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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

  1. Electromagnetic Trace Spaces ✓ ✏ ( Ω ⊂ R 3 bounded Lipschitz domain, Γ := ∂ Ω ) γ E : H ( curl , Ω) → H − 1 2 ( curl Γ , Γ) := H E (Γ) continuous & surjective ✒ ✑ γ M : H ( curl 2 , Ω) → H − 1 2 ( div Γ , Γ) := H M (Γ) ✓ ✏ H E (Γ) := H − 1 2 ( curl Γ , Γ) ↔ H − 1 L 2 2 ( div Γ , Γ) := H M (Γ) t (Γ) -duality: � [ Pairing ( v , η ) �→ � v , η � Γ := Γ v ( y ) · η ( y ) d S ( y ) ] ✒ ✑ EM compound trace operator γ := ( γ E , γ M ) = ( γ t , curl · × n ) , γ : H ( curl 2 , Ω) → H (Γ) := H E (Γ) × H M (Γ) . EM compound trace space H (Γ) : self-dual w.r.t. pairing ➣ � �� u � � v �� � � u � � v � := � u , ν � Γ − � v , µ � Γ , ∈ T (Γ) . , , µ ν µ ν Γ R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 3 / 29

  2. Stratton-Chu Representation Formula R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

  3. Stratton-Chu Representation Formula Representation formula: U ℓ ∈ H ( curl , Ω ℓ ) , − curl curl U ℓ − κ 2 ℓ U ℓ = 0 in Ω ℓ : R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

  4. Stratton-Chu Representation Formula Representation formula: U ℓ ∈ H ( curl , Ω ℓ ) , − curl curl U ℓ − κ 2 ℓ U ℓ = 0 in Ω ℓ : � U ℓ in Ω ℓ , S ℓ [ κ ℓ ] ( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ] ( γ ℓ E U ℓ ) = in R 3 \ Ω ℓ . 0 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

  5. Stratton-Chu Representation Formula Representation formula: U ℓ ∈ H ( curl , Ω ℓ ) , − curl curl U ℓ − κ 2 ℓ U ℓ = 0 in Ω ℓ : � U ℓ in Ω ℓ , S ℓ [ κ ℓ ] ( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ] ( γ ℓ E U ℓ ) = in R 3 \ Ω ℓ . 0 (Maxwell) single layer potential R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

  6. Stratton-Chu Representation Formula Representation formula: U ℓ ∈ H ( curl , Ω ℓ ) , − curl curl U ℓ − κ 2 ℓ U ℓ = 0 in Ω ℓ : � U ℓ in Ω ℓ , S ℓ [ κ ℓ ] ( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ] ( γ ℓ E U ℓ ) = in R 3 \ Ω ℓ . 0 (Maxwell) single layer potential x �∈ Γ , S ℓ [ κ ]( ν ) ( x ) = V ℓ [ κ ]( ν )( x ) + ∇ x V ℓ [ κ ]( div Γ ν )( x ) , R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

  7. Stratton-Chu Representation Formula Representation formula: U ℓ ∈ H ( curl , Ω ℓ ) , − curl curl U ℓ − κ 2 ℓ U ℓ = 0 in Ω ℓ : � U ℓ in Ω ℓ , S ℓ [ κ ℓ ] ( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ] ( γ ℓ E U ℓ ) = in R 3 \ Ω ℓ . 0 (Maxwell) single layer potential � V ℓ [ κ ]( ϕ )( x ) = Φ[ κ ]( x , y ) ϕ ( y ) d S ( y ) , Γ ℓ x �∈ Γ , S ℓ [ κ ]( ν ) ( x ) = V ℓ [ κ ]( ν )( x ) + ∇ x V ℓ [ κ ]( div Γ ν )( x ) , Φ[ κ ]( x , y ) = exp ( ı κ | x − y | ) , x � = y Helmholtz fundamental solution: 4 π | x − y | R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

  8. Stratton-Chu Representation Formula Representation formula: U ℓ ∈ H ( curl , Ω ℓ ) , − curl curl U ℓ − κ 2 ℓ U ℓ = 0 in Ω ℓ : � U ℓ in Ω ℓ , S ℓ [ κ ℓ ] ( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ] ( γ ℓ E U ℓ ) = in R 3 \ Ω ℓ . 0 (Maxwell) single layer potential � V ℓ [ κ ]( ϕ )( x ) = Φ[ κ ]( x , y ) ϕ ( y ) d S ( y ) , Γ ℓ x �∈ Γ , S ℓ [ κ ]( ν ) ( x ) = V ℓ [ κ ]( ν )( x ) + ∇ x V ℓ [ κ ]( div Γ ν )( x ) , "smoothing": V : H − 1 2 ( ∂ Ω ℓ ) → H 1 loc ( R 3 ) Φ[ κ ]( x , y ) = exp ( ı κ | x − y | ) , x � = y Helmholtz fundamental solution: 4 π | x − y | R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

  9. Stratton-Chu Representation Formula Representation formula: U ℓ ∈ H ( curl , Ω ℓ ) , − curl curl U ℓ − κ 2 ℓ U ℓ = 0 in Ω ℓ : � U ℓ in Ω ℓ , S ℓ [ κ ℓ ] ( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ] ( γ ℓ E U ℓ ) = in R 3 \ Ω ℓ . 0 (Maxwell) single layer potential (Maxwell) double layer potential � V ℓ [ κ ]( ϕ )( x ) = Φ[ κ ]( x , y ) ϕ ( y ) d S ( y ) , Γ ℓ x �∈ Γ , S ℓ [ κ ]( ν ) ( x ) = V ℓ [ κ ]( ν )( x ) + ∇ x V ℓ [ κ ]( div Γ ν )( x ) , D ℓ [ κ ]( v ) ( x ) = curl V ℓ ( v )( x ) . Φ[ κ ]( x , y ) = exp ( ı κ | x − y | ) , x � = y Helmholtz fundamental solution: 4 π | x − y | R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

  10. Stratton-Chu Representation Formula Representation formula: U ℓ ∈ H ( curl , Ω ℓ ) , − curl curl U ℓ − κ 2 ℓ U ℓ = 0 in Ω ℓ : � U ℓ in Ω ℓ , G ℓ [ κ ℓ ]( γ ℓ U ℓ ) := S ℓ [ κ ℓ ] ( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ] ( γ ℓ E U ℓ ) = in R 3 \ Ω ℓ . 0 (Maxwell) single layer potential (Maxwell) double layer potential � V ℓ [ κ ]( ϕ )( x ) = Φ[ κ ]( x , y ) ϕ ( y ) d S ( y ) , Γ ℓ x �∈ Γ , S ℓ [ κ ]( ν ) ( x ) = V ℓ [ κ ]( ν )( x ) + ∇ x V ℓ [ κ ]( div Γ ν )( x ) , D ℓ [ κ ]( v ) ( x ) = curl V ℓ ( v )( x ) . Φ[ κ ]( x , y ) = exp ( ı κ | x − y | ) , x � = y Helmholtz fundamental solution: 4 π | x − y | R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 4 / 29

  11. (Subdomain) Boundary Integral Operators R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  12. (Subdomain) Boundary Integral Operators S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  13. (Subdomain) Boundary Integral Operators S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  14. (Subdomain) Boundary Integral Operators S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  15. (Subdomain) Boundary Integral Operators S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  16. (Subdomain) Boundary Integral Operators S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  17. (Subdomain) Boundary Integral Operators S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  18. (Subdomain) Boundary Integral Operators S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  19. (Subdomain) Boundary Integral Operators S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) EFIE Op. R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  20. (Subdomain) Boundary Integral Operators G ℓ [ κ ℓ ]( γ ℓ U ℓ ) := S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) MFIE Op. R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  21. (Subdomain) Boundary Integral Operators G ℓ [ κ ℓ ]( γ ℓ U ℓ ) := S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) � � 1 � γ ℓ G ℓ [ κ ℓ ]( γ ℓ U ℓ ) = ( γ ℓ U ℓ ) = γ ℓ U ℓ 2 Id + A ℓ [ κ ℓ ] “on Γ ” . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  22. (Subdomain) Boundary Integral Operators G ℓ [ κ ℓ ]( γ ℓ U ℓ ) := S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) � � 1 � γ ℓ G ℓ [ κ ℓ ]( γ ℓ U ℓ ) = ( γ ℓ U ℓ ) = γ ℓ U ℓ 2 Id + A ℓ [ κ ℓ ] “on Γ ” . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  23. (Subdomain) Boundary Integral Operators G ℓ [ κ ℓ ]( γ ℓ U ℓ ) := S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) � � 1 � γ ℓ G ℓ [ κ ℓ ]( γ ℓ U ℓ ) = ( γ ℓ U ℓ ) = γ ℓ U ℓ 2 Id + A ℓ [ κ ℓ ] “on Γ ” . ✤ ✜ � � u � 1 � � u � u = ∈ H (Γ ℓ ) 2 Id − C ℓ [ κ ℓ ] S ℓ [ κ ℓ ] ⇔ γ ℓ G ℓ [ κ ℓ ]( u ) = ϕ = u . S ′ 1 2 Id + C ′ ℓ [ κ ℓ ] ℓ [ κ ℓ ] ϕ are Cauchy data � �� � ✣ ✢ R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  24. (Subdomain) Boundary Integral Operators G ℓ [ κ ℓ ]( γ ℓ U ℓ ) := S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) � � 1 � γ ℓ G ℓ [ κ ℓ ]( γ ℓ U ℓ ) = ( γ ℓ U ℓ ) = γ ℓ U ℓ 2 Id + A ℓ [ κ ℓ ] “on Γ ” . ✤ ✜ � � u � 1 � � u � u = ∈ H (Γ ℓ ) 2 Id − C ℓ [ κ ℓ ] S ℓ [ κ ℓ ] ⇔ γ ℓ G ℓ [ κ ℓ ]( u ) = ϕ = u . S ′ 1 2 Id + C ′ ℓ [ κ ℓ ] ℓ [ κ ℓ ] ϕ are Cauchy data � �� � ✣ ✢ R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  25. (Subdomain) Boundary Integral Operators G ℓ [ κ ℓ ]( γ ℓ U ℓ ) := S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . Compound traces of U : curl curl U − κ 2 ℓ U = 0 in Ω ℓ � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) � � 1 � γ ℓ G ℓ [ κ ℓ ]( γ ℓ U ℓ ) = ( γ ℓ U ℓ ) = γ ℓ U ℓ 2 Id + A ℓ [ κ ℓ ] “on Γ ” . ✤ ✜ � � u � 1 � � u � u = ∈ H (Γ ℓ ) 2 Id − C ℓ [ κ ℓ ] S ℓ [ κ ℓ ] ⇔ γ ℓ G ℓ [ κ ℓ ]( u ) = ϕ = u . S ′ 2 Id + C ′ 1 ℓ [ κ ℓ ] ℓ [ κ ℓ ] ϕ are Cauchy data � �� � ✣ ✢ R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  26. (Subdomain) Boundary Integral Operators G ℓ [ κ ℓ ]( γ ℓ U ℓ ) := S ℓ [ κ ℓ ]( γ ℓ M U ℓ ) − D ℓ [ κ ℓ ]( γ ℓ R.F.: E U ℓ ) = U ℓ in Ω ℓ . E S ℓ [ κ ℓ ] ( γ ℓ γ ℓ − γ ℓ E D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) E U ℓ ) = E U ℓ , γ ℓ M S ℓ [ κ ℓ ] ( γ ℓ γ ℓ M D ℓ [ κ ℓ ] ( γ ℓ γ ℓ M U ℓ ) − E U ℓ ) = M U ℓ . � �� 1 � � � γ ℓ � ( γ ℓ S ℓ [ κ ℓ ] ( γ ℓ 2 Id − C ℓ [ κ ℓ ] E U ℓ ) + M U ℓ ) E U ℓ = � 1 � γ ℓ M U ℓ S ′ ℓ [ κ ℓ ] ( γ ℓ 2 Id + C ′ ( γ ℓ E U ℓ ) + ℓ [ κ ℓ ] M U ℓ ) � � 1 � γ ℓ G ℓ [ κ ℓ ]( γ ℓ U ℓ ) = ( γ ℓ U ℓ ) = γ ℓ U ℓ 2 Id + A ℓ [ κ ℓ ] “on Γ ” . ✤ ✜ � � u � 1 � � u � u = ∈ H (Γ ℓ ) 2 Id − C ℓ [ κ ℓ ] S ℓ [ κ ℓ ] ⇔ γ ℓ G ℓ [ κ ℓ ]( u ) = ϕ = u . S ′ 1 2 Id + C ′ ℓ [ κ ℓ ] ℓ [ κ ℓ ] ϕ are Cauchy data � �� � ✣ ✢ (interior) Calderón projector P ℓ = 1 2 Id + A ℓ [ κ ℓ ] R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 5 / 29

  27. What Next ? Scattering at Composite Objects 1 BIE: Single Subdomain Setting 2 BIE: Composite Scatterer 3 Numerical Experiments 4 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 6 / 29

  28. Single Subdomain Setting (SSS) R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 7 / 29

  29. Single Subdomain Setting (SSS) c Ω 0 = Ω U inc n = Homogeneous scatterer: n 0 Γ Ω 1 n n 0 n R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 7 / 29

  30. Single Subdomain Setting (SSS) c Ω 0 = Ω U inc n = Homogeneous scatterer: n 0 Γ Ω 1 n n 0 n � for x ∈ Ω 0 , κ 0 κ ( x ) = κ 1 for x ∈ Ω 1 , curl curl U − κ ( x ) 2 U = 0 Electromagnetic scattering: R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 7 / 29

  31. Single Subdomain Setting (SSS) c Ω 0 = Ω U inc n = Homogeneous scatterer: n 0 Γ Ω 1 n n 0 n � for x ∈ Ω 0 , κ 0 κ ( x ) = κ 1 for x ∈ Ω 1 , curl curl U − κ ( x ) 2 U = 0 Electromagnetic scattering: − ∆ U − κ ( x ) 2 U = 0 Acoustic scattering: R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 7 / 29

  32. SSS: Helmholtz Transmission Problem R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

  33. SSS: Helmholtz Transmission Problem If U solves Helmholtz scattering trans- mission problem c Ω 0 = Ω n u inc n 0 Γ Ω n 0 n n R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

  34. SSS: Helmholtz Transmission Problem If U solves Helmholtz scattering trans- mission problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω n 0 n n R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

  35. SSS: Helmholtz Transmission Problem If U solves Helmholtz scattering trans- mission problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

  36. SSS: Helmholtz Transmission Problem If U solves Helmholtz scattering trans- mission problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . compact operator ( L 2 (Γ)) 2 �→ ( L 2 (Γ)) 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

  37. SSS: Helmholtz Transmission Problem If U solves Helmholtz scattering trans- mission problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . compact operator ( L 2 (Γ)) 2 �→ ( L 2 (Γ)) 2 � − δ K 1 � δ V 1 A 1 [ κ 0 ] − A 1 [ κ 0 ] = [difference BI-Ops] δ K ′ δ W 1 1 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

  38. SSS: Helmholtz Transmission Problem If U solves Helmholtz scattering trans- mission problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . compact operator ( L 2 (Γ)) 2 �→ ( L 2 (Γ)) 2 � − δ K 1 � δ V 1 A 1 [ κ 0 ] − A 1 [ κ 0 ] = [difference BI-Ops] δ K ′ δ W 1 1 � exp ( ı κ 0 | x − y | ) − exp ( ı κ 1 | x − y | ) δ V 1 ( ϕ )( x ) = ϕ ( y ) d S ( y ) . 4 π | x − y | Γ � �� � C 0 -kernel R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

  39. SSS: Helmholtz Transmission Problem If U solves Helmholtz scattering trans- mission problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . compact operator ( L 2 (Γ)) 2 �→ ( L 2 (Γ)) 2 ➤ 2nd-kind BIE � − δ K 1 � δ V 1 A 1 [ κ 0 ] − A 1 [ κ 0 ] = [difference BI-Ops] δ K ′ δ W 1 1 � exp ( ı κ 0 | x − y | ) − exp ( ı κ 1 | x − y | ) δ V 1 ( ϕ )( x ) = ϕ ( y ) d S ( y ) . 4 π | x − y | Γ � �� � C 0 -kernel R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 8 / 29

  40. SSS: Maxwell Transmission Problem R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

  41. SSS: Maxwell Transmission Problem Copy Helmholtz approach: If U solves Maxwell scattering problem c Ω 0 = Ω n u inc n 0 Γ Ω 1 n 0 n n R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

  42. SSS: Maxwell Transmission Problem Copy Helmholtz approach: If U solves Maxwell scattering problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω 1 n 0 n n R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

  43. SSS: Maxwell Transmission Problem Copy Helmholtz approach: If U solves Maxwell scattering problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω 1 n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

  44. SSS: Maxwell Transmission Problem Copy Helmholtz approach: If U solves Maxwell scattering problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω 1 n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . t (Γ)) 2 ? t (Γ)) 2 �→ ( L 2 compact operator ( L 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

  45. SSS: Maxwell Transmission Problem Copy Helmholtz approach: If U solves Maxwell scattering problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω 1 n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . t (Γ)) 2 ? t (Γ)) 2 �→ ( L 2 compact operator ( L 2 S 1 [ κ 1 ]( ϕ ) − S 1 [ κ 0 ]( ϕ ) = V 1 [ κ 0 ]( ϕ ) − V 1 [ κ 1 ]( ϕ )+ 1 0 ∇ Γ V 1 [ κ 0 ]( div Γ ϕ ) − 1 1 ∇ Γ V 1 [ κ 1 ]( div Γ ϕ ) . κ 2 κ 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

  46. SSS: Maxwell Transmission Problem Copy Helmholtz approach: If U solves Maxwell scattering problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Cancellation of singularities Γ Ω 1 n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . t (Γ)) 2 ? t (Γ)) 2 �→ ( L 2 compact operator ( L 2 S 1 [ κ 1 ]( ϕ ) − S 1 [ κ 0 ]( ϕ ) = V 1 [ κ 0 ]( ϕ ) − V 1 [ κ 1 ]( ϕ )+ 1 0 ∇ Γ V 1 [ κ 0 ]( div Γ ϕ ) − 1 1 ∇ Γ V 1 [ κ 1 ]( div Γ ϕ ) . κ 2 κ 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

  47. SSS: Maxwell Transmission Problem Copy Helmholtz approach: If U solves Maxwell scattering problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω 1 Remains hypersingular! n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . t (Γ)) 2 ? t (Γ)) 2 �→ ( L 2 compact operator ( L 2 S 1 [ κ 1 ]( ϕ ) − S 1 [ κ 0 ]( ϕ ) = V 1 [ κ 0 ]( ϕ ) − V 1 [ κ 1 ]( ϕ )+ 1 0 ∇ Γ V 1 [ κ 0 ]( div Γ ϕ ) − 1 1 ∇ Γ V 1 [ κ 1 ]( div Γ ϕ ) . κ 2 κ 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

  48. SSS: Maxwell Transmission Problem Copy Helmholtz approach: If U solves Maxwell scattering problem c Ω 0 = Ω ( 1 2 Id + A 1 [ κ 1 ]) γ 1 U = 0 , n u inc ( 1 + 2 Id + A 0 [ κ 0 ]) γ 0 U = . . . . n 0 Γ Ω 1 n 0 n n ( Id + ( A 1 [ κ 0 ] − A 1 [ κ 1 ])) u = . . . . t (Γ)) 2 ? NO! t (Γ)) 2 �→ ( L 2 compact operator ( L 2 S 1 [ κ 1 ]( ϕ ) − S 1 [ κ 0 ]( ϕ ) = V 1 [ κ 0 ]( ϕ ) − V 1 [ κ 1 ]( ϕ )+ 1 0 ∇ Γ V 1 [ κ 0 ]( div Γ ϕ ) − 1 1 ∇ Γ V 1 [ κ 1 ]( div Γ ϕ ) . κ 2 κ 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 9 / 29

  49. Homogeneous Scatterer: Müller Formulation R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  50. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  51. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  52. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , ( − κ 2 1 V 1 [ κ 1 ] + curl Γ V 1 [ κ 1 ] curl Γ )( γ 1 E U ) + ( 1 2 Id − C ′ 1 [ κ 1 ])( γ 1 Int ➁ : M U ) = 0 , R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  53. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , ( − κ 2 1 V 1 [ κ 1 ] + curl Γ V 1 [ κ 1 ] curl Γ )( γ 1 E U ) + ( 1 2 Id − C ′ 1 [ κ 1 ])( γ 1 Int ➁ : M U ) = 0 , R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  54. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , ( − κ 2 1 V 1 [ κ 1 ] + curl Γ V 1 [ κ 1 ] curl Γ )( γ 1 E U ) + ( 1 2 Id − C ′ 1 [ κ 1 ])( γ 1 Int ➁ : M U ) = 0 , E U ) + ( V 1 [ κ 0 ] + κ − 2 ( 1 2 Id − C 1 [ κ 0 ])( γ 1 0 ∇ Γ V 1 [ κ 0 ] div Γ )( γ 1 Ext ➂ : M U ) = . . . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  55. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , ( − κ 2 1 V 1 [ κ 1 ] + curl Γ V 1 [ κ 1 ] curl Γ )( γ 1 E U ) + ( 1 2 Id − C ′ 1 [ κ 1 ])( γ 1 Int ➁ : M U ) = 0 , E U ) + ( V 1 [ κ 0 ] + κ − 2 ( 1 2 Id − C 1 [ κ 0 ])( γ 1 0 ∇ Γ V 1 [ κ 0 ] div Γ )( γ 1 Ext ➂ : M U ) = . . . ( κ 2 0 V 1 [ κ 0 ] − curl Γ V 1 [ κ 0 ] curl Γ )( γ 1 E U ) + ( 1 2 Id + C ′ 1 [ κ 0 ])( γ 1 Ext ➃ : M U ) = . . . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  56. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , ( − κ 2 1 V 1 [ κ 1 ] + curl Γ V 1 [ κ 1 ] curl Γ )( γ 1 E U ) + ( 1 2 Id − C ′ 1 [ κ 1 ])( γ 1 Int ➁ : M U ) = 0 , E U ) + ( V 1 [ κ 0 ] + κ − 2 ( 1 2 Id − C 1 [ κ 0 ])( γ 1 0 ∇ Γ V 1 [ κ 0 ] div Γ )( γ 1 Ext ➂ : M U ) = . . . ( κ 2 0 V 1 [ κ 0 ] − curl Γ V 1 [ κ 0 ] curl Γ )( γ 1 E U ) + ( 1 2 Id + C ′ 1 [ κ 0 ])( γ 1 Ext ➃ : M U ) = . . . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  57. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , ( − κ 2 1 V 1 [ κ 1 ] + curl Γ V 1 [ κ 1 ] curl Γ )( γ 1 E U ) + ( 1 2 Id − C ′ 1 [ κ 1 ])( γ 1 Int ➁ : M U ) = 0 , E U ) + ( V 1 [ κ 0 ] + κ − 2 ( 1 2 Id − C 1 [ κ 0 ])( γ 1 0 ∇ Γ V 1 [ κ 0 ] div Γ )( γ 1 Ext ➂ : M U ) = . . . ( κ 2 0 V 1 [ κ 0 ] − curl Γ V 1 [ κ 0 ] curl Γ )( γ 1 E U ) + ( 1 2 Id + C ′ 1 [ κ 0 ])( γ 1 Ext ➃ : M U ) = . . . κ 2 Idea: i -weighted sum of Calderón identities κ 2 1 · ➀ + κ 2 0 · ➂ , ➁ + ➃ . R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  58. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , ( − κ 2 1 V 1 [ κ 1 ] + curl Γ V 1 [ κ 1 ] curl Γ )( γ 1 E U ) + ( 1 2 Id − C ′ 1 [ κ 1 ])( γ 1 Int ➁ : M U ) = 0 , E U ) + ( V 1 [ κ 0 ] + κ − 2 ( 1 2 Id − C 1 [ κ 0 ])( γ 1 0 ∇ Γ V 1 [ κ 0 ] div Γ )( γ 1 Ext ➂ : M U ) = . . . ( κ 2 0 V 1 [ κ 0 ] − curl Γ V 1 [ κ 0 ] curl Γ )( γ 1 E U ) + ( 1 2 Id + C ′ 1 [ κ 0 ])( γ 1 Ext ➃ : M U ) = . . . κ 2 Idea: i -weighted sum of Calderón identities κ 2 1 · ➀ + κ 2 0 · ➂ , ➁ + ➃ . � κ 2 �� � �� � γ 1 � κ 2 1 + κ 2 1 C 1 [ κ 1 ] − κ 2 − κ 2 1 V 1 [ κ 1 ] + κ 2 0 C 1 [ κ 0 ] 0 V 1 [ κ 0 ] E U Id 0 0 + = . . 2 1 C ′ 0 C ′ − κ 2 1 V 1 [ κ 1 ] + κ 2 − κ 2 1 [ κ 1 ] + κ 2 γ 1 0 V 1 [ κ 0 ] 1 [ κ 0 ] M U 0 Id R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  59. Homogeneous Scatterer: Müller Formulation Details: Maxwell Calderón identities (w.r.t Ω 1 ) E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , ( − κ 2 1 V 1 [ κ 1 ] + curl Γ V 1 [ κ 1 ] curl Γ )( γ 1 E U ) + ( 1 2 Id − C ′ 1 [ κ 1 ])( γ 1 Int ➁ : M U ) = 0 , E U ) + ( V 1 [ κ 0 ] + κ − 2 ( 1 2 Id − C 1 [ κ 0 ])( γ 1 0 ∇ Γ V 1 [ κ 0 ] div Γ )( γ 1 Ext ➂ : M U ) = . . . ( κ 2 0 V 1 [ κ 0 ] − curl Γ V 1 [ κ 0 ] curl Γ )( γ 1 E U ) + ( 1 2 Id + C ′ 1 [ κ 0 ])( γ 1 Ext ➃ : M U ) = . . . κ 2 Idea: i -weighted sum of Calderón identities κ 2 1 · ➀ + κ 2 0 · ➂ , ➁ + ➃ . � κ 2 �� � �� � γ 1 � κ 2 1 + κ 2 1 C 1 [ κ 1 ] − κ 2 − κ 2 1 V 1 [ κ 1 ] + κ 2 0 C 1 [ κ 0 ] 0 V 1 [ κ 0 ] E U Id 0 0 + = . . 2 1 C ′ 0 C ′ − κ 2 1 V 1 [ κ 1 ] + κ 2 − κ 2 1 [ κ 1 ] + κ 2 γ 1 0 V 1 [ κ 0 ] 1 [ κ 0 ] M U 0 Id Cancellation of hypersingular operators! R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  60. Homogeneous Scatterer: Müller Formulation No cancellation in kernels! Details: Maxwell Calderón identities (w.r.t Ω 1 ) Fredholm, index = 0, in ( L 2 t (Γ)) 2 E U ) − ( V 1 [ κ 1 ] + κ − 2 ( 1 2 Id + C 1 [ κ 1 ])( γ 1 1 ∇ Γ V 1 [ κ 1 ] div Γ )( γ 1 Int ➀ : M U ) = 0 , D. M ITREA , M. M ITREA , AND J. P IPHER , Vector potential theory on nonsmooth domains in R 3 and applications to ( − κ 2 1 V 1 [ κ 1 ] + curl Γ V 1 [ κ 1 ] curl Γ )( γ 1 E U ) + ( 1 2 Id − C ′ 1 [ κ 1 ])( γ 1 Int ➁ : M U ) = 0 , electromagnetic scattering , J. Fourier Anal. Appl., 3 (1997), pp. 131–192. E U ) + ( V 1 [ κ 0 ] + κ − 2 ( 1 2 Id − C 1 [ κ 0 ])( γ 1 0 ∇ Γ V 1 [ κ 0 ] div Γ )( γ 1 Ext ➂ : M U ) = . . . ( κ 2 0 V 1 [ κ 0 ] − curl Γ V 1 [ κ 0 ] curl Γ )( γ 1 E U ) + ( 1 2 Id + C ′ 1 [ κ 0 ])( γ 1 Ext ➃ : M U ) = . . . κ 2 Idea: i -weighted sum of Calderón identities κ 2 1 · ➀ + κ 2 0 · ➂ , ➁ + ➃ . � κ 2 �� � �� � γ 1 � κ 2 1 + κ 2 1 C 1 [ κ 1 ] − κ 2 − κ 2 1 V 1 [ κ 1 ] + κ 2 0 C 1 [ κ 0 ] 0 V 1 [ κ 0 ] E U Id 0 0 + = . . 2 1 C ′ 0 C ′ − κ 2 1 V 1 [ κ 1 ] + κ 2 − κ 2 1 [ κ 1 ] + κ 2 γ 1 0 V 1 [ κ 0 ] 1 [ κ 0 ] M U 0 Id Cancellation of hypersingular operators! R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 10 / 29

  61. Composite Scattering Challenge R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 11 / 29

  62. Composite Scattering Challenge n 1 U inc c Ω 0 = Ω Γ n 0 n u inc Ω 1 n 3 n 0 n 1 n 1 n 0 Γ Ω 3 Ω 1 n n 3 n 2 n 0 Ω 2 n Ω 0 n 0 n 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 11 / 29

  63. Composite Scattering Challenge n 1 U inc c Ω 0 = Ω Γ n 0 n u inc Ω 1 n 3 n 0 n 1 n 1 n 0 Γ Ω 3 Ω 1 n n 3 n 2 n 0 Ω 2 n Ω 0 n 0 n 2 Calderón identities set in same space: A 0 [ κ 0 ] , A 1 [ κ 1 ] : H ( γ ) → H (Γ) R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 11 / 29

  64. Composite Scattering Challenge n 1 U inc c Ω 0 = Ω Γ n 0 n u inc Ω 1 n 3 n 0 n 1 n 1 n 0 Γ Ω 3 Ω 1 n n 3 n 2 n 0 Ω 2 n Ω 0 n 0 n 2 Calderón identities set in same Local Calderón identities in local trace spaces H (Γ k ) space: ? A 0 [ κ 0 ] , A 1 [ κ 1 ] : H ( γ ) → H (Γ) R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 11 / 29

  65. Müller Formulation: Multi-Potential Approach R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

  66. Müller Formulation: Multi-Potential Approach Multi-potential representation, U solves Maxwell TP: (1) U = − D 0 [ κ 0 ]( γ 0 E U ) + S 0 [ κ 0 ]( γ 0 M U ) (2) − D 1 [ κ 1 ]( γ 1 E U ) + S 1 [ κ 1 ]( γ 1 M U ) + . . . , R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

  67. Müller Formulation: Multi-Potential Approach Multi-potential representation, U solves Maxwell TP: ( κ 2 1 χ 1 + κ 2 0 χ 0 ) U = − κ 2 0 D 0 [ κ 0 ]( γ 0 E U ) + κ 2 0 S 0 [ κ 0 ]( γ 0 M U ) (1) − κ 2 1 D 1 [ κ 1 ]( γ 1 E U ) + κ 2 1 S 1 [ κ 1 ]( γ 1 M U ) + . . . , U = − D 0 [ κ 0 ]( γ 0 E U ) + S 0 [ κ 0 ]( γ 0 M U ) (2) − D 1 [ κ 1 ]( γ 1 E U ) + S 1 [ κ 1 ]( γ 1 M U ) + . . . , R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

  68. Müller Formulation: Multi-Potential Approach Multi-potential representation, U solves Maxwell TP: ( κ 2 1 χ 1 + κ 2 0 χ 0 ) U = − κ 2 0 D 0 [ κ 0 ]( γ 0 E U ) + κ 2 0 S 0 [ κ 0 ]( γ 0 M U ) (1) − κ 2 1 D 1 [ κ 1 ]( γ 1 E U ) + κ 2 1 S 1 [ κ 1 ]( γ 1 M U ) + . . . , U = − D 0 [ κ 0 ]( γ 0 E U ) + S 0 [ κ 0 ]( γ 0 M U ) (2) − D 1 [ κ 1 ]( γ 1 E U ) + S 1 [ κ 1 ]( γ 1 M U ) + . . . , Apply traces γ 1 E , γ 0 γ 1 M , γ 0 E to (1) & M on (2) [signs adjusted] R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

  69. Müller Formulation: Multi-Potential Approach Multi-potential representation, U solves Maxwell TP: ( κ 2 1 χ 1 + κ 2 0 χ 0 ) U = − κ 2 0 D 0 [ κ 0 ]( γ 0 E U ) + κ 2 0 S 0 [ κ 0 ]( γ 0 M U ) (1) − κ 2 1 D 1 [ κ 1 ]( γ 1 E U ) + κ 2 1 S 1 [ κ 1 ]( γ 1 M U ) + . . . , U = − D 0 [ κ 0 ]( γ 0 E U ) + S 0 [ κ 0 ]( γ 0 M U ) (2) − D 1 [ κ 1 ]( γ 1 E U ) + S 1 [ κ 1 ]( γ 1 M U ) + . . . , Apply traces γ 1 E , γ 0 γ 1 M , γ 0 E to (1) & M on (2) [signs adjusted] � 1 � � γ 1 � 2 ( κ 2 0 + κ 2 1 ) Id + κ 2 0 C 0 [ κ 0 ] + κ 2 − κ 2 0 S 0 [ κ 0 ] + κ 2 E U 1 C 1 [ κ 1 ] 1 S 1 [ κ 1 ] = . . . S ′ 0 [ κ 0 ] − S ′ Id − C ′ 0 [ κ 0 ] − C ′ γ 1 1 [ κ 1 ] 1 [ κ 1 ] M U R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

  70. Müller Formulation: Multi-Potential Approach Multi-potential representation, U solves Maxwell TP: ( κ 2 1 χ 1 + κ 2 0 χ 0 ) U = − κ 2 0 D 0 [ κ 0 ]( γ 0 E U ) + κ 2 0 S 0 [ κ 0 ]( γ 0 M U ) (1) − κ 2 1 D 1 [ κ 1 ]( γ 1 E U ) + κ 2 1 S 1 [ κ 1 ]( γ 1 M U ) + . . . , U = − D 0 [ κ 0 ]( γ 0 E U ) + S 0 [ κ 0 ]( γ 0 M U ) (2) − D 1 [ κ 1 ]( γ 1 E U ) + S 1 [ κ 1 ]( γ 1 M U ) + . . . , Apply traces γ 1 E , γ 0 γ 1 M , γ 0 E to (1) & M on (2) [signs adjusted] � 1 � � γ 1 � 2 ( κ 2 0 + κ 2 1 ) Id + κ 2 0 C 0 [ κ 0 ] + κ 2 − κ 2 0 S 0 [ κ 0 ] + κ 2 E U 1 C 1 [ κ 1 ] 1 S 1 [ κ 1 ] = . . . S ′ 0 [ κ 0 ] − S ′ Id − C ′ 0 [ κ 0 ] − C ′ γ 1 1 [ κ 1 ] 1 [ κ 1 ] M U � κ 2 �� � �� � γ 1 κ 2 1 + κ 2 1 C 1 [ κ 1 ] − κ 2 − κ 2 1 V 1 [ κ 1 ] + κ 2 � 0 C 1 [ κ 0 ] 0 V 1 [ κ 0 ] E U 0 Id 0 + = . . . 2 − κ 2 1 V 1 [ κ 1 ] + κ 2 − κ 2 1 [ κ 1 ] + κ 2 γ 1 0 V 1 [ κ 0 ] 1 C ′ 0 C ′ 1 [ κ 0 ] M U 0 Id R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 12 / 29

  71. What Next ? Scattering at Composite Objects 1 BIE: Single Subdomain Setting 2 BIE: Composite Scatterer 3 Numerical Experiments 4 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 13 / 29

  72. Composite EM Scattering n 1 U inc n 0 Γ Ω 1 n 3 n 1 n 0 n 1 Ω 3 n 3 n 2 Ω 2 Ω 0 n 0 n 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 14 / 29

  73. Composite EM Scattering curl curl U − κ ( x ) 2 U = 0 , n 1 U inc κ ( x ) = κ i > 0 in Ω i . n 0 Γ Ω 1 n 3 n 1 n 0 n 1 Ω 3 n 3 n 2 Ω 2 Ω 0 n 0 n 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 14 / 29

  74. Composite EM Scattering curl curl U − κ ( x ) 2 U = 0 , n 1 U inc κ ( x ) = κ i > 0 in Ω i . n 0 Γ Ω 1 n 3 Boundaries Γ k := ∂ Ω k n 1 n 0 n 1 Ω 3 Interfaces Γ k ℓ = ∂ Ω k ∩ ∂ Ω ℓ n 3 n 2 Skeleton: Ω 2 � � Ω 0 n 0 Σ := ∂ Ω k = Γ k ℓ . n 2 k k � = ℓ R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 14 / 29

  75. Skeleton Trace Spaces R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

  76. Skeleton Trace Spaces U inc n 1 n 0 Γ Ω 1 n 3 n 1 n 0 n 1 Ω 3 n 3 n 2 Ω 2 Ω 0 n 0 n 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

  77. Skeleton Trace Spaces Multi-trace space (electric/magnetic): U inc n 1 n 0 Γ MT (Σ) := H (Γ 0 ) × · · · × H (Γ N ) ∼ = MT E (Σ) × MT M (Σ) . Ω 1 n 3 n 1 n 0 n 1 Ω 3 n 3 n 2 Ω 2 Ω 0 n 0 n 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

  78. Skeleton Trace Spaces Multi-trace space (electric/magnetic): U inc n 1 n 0 Γ H (Γ 0 ) × · · · × H (Γ N ) ∼ MT (Σ) := = MT E (Σ) × MT M (Σ) . Ω 1 n 3 n 1 n 0 n 1 Ω 3 n 3 n 2 Ω 2 Ω 0 n 0 H (Γ l ) = H E (Γ ℓ ) × H M (Γ ℓ ) n 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

  79. Skeleton Trace Spaces Multi-trace space (electric/magnetic): U inc n 1 n 0 Γ MT (Σ) := H (Γ 0 ) × · · · × H (Γ N ) ∼ = MT E (Σ) × MT M (Σ) . Ω 1 n 3 n 1 n 0 n 1 Ω 3 γ Σ : � n 3 n 2 ℓ H loc ( curl 2 , Ω ℓ ) → MT (Σ) , Ω 2 Multi-trace operator: γ Σ := γ 0 × · · · × γ N ∼ Ω 0 = γ Σ E × γ Σ n 0 M n 2 R. Hiptmair (SAM, ETH Zürich) 2nd-kind BIE for EM Scattering RICAM, Oct 19, 2016 15 / 29

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