MIXED INTERIOR TRANSMISSION EIGENVALUES joint work with Jijun Liu - PowerPoint PPT Presentation

1.2 1.2 1.2 1.2 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 -0.2 -0.2 -0.2 -0.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 MIXED INTERIOR TRANSMISSION EIGENVALUES joint work with Jijun Liu CMMSE 2019 (MS 23) | July 2, 2019 Andreas Kleefeld J¨ ulich Supercomputing Centre, Germany Member of the Helmholtz Association

TABLE OF CONTENTS Part 1: Introduction & motivation Part 2: Some theory Part 3: Boundary integral equations Part 4: Numerical results Part 5: Summary & outlook CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

Part I: Introduction & motivation CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

INTRODUCTION & MOTIVATION A general physical configuration Obstacle D is located in perfect D 1 conducting substrate D 2 with boundary Γ 2 ⊂ Γ . 1 Remaining part of boundary Γ 1 = Γ \ Γ 2 D contacts with surface of background dielectric medium D 1 . 2 We assume Γ = Γ 1 ∪ Γ 2 , Γ 1 � = ∅ , and D 2 Γ 2 � = ∅ . Scattering problem for isotropic inhomogeneous media (TE mode electromagnetic scattering) leads to . . . CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

INTRODUCTION & MOTIVATION A general physical configuration . . . (acoustic) interior transmission problem with mixed boundary condition:  ∆ u + k 2 u = 0 , x ∈ D ,   ∆ v + k 2 nv = 0 ,  x ∈ D ,  (1) ∂ u ∂ν = ∂ v u = v , ∂ν , x ∈ Γ 1 , (transmission condition)    u = v = 0 , x ∈ Γ 2 . (hom. Dirichlet condition)  Here, n � = 1 is the real-valued index of refraction (constant). Find k � = 0 and non-trivial ( u , v ) such that (1) is satisfied. Such k will be called mixed interior transmission eigenvalues (MITEs). CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

INTRODUCTION & MOTIVATION Goal This is a non-standard eigenvalue problem. It is neither elliptic nor self-adjoint. How to solve this problem numerically? No results for the computation of MITEs have yet been reported. CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

Part II: Some theory CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

SOME THEORY A short review The set of MITEs is at most discrete. Does not accumulate at zero. There exists an infinite number of real MITEs. Only accumulation point is ∞ . Nothing is known for complex-valued MITEs. CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

Part III: Boundary integral equations CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

BOUNDARY INTEGRAL EQUATIONS Preliminaries Fundamental solution: Φ k ( x , y ) = i H ( 1 ) 0 ( k | x − y | ) / 4 , x � = y . Single- and double-layer potentials over Γ given for x / ∈ Γ by � SL Γ k [ ψ ] ( x ) = Φ k ( x , y ) ψ ( y ) d s ( y ) , Γ � DL Γ k [ ψ ] ( x ) = ∂ ν ( y ) Φ k ( x , y ) ψ ( y ) d s ( y ) . Γ Green’s representation theorem: u ( x ) = SL Γ k [ ∂ ν u | Γ ] ( x ) − DL Γ k [ u | Γ ] ( x ) , x ∈ D . CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

BOUNDARY INTEGRAL EQUATIONS Preliminaries Γ is disjoint union of Γ 1 and Γ 2 . Hence, SL Γ 1 k [ ∂ ν u | Γ 1 ] ( x ) + SL Γ 2 u ( x ) = k [ ∂ ν u | Γ 2 ] ( x ) DL Γ 1 k [ u | Γ 1 ] ( x ) − DL Γ 2 − k [ u | Γ 2 ] ( x ) , x ∈ D , (2) SL Γ 1 k √ n [ ∂ ν v | Γ 1 ] ( x ) + SL Γ 2 v ( x ) = k √ n [ ∂ ν v | Γ 2 ] ( x ) DL Γ 1 k √ n [ v | Γ 1 ] ( x ) − DL Γ 2 − k √ n [ v | Γ 2 ] ( x ) , x ∈ D . (3) Using u | Γ 2 = v | Γ 2 = 0, equations (2) and (3) can be simplified to u ( x ) = SL Γ 1 k [ ∂ ν u | Γ 1 ] ( x ) + SL Γ 2 k [ ∂ ν u | Γ 2 ] ( x ) − DL Γ 1 k [ u | Γ 1 ] ( x ) , x ∈ D , (4) v ( x ) = SL Γ 1 k √ n [ ∂ ν v | Γ 1 ] ( x ) + SL Γ 2 k √ n [ ∂ ν v | Γ 2 ] ( x ) − DL Γ 1 k √ n [ v | Γ 1 ] ( x ) , x ∈ D . (5) CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

BOUNDARY INTEGRAL EQUATIONS Preliminaries Boundary integral operators: � Γ i → Γ j [ ψ | Γ i ] ( x ) = Φ k ( x , y ) ψ ( y ) d s ( y ) , x ∈ Γ j , S k Γ i � Γ i → Γ j [ ψ | Γ i ] ( x ) = ∂ ν i ( y ) Φ k ( x , y ) ψ ( y ) d s ( y ) , x ∈ Γ j , K k Γ i � Γ i → Γ j [ ψ | Γ i ] ( x ) K ⊤ x ∈ Γ j , = ∂ ν j ( x ) Φ k ( x , y ) ψ ( y ) d s ( y ) , k Γ i � Γ i → Γ j [ ψ | Γ i ] ( x ) = ∂ ν j ( x ) ∂ ν i ( y ) Φ k ( x , y ) ψ ( y ) d s ( y ) , x ∈ Γ j , T k Γ i where i , j ∈ { 1 , 2 } . CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

BOUNDARY INTEGRAL EQUATIONS Derivation of first boundary integral equation D ∋ x → x ∈ Γ 1 in (4) and (5) and jump relations: � � [ u | Γ 1 ] − 1 u | Γ 1 = S Γ 1 → Γ 1 [ ∂ ν u | Γ 1 ] + S Γ 2 → Γ 1 K Γ 1 → Γ 1 [ ∂ ν u | Γ 2 ] − 2 u | Γ 1 , (6) k k k � � [ v | Γ 1 ] − 1 v | Γ 1 = S Γ 1 → Γ 1 [ ∂ ν v | Γ 1 ] + S Γ 2 → Γ 1 K Γ 1 → Γ 1 [ ∂ ν v | Γ 2 ] − 2 v | Γ 1 . (7) k √ n k √ n k √ n Difference of (6) and (7), u | Γ 1 = v | Γ 1 and ∂ ν u | Γ 1 = ∂ ν v | Γ 1 : � � S Γ 1 → Γ 1 − S Γ 1 → Γ 1 [ ∂ ν u | Γ 1 ] + S Γ 2 → Γ 1 0 = [ ∂ ν u | Γ 2 ] k √ n k k � � S Γ 2 → Γ 1 K Γ 1 → Γ 1 − K Γ 1 → Γ 1 − [ ∂ ν v | Γ 2 ] − [ u | Γ 1 ] . (8) k √ n k √ n k CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

BOUNDARY INTEGRAL EQUATIONS Derivation of second boundary integral equation D ∋ x → x ∈ Γ 2 in (4) and (5): u | Γ 2 = S Γ 1 → Γ 2 [ ∂ ν u | Γ 1 ] + S Γ 2 → Γ 2 [ ∂ ν u | Γ 2 ] − K Γ 1 → Γ 2 [ u | Γ 1 ] , (9) k k k v | Γ 2 = S Γ 1 → Γ 2 [ ∂ ν v | Γ 1 ] + S Γ 2 → Γ 2 [ ∂ ν v | Γ 2 ] − K Γ 1 → Γ 2 [ v | Γ 1 ] . (10) k √ n k √ n k √ n Difference of (9) and (10), u | Γ 2 = v | Γ 2 = 0, u | Γ 1 = v | Γ 1 and ∂ ν u | Γ 1 = ∂ ν v | Γ 1 : � � S Γ 1 → Γ 2 − S Γ 1 → Γ 2 [ ∂ ν u | Γ 1 ] + S Γ 2 → Γ 2 0 = [ ∂ ν u | Γ 2 ] k √ n k k � � S Γ 2 → Γ 2 K Γ 1 → Γ 2 − K Γ 1 → Γ 2 − [ ∂ ν v | Γ 2 ] − [ u | Γ 1 ] . (11) k √ n k √ n k CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

BOUNDARY INTEGRAL EQUATIONS Derivation of third boundary integral equation Normal derivative of (4) and (5), D ∋ x → x ∈ Γ 1 , and jump relations: Γ 1 → Γ 1 [ ∂ ν u | Γ 1 ] + 1 Γ 2 → Γ 1 [ ∂ ν u | Γ 2 ] − T Γ 1 → Γ 1 ∂ ν u | Γ 1 = K ⊤ 2 ∂ ν u | Γ 1 + K ⊤ [ u | Γ 1 ] , (12) k k k Γ 1 → Γ 1 [ ∂ ν v | Γ 1 ] + 1 Γ 2 → Γ 1 [ ∂ ν v | Γ 2 ] − T Γ 1 → Γ 1 ∂ ν v | Γ 1 = K ⊤ 2 ∂ ν v | Γ 1 + K ⊤ [ v | Γ 1 ] . (13) k √ n k √ n k √ n Difference of (12) and (13), u | Γ 1 = v | Γ 1 and ∂ ν u | Γ 1 = ∂ ν v | Γ 1 : Γ 1 → Γ 1 − K ⊤ Γ 2 → Γ 1 [ ∂ ν u | Γ 2 ] � Γ 1 → Γ 1 � K ⊤ [ ∂ ν u | Γ 1 ] + K ⊤ 0 = k √ n k k Γ 2 → Γ 1 [ ∂ ν v | Γ 2 ] − � � K ⊤ T Γ 1 → Γ 1 − T Γ 1 → Γ 1 − [ u | Γ 1 ] . (14) k √ n k √ n k CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

BOUNDARY INTEGRAL EQUATIONS Derivation of fourth boundary integral equation Normal derivative of (4) and (5), D ∋ x → x ∈ Γ 2 , and jump relations: Γ 2 → Γ 2 [ ∂ ν u | Γ 2 ] + 1 Γ 1 → Γ 2 [ ∂ ν u | Γ 1 ] + K ⊤ 2 ∂ ν u | Γ 2 − T Γ 1 → Γ 2 ∂ ν u | Γ 2 = K ⊤ [ u | Γ 1 ] , (15) k k k Γ 2 → Γ 2 [ ∂ ν v | Γ 2 ] + 1 Γ 1 → Γ 2 [ ∂ ν v | Γ 1 ] + K ⊤ 2 ∂ ν v | Γ 2 − T Γ 1 → Γ 2 ∂ ν v | Γ 2 = K ⊤ k √ n k √ n [ v | Γ 1 ] . (16) k √ n Equations (15) and (16) can be rewritten as [ u | Γ 1 ] − 1 Γ 1 → Γ 2 [ ∂ ν u | Γ 1 ] + K ⊤ Γ 2 → Γ 2 [ ∂ ν u | Γ 2 ] − T Γ 1 → Γ 2 0 = K ⊤ 2 ∂ ν u | Γ 2 , (17) k k k [ v | Γ 1 ] − 1 Γ 1 → Γ 2 [ ∂ ν v | Γ 1 ] + K ⊤ Γ 2 → Γ 2 [ ∂ ν v | Γ 2 ] − T Γ 1 → Γ 2 0 = K ⊤ 2 ∂ ν v | Γ 2 . k √ n k √ n (18) k √ n CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld

BOUNDARY INTEGRAL EQUATIONS Derivation of fourth boundary integral equation Difference of (17) and (18), u | Γ 1 = v | Γ 1 and ∂ ν u | Γ 1 = ∂ ν v | Γ 1 : Γ 1 → Γ 2 − K ⊤ Γ 2 → Γ 2 [ ∂ ν u | Γ 2 ] � Γ 1 → Γ 2 � K ⊤ [ ∂ ν u | Γ 1 ] + K ⊤ 0 = k √ n k k [ u | Γ 1 ] − 1 Γ 2 → Γ 2 [ ∂ ν v | Γ 2 ] − � � T Γ 1 → Γ 2 − T Γ 1 → Γ 2 K ⊤ − 2 ∂ ν u | Γ 2 k √ n k √ n k 1 2 ∂ ν v | Γ 2 . + (19) CMMSE 2019 (MS 23) | July 2, 2019 Member of the Helmholtz Association Andreas Kleefeld