk[κk] + ϑ2Di[ϑ])(uk) + (κ2 kVk[κk] − ϑ2Vi[ϑ])(νk)

+ ∇(Vk[κk] − Vk[ϑ])(divΓνk)

  • γj

×

N

  • k=0

∇(Vi[κi] − Vi[ϑ])(curlΓ ui) + (κ2

i Vi[κi] − ϑ2Vi[ϑ])(ui × ni) +

curl(Vi[κi] − Vi[ϑ])( νi)

           

N j=0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-131
SLIDE 131

Regularized Skeleton BIE

  • C0-kernels

For u = u0

ν0

  • , . . . ,

uN

νN

  • ∈ ST (Σ)

(M − M∗)(u) =  γΣ

E

N

k=0

  • κ2

kGk[κk] − ϑ2Gk[ϑ]

  • (uk)

γΣ

M

N

k=0 (Gk[κk] − Gk[ϑ]) (uk)

  =              γj

E

N

  • k=0

( κ2

kDk[κk] + ϑ2Di[ϑ])(uk) + (κ2 kVk[κk] − ϑ2Vi[ϑ])(νk)

+ ∇(Vk[κk] − Vk[ϑ])(divΓνk)

  • γj

×

N

  • k=0

∇(Vi[κi] − Vi[ϑ])(curlΓ ui) + (κ2

i Vi[κi] − ϑ2Vi[ϑ])(ui × ni) +

curl(Vi[κi] − Vi[ϑ])( νi)

           

N j=0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-132
SLIDE 132

Regularized Skeleton BIE

  • smoothing

For u = u0

ν0

  • , . . . ,

uN

νN

  • ∈ ST (Σ)

(M − M∗)(u) =  γΣ

E

N

k=0

  • κ2

kGk[κk] − ϑ2Gk[ϑ]

  • (uk)

γΣ

M

N

k=0 (Gk[κk] − Gk[ϑ]) (uk)

  =              γj

E

N

  • k=0

( κ2

kDk[κk] + ϑ2Di[ϑ])(uk) + (κ2 kVk[κk] − ϑ2Vi[ϑ])(νk)

+ ∇(Vk[κk] − Vk[ϑ])(divΓνk)

  • γj

×

N

  • k=0

∇(Vi[κi] − Vi[ϑ])(curlΓ ui) + (κ2

i Vi[κi] − ϑ2Vi[ϑ])(ui × ni) +

curl(Vi[κi] − Vi[ϑ])( νi)

           

N j=0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-133
SLIDE 133

Regularized Skeleton BIE

  • Cauchy singular

For u = u0

ν0

  • , . . . ,

uN

νN

  • ∈ ST (Σ)

(M − M∗)(u) =  γΣ

E

N

k=0

  • κ2

kGk[κk] − ϑ2Gk[ϑ]

  • (uk)

γΣ

M

N

k=0 (Gk[κk] − Gk[ϑ]) (uk)

  =              γj

E

N

  • k=0

( κ2

kDk[κk] + ϑ2Di[ϑ])(uk) + (κ2 kVk[κk] − ϑ2Vi[ϑ])(νk)

+ ∇(Vk[κk] − Vk[ϑ])(divΓνk)

  • γj

×

N

  • k=0

∇(Vi[κi] − Vi[ϑ])(curlΓ ui) + (κ2

i Vi[κi] − ϑ2Vi[ϑ])(ui × ni) +

curl(Vi[κi] − Vi[ϑ])( νi)

           

N j=0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-134
SLIDE 134

Regularized Skeleton BIE

  • C0-kernels
  • smoothing
  • Cauchy singular

For u = u0

ν0

  • , . . . ,

uN

νN

  • ∈ ST (Σ)

(M − M∗)(u) =  γΣ

E

N

k=0

  • κ2

kGk[κk] − ϑ2Gk[ϑ]

  • (uk)

γΣ

M

N

k=0 (Gk[κk] − Gk[ϑ]) (uk)

  =              γj

E

N

  • k=0

( κ2

kDk[κk] + ϑ2Di[ϑ])(uk) + (κ2 kVk[κk] − ϑ2Vi[ϑ])(νk)

+ ∇(Vk[κk] − Vk[ϑ])(divΓνk)

  • γj

×

N

  • k=0

∇(Vi[κi] − Vi[ϑ])(curlΓ ui) + (κ2

i Vi[κi] − ϑ2Vi[ϑ])(ui × ni) +

curl(Vi[κi] − Vi[ϑ])( νi)

           

N j=0

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-135
SLIDE 135

Regularized Skeleton BIE

  • C0-kernels
  • smoothing
  • Cauchy singular

For u = u0

ν0

  • , . . . ,

uN

νN

  • ∈ ST (Σ)

(M − M∗)(u) =  γΣ

E

N

k=0

  • κ2

kGk[κk] − ϑ2Gk[ϑ]

  • (uk)

γΣ

M

N

k=0 (Gk[κk] − Gk[ϑ]) (uk)

  =              γj

E

N

  • k=0

( κ2

kDk[κk] + ϑ2Di[ϑ])(uk) + (κ2 kVk[κk] − ϑ2Vi[ϑ])(νk)

+ ∇(Vk[κk] − Vk[ϑ])(divΓνk)

  • γj

×

N

  • k=0

∇(Vi[κi] − Vi[ϑ])(curlΓ ui) + (κ2

i Vi[κi] − ϑ2Vi[ϑ])(ui × ni) +

curl(Vi[κi] − Vi[ϑ])( νi)

           

N j=0

Hypersingular BI-Ops. gone!

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 18 / 29

slide-136
SLIDE 136

Skeleton BIE: Lifting into L2

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 19 / 29

slide-137
SLIDE 137

Skeleton BIE: Lifting into L2

L2 tangential multi-trace/single-trace spaces:

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 19 / 29

slide-138
SLIDE 138

Skeleton BIE: Lifting into L2

L2 tangential multi-trace/single-trace spaces: ML(Σ) :=

N

  • i=0
  • L2

t (Γi) × L2 t (Γi)

  • ∼

=

  • i,j

L2

t (Γij)

  • ×
  • i,j

L2

t (Γij)

  • ,
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 19 / 29

slide-139
SLIDE 139

Skeleton BIE: Lifting into L2

L2 tangential multi-trace/single-trace spaces: ML(Σ) :=

N

  • i=0
  • L2

t (Γi) × L2 t (Γi)

  • ∼

=

  • i,j

L2

t (Γij)

  • ×
  • i,j

L2

t (Γij)

  • ,

SL(Σ) := u0 ν0

  • , . . . ,

uL νN

  • ∈ ML(Σ)
  • uj |Γij = ui |Γij , νj |Γij = − νi |Γij ∀j < i ∈ {0, . . . , N}
  • .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 19 / 29

slide-140
SLIDE 140

Skeleton BIE: Lifting into L2

L2 tangential multi-trace/single-trace spaces: ML(Σ) :=

N

  • i=0
  • L2

t (Γi) × L2 t (Γi)

  • ∼

=

  • i,j

L2

t (Γij)

  • ×
  • i,j

L2

t (Γij)

  • ,

SL(Σ) := u0 ν0

  • , . . . ,

uL νN

  • ∈ ML(Σ)
  • uj |Γij = ui |Γij , νj |Γij = − νi |Γij ∀j < i ∈ {0, . . . , N}
  • .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 19 / 29

slide-141
SLIDE 141

Skeleton BIE: Lifting into L2

L2 tangential multi-trace/single-trace spaces: ML(Σ) :=

N

  • i=0
  • L2

t (Γi) × L2 t (Γi)

  • ∼

=

  • i,j

L2

t (Γij)

  • ×
  • i,j

L2

t (Γij)

  • ,

SL(Σ) := u0 ν0

  • , . . . ,

uL νN

  • ∈ ML(Σ)
  • uj |Γij = ui |Γij , νj |Γij = − νi |Γij ∀j < i ∈ {0, . . . , N}
  • .

☛ ✡ ✟ ✠ M − M∗ : ML(Σ) → ML(Σ) continuous

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 19 / 29

slide-142
SLIDE 142

Skeleton BIE: Lifting into L2

L2 tangential multi-trace/single-trace spaces: ML(Σ) :=

N

  • i=0
  • L2

t (Γi) × L2 t (Γi)

  • ∼

=

  • i,j

L2

t (Γij)

  • ×
  • i,j

L2

t (Γij)

  • ,

SL(Σ) := u0 ν0

  • , . . . ,

uL νN

  • ∈ ML(Σ)
  • uj |Γij = ui |Γij , νj |Γij = − νi |Γij ∀j < i ∈ {0, . . . , N}
  • .

☛ ✡ ✟ ✠ M − M∗ : ML(Σ) → ML(Σ) continuous

?

Dκ − (M − M∗) : SL(Σ) → SL(Σ) Fredholm operator?

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 19 / 29

slide-143
SLIDE 143

Trace Complement Testing

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-144
SLIDE 144

Trace Complement Testing

Variational 2nd-kind skeleton BIE: seek u ∈ SL(Σ)

  • (Dκ − (M − M∗))u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ ML(Σ)
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-145
SLIDE 145

Trace Complement Testing

Variational 2nd-kind skeleton BIE: seek u ∈ SL(Σ)

  • (Dκ − (M − M∗))u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ ML(Σ)
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-146
SLIDE 146

Trace Complement Testing

Variational 2nd-kind skeleton BIE: seek u ∈ SL(Σ)

  • (Dκ − (M − M∗))u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ ML(Σ)

☛ ✡ ✟ ✠

  • (Dκ − (M − M∗))u, v

= 0 ∀u, v ∈ SL(Σ)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-147
SLIDE 147

Trace Complement Testing

Variational 2nd-kind skeleton BIE: seek u ∈ SL(Σ)

  • (Dκ − (M − M∗))u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ SL⊥(Σ)

☛ ✡ ✟ ✠

  • (Dκ − (M − M∗))u, v

= 0 ∀u, v ∈ SL(Σ) Complement test space SL⊥(Σ): ML(Σ) = SL(Σ) ⊕ SL⊥(Σ)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-148
SLIDE 148

Trace Complement Testing

Variational 2nd-kind skeleton BIE: seek u ∈ SL(Σ)

  • (Dκ − (M − M∗))u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ SL⊥(Σ)

☛ ✡ ✟ ✠

  • (Dκ − (M − M∗))u, v

= 0 ∀u, v ∈ SL(Σ) Complement test space SL⊥(Σ): ML(Σ) = SL(Σ) ⊕ SL⊥(Σ) “Signed copy” isomorphisms: J : (L2

t (Σ))2 → SL(Σ), J⊥ : (L2 t (Σ))2 → SL⊥(Σ)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-149
SLIDE 149

Trace Complement Testing

Variational 2nd-kind skeleton BIE: seek u ∈ SL(Σ)

  • (Dκ − (M − M∗))u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ SL⊥(Σ)

☛ ✡ ✟ ✠

  • (Dκ − (M − M∗))u, v

= 0 ∀u, v ∈ SL(Σ) Complement test space SL⊥(Σ): ML(Σ) = SL(Σ) ⊕ SL⊥(Σ) “Signed copy” isomorphisms: J : (L2

t (Σ))2 → SL(Σ), J⊥ : (L2 t (Σ))2 → SL⊥(Σ)

Ω0

Σ Ω1 Ω+ Ω− Γ

nΓ

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-150
SLIDE 150

Trace Complement Testing

Variational 2nd-kind skeleton BIE: seek u ∈ SL(Σ)

  • (Dκ − (M − M∗))u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ SL⊥(Σ)

☛ ✡ ✟ ✠

  • (Dκ − (M − M∗))u, v

= 0 ∀u, v ∈ SL(Σ) Complement test space SL⊥(Σ): ML(Σ) = SL(Σ) ⊕ SL⊥(Σ) “Signed copy” isomorphisms: J : (L2

t (Σ))2 → SL(Σ), J⊥ : (L2 t (Σ))2 → SL⊥(Σ)

SL(Σ) by distributing values through J

Ω+ Ω− u + + Ω+ Ω− ϕ − +

SL⊥(Σ) by distributing values through J⊥

Ω+ Ω− u − + Ω+ Ω− ϕ + +

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-151
SLIDE 151

Trace Complement Testing

Variational 2nd-kind skeleton BIE: seek u ∈ SL(Σ)

  • (Dκ − (M − M∗))u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ SL⊥(Σ)

☛ ✡ ✟ ✠

  • (Dκ − (M − M∗))u, v

= 0 ∀u, v ∈ SL(Σ) Complement test space SL⊥(Σ): ML(Σ) = SL(Σ) ⊕ SL⊥(Σ) “Signed copy” isomorphisms: J : (L2

t (Σ))2 → SL(Σ), J⊥ : (L2 t (Σ))2 → SL⊥(Σ)

Variational 2nd-kind skeleton BIE on tangential skeleton fields: seek u ∈ (L2

t (Σ))2

  • (Dκ − (M − M∗))Ju, J⊥v
  • =
  • Dκ0γΣUinc, J⊥v
  • v ∈ (L2

t (Σ))2 .

➤ interface-oriented reformulation possible in (L2

t (Σ))2

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-152
SLIDE 152

Trace Complement Testing

Variational 2nd-kind skeleton BIE: seek u ∈ SL(Σ)

  • (Dκ − (M − M∗))u, v

=

  • Dκ0γΣUinc, v
  • ∀v ∈ SL⊥(Σ)

☛ ✡ ✟ ✠

  • (Dκ − (M − M∗))u, v

= 0 ∀u, v ∈ SL(Σ) Complement test space SL⊥(Σ): ML(Σ) = SL(Σ) ⊕ SL⊥(Σ) “Signed copy” isomorphisms: J : (L2

t (Σ))2 → SL(Σ), J⊥ : (L2 t (Σ))2 → SL⊥(Σ)

Variational 2nd-kind skeleton BIE on tangential skeleton fields: seek u ∈ (L2

t (Σ))2

  • (Dκ − (M − M∗))Ju, J⊥v
  • =
  • Dκ0γΣUinc, J⊥v
  • v ∈ (L2

t (Σ))2 .

➤ interface-oriented reformulation possible in (L2

t (Σ))2

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 20 / 29

slide-153
SLIDE 153

What Next ?

1

Scattering at Composite Objects

2

BIE: Single Subdomain Setting

3

BIE: Composite Scatterer

4

Numerical Experiments

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 21 / 29

slide-154
SLIDE 154

Galerkin Boundary Element Discretization

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 22 / 29

slide-155
SLIDE 155

Galerkin Boundary Element Discretization

Σh ˆ = Skeleton mesh, compatible with interfaces

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 22 / 29

slide-156
SLIDE 156

Galerkin Boundary Element Discretization

Σh ˆ = Skeleton mesh, compatible with interfaces Th ⊂ L2

t (Σ)

ˆ = Boundary element space of piecewise constant tangential vectorfields on Σh

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 22 / 29

slide-157
SLIDE 157

Galerkin Boundary Element Discretization

Σh ˆ = Skeleton mesh, compatible with interfaces Th ⊂ L2

t (Σ)

ˆ = Boundary element space of piecewise constant tangential vectorfields on Σh Note: Lifting to SL(Σ) allows use of discontinuous BE for Galerkin discretization!

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 22 / 29

slide-158
SLIDE 158

Galerkin Boundary Element Discretization

Σh ˆ = Skeleton mesh, compatible with interfaces Th ⊂ L2

t (Σ)

ˆ = Boundary element space of piecewise constant tangential vectorfields on Σh Note: Lifting to SL(Σ) allows use of discontinuous BE for Galerkin discretization! Discrete variational 2nd-kind skeleton BIE: seek uh ∈ Th × Th

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 22 / 29

slide-159
SLIDE 159

Galerkin Boundary Element Discretization

Σh ˆ = Skeleton mesh, compatible with interfaces Th ⊂ L2

t (Σ)

ˆ = Boundary element space of piecewise constant tangential vectorfields on Σh Note: Lifting to SL(Σ) allows use of discontinuous BE for Galerkin discretization! Discrete variational 2nd-kind skeleton BIE: seek uh ∈ Th × Th

  • (Dκ − (M − M∗))Juh, J⊥vh
  • =
  • Dκ0γΣUinc, J⊥vh
  • vh ∈ Th × Th .
  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 22 / 29

slide-160
SLIDE 160

Galerkin Boundary Element Discretization

Σh ˆ = Skeleton mesh, compatible with interfaces Th ⊂ L2

t (Σ)

ˆ = Boundary element space of piecewise constant tangential vectorfields on Σh Note: Lifting to SL(Σ) allows use of discontinuous BE for Galerkin discretization! Discrete variational 2nd-kind skeleton BIE: seek uh ∈ Th × Th

  • (Dκ − (M − M∗))Juh, J⊥vh
  • =
  • Dκ0γΣUinc, J⊥vh
  • vh ∈ Th × Th .

If stable & orthonormal basis of Th ➣ h-uniformly well-conditioned linear systems.

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 22 / 29

slide-161
SLIDE 161

Galerkin Boundary Element Discretization

Σh ˆ = Skeleton mesh, compatible with interfaces Th ⊂ L2

t (Σ)

ˆ = Boundary element space of piecewise constant tangential vectorfields on Σh Note: Lifting to SL(Σ) allows use of discontinuous BE for Galerkin discretization! Discrete variational 2nd-kind skeleton BIE: seek uh ∈ Th × Th

  • (Dκ − (M − M∗))Juh, J⊥vh
  • =
  • Dκ0γΣUinc, J⊥vh
  • vh ∈ Th × Th .

If stable & orthonormal basis of Th ➣ h-uniformly well-conditioned linear systems. ?

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 22 / 29

slide-162
SLIDE 162

Experiment I

◮ Two hemispheres, (κ0, κ1, κ2) = (2, 3, 1), ◮ Σh ˆ

= conforming, piecewise flat triangulation

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 23 / 28

slide-163
SLIDE 163

Experiment II

◮ Two hemispheres plus box, (κ0, κ1, κ2) = (2, 3, 1, 4), ◮ Σh ˆ

= conforming, piecewise flat triangulation

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 24 / 28

slide-164
SLIDE 164

h-Convergence

100 101 102 10−3 10−2 10−1 100 101

−0.7 −1.2 −1.1 −1.8

h−1

M

Error in L2

t(Σ) and H −1

2

t (Σ)-norm

Experiment I 100 101 102 10−2 10−1 100 101 102

−0.6 −0.7 −0.8 −0.6 −1.0 −1.2

h−1

M

Error in L2

t(Σ) and H −1

2

t (Σ)-norm

Experiment II

  • γΣ

EU

  • L2(Σ)
  • γΣ

MU

  • L2(Σ)
  • γΣ

EU

  • H− 1

2 ([)Σ]

  • γΣ

MU

  • H− 1

2 ([)Σ]

2nd-kind BIE-BEM 1st-kind PMCHWT

[“Exact”, 2nd-kind on refined mesh]

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 25 / 28

slide-165
SLIDE 165

Conditioning

101 102 101 102 103 104 105 106

  • 0.03

3.33

h−1

M

Euclidean condition number of GM Experiment I first-kind second-kind 101 102 101 102 103 104 105

2.50 0.20

h−1

M

Euclidean condition number of GM Experiment II first kind second kind

Euclidean condition numbers of Galerkin matrices (diagonally scaled)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 26 / 28

slide-166
SLIDE 166

GMRES Convergence

200 400 600 800 1000 1200 1400 1600 1800 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 number of iterations 2-norm of residual Experiment I first kind, |TM| = 44 second kind, |TM| = 44 first kind, |TM| = 176 second kind, |TM| = 176 first kind, |TM| = 704 second kind, |TM| = 704 first kind, |TM| = 2816 second kind, |TM| = 2816 first kind, |TM| = 11264 second kind, |TM| = 11264 200 400 600 800 1000 1200 1400 1600 1800 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 number of iterations 2-norm of residual Experiment II first kind, |TM| = 140 second kind, |TM| = 140 first kind, |TM| = 560 second kind, |TM| = 560 first kind, |TM| = 2240 second kind, |TM| = 2240 first kind, |TM| = 8960 second kind, |TM| = 8960

Euclidean condition numbers of Galerkin matrices (diagonally scaled)

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 27 / 28

slide-167
SLIDE 167

Experiment I: Robustness

6 8 10 12 14 16 18 20 22 24 101 102 103 104

0.05

κ0 numbers of iterations N of GMRES s.t. rN

r0 < 10−6

second kind first kind

cond(κ0)

6 8 10 12 14 16 18 20 22 24 101 102 103 104

0.05

κ1 numbers of iterations N of GMRES s.t. rN

r0 < 10−6

second kind first kind

cond(κ1) κ-dependence of Euclidean condition numbers

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 28 / 28

slide-168
SLIDE 168

Conclusion

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-169
SLIDE 169

Conclusion

Appraisal of second-kind single trace formulations:

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-170
SLIDE 170

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-171
SLIDE 171

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy ◮ Requires merely L2-BEM

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-172
SLIDE 172

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy ◮ Requires merely L2-BEM ◮ h-uniformly well conditioned Galerkin matrices

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-173
SLIDE 173

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy ◮ Requires merely L2-BEM ◮ h-uniformly well conditioned Galerkin matrices

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-174
SLIDE 174

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy ◮ Requires merely L2-BEM ◮ h-uniformly well conditioned Galerkin matrices ◮ BIE theory wide open ?

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-175
SLIDE 175

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy ◮ Requires merely L2-BEM ◮ h-uniformly well conditioned Galerkin matrices ◮ BIE theory wide open ? ◮ Stability of BEM even wider open ???

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-176
SLIDE 176

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy ◮ Requires merely L2-BEM ◮ h-uniformly well conditioned Galerkin matrices ◮ BIE theory wide open ? ◮ Stability of BEM even wider open ??? ◮ Non-robustness: small contrasts only!

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-177
SLIDE 177

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy ◮ Requires merely L2-BEM ◮ h-uniformly well conditioned Galerkin matrices ◮ BIE theory wide open ? ◮ Stability of BEM even wider open ??? ◮ Non-robustness: small contrasts only! ◮ Treatment of essential boundary conditions ?

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-178
SLIDE 178

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy ◮ Requires merely L2-BEM ◮ h-uniformly well conditioned Galerkin matrices ◮ BIE theory wide open ? ◮ Stability of BEM even wider open ??? ◮ Non-robustness: small contrasts only! ◮ Treatment of essential boundary conditions ?

  • X. CLAEYS, A single trace integral formulation of the second-kind

for multiple sub-domain scattering, Report 2011-14, SAM, ETH Zürich, 2011. [Helmholtz]

  • X. CLAEYS, R. HIPTMAIR, AND E. SPINDLER, A second-kind

Galerkin boundary element method for scattering at composite

  • bjects, BIT, 55 (2015), pp. 33–57. [Helmholtz]
  • X. CLAEYS, R. HIPTMAIR, AND E. SPINDLER, Second kind

boundary integral equation for multi-subdomain diffusion problems, Tech. Rep. 2016-44, Seminar for Applied Mathematics, ETH Zürich, 2016. [pure diffusion] , Second-kind boundary integral equations for electromagnetic scattering at composite objects, Report 2016-43, SAM, ETH Zurich, 2016.

  • E. SPINDLER, Second Kind Single-Trace Boundary Integral

Formulations for Scattering at Composite Objects, ETH Dissertation no. 23620, ETH Zurich, 2016.

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29

slide-179
SLIDE 179

Conclusion

Appraisal of second-kind single trace formulations:

◮ (Empiric) competitive accuracy ◮ Requires merely L2-BEM ◮ h-uniformly well conditioned Galerkin matrices ◮ BIE theory wide open ? ◮ Stability of BEM even wider open ??? ◮ Non-robustness: small contrasts only! ◮ Treatment of essential boundary conditions ?

  • X. CLAEYS, A single trace integral formulation of the second-kind

for multiple sub-domain scattering, Report 2011-14, SAM, ETH Zürich, 2011. [Helmholtz]

  • X. CLAEYS, R. HIPTMAIR, AND E. SPINDLER, A second-kind

Galerkin boundary element method for scattering at composite

  • bjects, BIT, 55 (2015), pp. 33–57. [Helmholtz]
  • X. CLAEYS, R. HIPTMAIR, AND E. SPINDLER, Second kind

boundary integral equation for multi-subdomain diffusion problems, Tech. Rep. 2016-44, Seminar for Applied Mathematics, ETH Zürich, 2016. [pure diffusion] , Second-kind boundary integral equations for electromagnetic scattering at composite objects, Report 2016-43, SAM, ETH Zurich, 2016.

  • E. SPINDLER, Second Kind Single-Trace Boundary Integral

Formulations for Scattering at Composite Objects, ETH Dissertation no. 23620, ETH Zurich, 2016.

THANK YOU

  • R. Hiptmair (SAM, ETH Zürich)

2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 29 / 29