Explicit bounds for electromagnetic transmission problems Andrea - PowerPoint PPT Presentation

MAFELAP , B RUNEL U NIVERSITY , 18–21 J UNE 2019 Explicit bounds for electromagnetic transmission problems Andrea Moiola Joint work with E.A. Spence (Bath)

Maxwell equations in heterogeneous media Given: ◮ wavenumber k > 0 ◮ sources J , K ∈ H ( div 0 ; R 3 ) , compactly supported ◮ ǫ 0 , µ 0 > 0 ◮ ǫ, µ ∈ L ∞ ( R 3 ; SPD ) such that � � is bounded and Lipschitz Ω i := int supp( ǫ − ǫ 0 I ) ∪ supp( µ − µ 0 I ) Find E , H ∈ H loc ( curl ; R 3 ) such that in R 3 , i k ǫ E + ∇ × H = J ǫ, µ in R 3 , − i k µ H + ∇ × E = K Ω i ( E , H ) satisfy Silver–Müller radiation condition ǫ = ǫ 0 |√ ǫ 0 E − √ µ 0 H × | x | | = O | x |→∞ ( | x | − 2 ) . x µ = µ 0 Special case: “transmission problem”, i.e. homogeneous scatterer � � in Ω i ǫ i µ i 0 < ǫ i , ǫ 0 , µ i , µ 0 constant . ǫ = µ = in Ω o := R 3 \ Ω i ǫ 0 µ 0 2

Wave scattering The example we have in mind is incident wave E Inc , H Inc hitting Ω i : → BVP with data J = i k 2 ( ǫ 0 − ǫ ) E Inc , supported on Ω i : K = i k 2 ( µ − µ 0 ) H Inc . Total field Scattered field Incoming field � ǫ 0 A e i k √ ǫ 0 µ 0 x · d E Inc = µ 0 E + E Inc E H Inc = d × A e i k √ ǫ 0 µ 0 x · d H + H Inc H datum BVP solution physical field 3

Goal and motivation If ǫ, µ are sufficiently regular then the problem is well-posed. From Fredholm theory we have � E � J � �� � �� � � � � ≤ C � � � � H K � � � � Ω i / o Ω i / o Goal: find out how C = C ( k , ǫ, µ ) depends on k , ǫ and µ . Why? In FEM & BEM analysis and in UQ for time-harmonic problems, explicit parameter dependence allows to control: ◮ Quasi-optimality & pollution effect ◮ Gmres iteration numbers ◮ Matrix compression ◮ hp -FEM & BEM (Melenk–Sauter) ◮ Shape differentiation & uncertainty quantification ◮ . . . 4

Who cares? L AFONTAINE , S PENCE , W UNSCH , arXiv 2019: (Helmholtz) 5

What about Helmholtz? [M. & S. 2019] Simplest heterogeneous Helmholtz problem: find u ∈ H 1 loc ( R d ) s.t. � ∆ u + k 2 n u = f constant in Ω i , in R d n i f ∈ L 2 ( R d ) , n = in Ω o . +Sommerfeld radiation c. 1 ◮ If 0 < n i < 1 , Ω i star-shaped Ω i ∪ supp f ⊂ B R � � � 2 � � √ n u 2 R + d − 1 4 R 2 + 1 L 2 ( B R ) + k 2 � � 2 � �∇ u � 2 � f � 2 L 2 ( B R ) ≤ L 2 ( B R ) n i k Fully explicit, k -independent, shape-robust estimate. (For d = 2 it implies bounds for Maxwell TE/TM modes.) ◮ If n i > 1 , Ω i strictly convex & C ∞ : superalgebraic blow up in k , quasi-resonances, ray trapping, creeping waves. . . Dependence on parameters is complicated! Monotonicity of n & shape of Ω i are crucial. 6

Wavenumber-explicit bounds: a bit of history ◮ M ORAWETZ 1960 S /70 S : introduced main tools (multipliers) ◮ M ELENK 1995: 1st k -explicit bound for Helmholtz, bdd dom. ◮ C HANDLER -W ILDE , M ONK 2008: unbounded domains ◮ H IPTMAIR , M OIOLA , P ERUGIA 2011: Maxwell, bdd dom. homogeneous coeff. heterogeneous coeff. ◮ M OIOLA , S PENCE 2019: Helmholtz & piecewise-constant n ◮ G RAHAM , P EMBERY , S PENCE 2019: Helmholtz & general coeff. ◮ V ERFÜRTH 2019: Maxwell & impedance Plenty of other related contributions exist! B ARUCQ , C HAUMONT -F RELET , F ENG , H ETMANIUK , L ORTON , P ETERSEIM , S AUTER , T ORRES , W IENERS &W OHLMUTH , [your name here], . . . Our goal: extend [G RAHAM , P EMBERY , S PENCE 2019] to Maxwell eq.s. 7

Bound #1: transmission problem Single homogeneous scatterer: � � in Ω i in Ω i ǫ i µ i 0 < ǫ i , ǫ 0 , µ i , µ 0 constant . ǫ = , µ = in Ω o in Ω o ǫ 0 µ 0 If ǫ i ≤ ǫ 0 , µ i ≤ µ 0 , Ω i star-shaped , Ω i ∪ supp J ∪ supp K ⊂ B R , then � ǫ 0 �� + µ 0 � ǫ i � E � 2 B R + µ i � H � 2 ǫ 0 � K � 2 B R + µ 0 � J � 2 4 R 2 ≤ . B R B R ǫ i µ i �·� B R = �·� L 2 ( B R ) Equivalent to wavenumber-independent H ( curl ; B R ) bound for E . If ǫ i is (constant) SPD matrix, same holds if max eig ( ǫ i ) ≤ ǫ 0 and with ǫ i substituted by min eig ( ǫ i ) in the bound. Same for µ i . 8

Bound #2: more general ǫ, µ Assume ǫ, µ ∈ W 1 , ∞ (Ω i ; SPD ) , Ω i Lipschitz, ◮ Ω i star-shaped ◮ � ǫ i � L ∞ ( ∂ Ω i ) ≤ ǫ 0 , � µ i � L ∞ ( ∂ Ω i ) ≤ µ 0 , i.e. jumps are “upwards” on ∂ Ω i � � � � ◮ ǫ ∗ := ess inf x ∈ Ω i ǫ + ( x · ∇ ) ǫ > 0 , µ ∗ := ess inf x ∈ Ω i µ + ( x · ∇ ) µ > 0 “weak monotonicity” in radial direction, avoid trapping of rays ◮ “extra regularity” ( E , H ∈ H 1 (Ω i ∪ Ω o ) 3 or ǫ, µ ∈ C 1 (Ω i ) or W 1 , ∞ ( R 3 ) ) Then we have explicit wavenumber-independent bound: ǫ ∗ � E � 2 B R + µ ∗ � H � 2 B R � � ǫ � 2 � � µ � 2 � � + ǫ 0 µ 0 + ǫ 0 µ 0 L ∞ ( B R ) � K � 2 L ∞ ( B R ) � J � 2 ≤ 4 R 2 B R + 4 R 2 B R . ǫ ∗ µ ∗ µ ∗ ǫ ∗ Expect (from Helmholtz analogy) superalgebraic blow up in k if any of the first 3 assumptions is lifted. Similar results when R 3 is truncated with impedance BCs. 9

How our bound was obtained First consider smooth case E , H ∈ C 1 ( R 3 ; C 3 ) . (i) Multiply the 2 PDEs by the “test fields” (Morawetz multipliers) ( ǫ E × x + R √ ǫµ H ) ( µ H × x − R √ ǫµ E ) in B R ⊃ Ω i , & ( ǫ 0 E × x + r √ ǫ 0 µ 0 H ) ( µ 0 H × x − r √ ǫ 0 µ 0 E ) in R 3 \ B R , & R 3 \ B R , (ii) integrate by parts in Ω i , and B R \ Ω i (iii) sum 3 contributions, (iv) take ℜ eal part, (v) have fun! � � � � � Using PDEs & E · ǫ + ( x · ∇ ) ǫ E + H · µ + ( x · ∇ ) µ H ∇· [ ǫ E ]= ∇· [ µ H ]= 0 B R � �� � � �� � ≥ ǫ ∗ by assumpt. ≥ µ ∗ by assumpt. � � K · ( ǫ E × x + √ ǫ 0 µ 0 R H ) + J · ( µ H × x − √ ǫ 0 µ 0 R E ) � = 2 ℜ B R � � [ terms from IBP ] [ terms from IBP ] + + � �� � � �� � ∂ Ω i ∂ B R ≤ 0 by ǫ i � ǫ 0 ,µ i � µ 0 , ≤ 0 by S–M radiation c. n · x ≥ 0 , [ [ E T , H T , ( ǫ E ) N , ( µ H ) N ] ]= 0 Conclude by Cauchy–Schwarz. 10

Rough coefficients, regularity and density Proof in previous slide only uses elementary results if E , H ∈ C 1 ( R 3 ; C 3 ) . For general case we need density of inclusion � � C ∞ ( D ) 3 ⊂ v ∈ H ( curl ; D ) , ∇· [ A v ] ∈ L 2 ( D ) , A v · ˆ n ∈ L 2 ( ∂ D ) , v T ∈ L 2 T ( ∂ D ) for A = ǫ & A = µ , D Lipschitz bdd. If A ∈ C 1 (Ω i ; SPD ) , this density is non-trivial but follows from regularity results for layer potentials on manifolds [M ITREA , T AYLOR 1999]. ◮ Equivalent step for Helmholtz was much simpler. ◮ Constant scalar ǫ & µ : density proved in C OSTABEL , D AUGE 1998. ◮ If E , H ∈ H 1 loc ( R 3 ; C 3 ) then no density is needed. E.g. ensured if ǫ, µ ∈ W 1 , ∞ ( R 3 ; SPD ) (no jumps). ◮ What about A ∈ W 1 , ∞ (Ω i ; SPD ) ? 11

Summary Time-harmonic Maxwell eq.s in R 3 with heterogeneous inclusion: ◮ fully explicit bounds on � E � H ( curl , B R ) if ǫ, µ “radially growing” ◮ also for impedance BVPs in star-shaped domains ◮ extends Helmholtz results from [G RAHAM , P EMBERY , S PENCE 2019] Some open questions: ◮ resonance-free strip in complex k plane? ◮ presence of quasi-resonances blow up for “wrong” coefficients? ◮ rougher ( W 1 , ∞ (Ω i ; SPD ) , L ∞ ) coefficients? ◮ relation with shape-differentiation and UQ? Preprint coming soon. . . Thank you! 12

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