Convergence of Boundary Element Methods on Fractals Simon - PowerPoint PPT Presentation

B OUNDARY E LEMENT M ETHODS , O BERWOLFACH , 6 F EBRUARY 2020 Convergence of Boundary Element Methods on Fractals Simon Chandler-Wilde http://www.personal.reading.ac.uk/~sms03snc/ Joint work with D.P . Hewett (UCL), A. Moiola (Pavia), J. Besson (ENSTA)

Acoustic wave scattering by a planar screen u satisfies Helmholtz equation ∆ u + k 2 u = 0 , with wavenumber k > 0 . Scattering: incoming wave u i hits flat screen Γ and generates field u . Γ bounded subset of Γ ∞ := { x ∈ R n : x n = 0 } ∼ = R n − 1 , n = 2 , 3 ∆ u + k 2 u = 0 D := R n \ Γ x 3 x 2 u i ( x ) = e i k d · x Γ u = − u i x 1 u satisfies Sommerfeld radiation condition (SRC) at infinity � r ( 1 − n ) / 2 � (i.e. ∂ r u − iku = o uniformly as r = | x | → ∞ ). 2

Acoustic wave scattering by a planar screen u satisfies Helmholtz equation ∆ u + k 2 u = 0 , with wavenumber k > 0 . Scattering: incoming wave u i hits flat screen Γ and generates field u . Γ bounded subset of Γ ∞ := { x ∈ R n : x n = 0 } ∼ = R n − 1 , n = 2 , 3 ∆ u + k 2 u = 0 D := R n \ Γ x 3 x 2 u i ( x ) = e i k d · x Γ u = − u i x 1 u satisfies Sommerfeld radiation condition (SRC) at infinity � r ( 1 − n ) / 2 � (i.e. ∂ r u − iku = o uniformly as r = | x | → ∞ ). 2

Acoustic wave scattering by a planar screen u satisfies Helmholtz equation ∆ u + k 2 u = 0 , with wavenumber k > 0 . Scattering: incoming wave u i hits flat screen Γ and generates field u . Γ bounded subset of Γ ∞ := { x ∈ R n : x n = 0 } ∼ = R n − 1 , n = 2 , 3 ∆ u + k 2 u = 0 D := R n \ Γ x 3 x 2 u i ( x ) = e i k d · x Γ u = − u i x 1 u satisfies Sommerfeld radiation condition (SRC) at infinity � r ( 1 − n ) / 2 � (i.e. ∂ r u − iku = o uniformly as r = | x | → ∞ ). 2

Scattering by Lipschitz and rough screens u tot = u + u i Incident field is plane wave u i ( x ) = e i k d · x , | d | = 1 . Classical problem when Γ is open and Lipschitz. What happens for arbitrary (rougher than Lipschitz, e.g. fractal) Γ ? 3

Scattering by Lipschitz and rough screens u tot = u + u i Incident field is plane wave u i ( x ) = e i k d · x , | d | = 1 . Classical problem when Γ is open and Lipschitz. What happens for arbitrary (rougher than Lipschitz, e.g. fractal) Γ ? 3

Waves and fractals: applications Wideband fractal antennas (Figures from http://www.antenna-theory.com/antennas/fractal.php ) 4

Waves and fractals: applications Wideband fractal antennas (Figures from http://www.antenna-theory.com/antennas/fractal.php ) Scattering by ice crystals in atmospheric physics e.g. C. Westbrook Fractal apertures in laser optics e.g. J. Christian 4

Scattering by fractal screens · · · Lots of mathematical challenges: ◮ How to formulate well-posed BVPs? (What is the right function space setting? How to impose BCs?) ◮ Do solutions on prefractals converge to solutions on fractals? ◮ Do BEM solutions on prefractals converge? Ideas and analysis relevant to BEM for any BIE/ Ψ DO on fractals or other rough sets – e.g. fractional Laplacian on rough sets? Previous BEM computations on sequences of prefractals, e.g. Jones, Ma, Rokhlin 1994, Panagiotopoulos, Panagouli 1996, but no proof that these converge to right limit. 5

Scattering by fractal screens · · · Lots of mathematical challenges: ◮ How to formulate well-posed BVPs? (What is the right function space setting? How to impose BCs?) ◮ Do solutions on prefractals converge to solutions on fractals? ◮ Do BEM solutions on prefractals converge? Ideas and analysis relevant to BEM for any BIE/ Ψ DO on fractals or other rough sets – e.g. fractional Laplacian on rough sets? Previous BEM computations on sequences of prefractals, e.g. Jones, Ma, Rokhlin 1994, Panagiotopoulos, Panagouli 1996, but no proof that these converge to right limit. 5

Our method is: solve by piecewise constant BEM on sequence of prefractals : results for Cantor set Γ 1 and Re u h 1 6

Our method is: solve by piecewise constant BEM on sequence of prefractals : results for Cantor set Γ 2 and Re u h 2 6

Our method is: solve by piecewise constant BEM on sequence of prefractals : results for Cantor set Γ 3 and Re u h 3 6

Our method is: solve by piecewise constant BEM on sequence of prefractals : results for Cantor set Γ 4 and Re u h 4 6

Our method is: solve by piecewise constant BEM on sequence of prefractals : results for Cantor set Γ 5 and Re u h 5 6

Outline ◮ Sobolev spaces on rough sets ◮ BVPs and BIEs ◮ open screens ◮ compact screens ◮ Abstract convergence framework, using Mosco convergence ◮ Prefractal to fractal convergence ◮ Convergence of BEM on sequences of prefractals ◮ Numerical examples ◮ Cantor set ◮ Cantor dust: dependence on Hausdorff dimension ◮ Fractal apertures 7

Sobolev spaces on rough subsets of R n − 1 We need Sobolev spaces on Γ ⊂ R n − 1 . For s ∈ R let � � � u ( ξ ) | 2 d ξ < ∞ H s ( R n − 1 )= u ∈ S ∗ ( R n − 1 ) : � u � 2 R n − 1 ( 1 + | ξ | 2 ) s | ˆ H s ( R n − 1 ) := For Γ ⊂ R n − 1 open and F ⊂ R n − 1 closed define [M C L EAN ] H s (Γ) := { u | Γ : u ∈ H s ( R n − 1 ) } restriction H s ( R n − 1 ) � H s (Γ) := C ∞ 0 (Γ) closure H s F := { u ∈ H s ( R n − 1 ) : supp u ⊂ F } support When Γ is Lipschitz it holds that For general open Γ = ( H − s (Γ)) ∗ with equal norms H s (Γ) ∼ ◮ � ◮ � � H s (Γ) ∼ � ◮ s ∈ N ⇒ � u � 2 Γ | ∂ α u | 2 L IPSCHITZ ◮ × | α |≤ s ◮ � ( ∼ H s (Γ) = H s = H s 00 (Γ) , s ≥ 0 ) ◮ × IS Γ ◮ H ± 1 / 2 = { 0 } L UXURY ! ◮ × ∂ Γ ◮ { H s (Γ) } s ∈ R and { � H s (Γ) } s ∈ R ◮ × are interpolation scales. 8

Sobolev spaces on rough subsets of R n − 1 We need Sobolev spaces on Γ ⊂ R n − 1 . For s ∈ R let � � � u ( ξ ) | 2 d ξ < ∞ H s ( R n − 1 )= u ∈ S ∗ ( R n − 1 ) : � u � 2 R n − 1 ( 1 + | ξ | 2 ) s | ˆ H s ( R n − 1 ) := For Γ ⊂ R n − 1 open and F ⊂ R n − 1 closed define [M C L EAN ] H s (Γ) := { u | Γ : u ∈ H s ( R n − 1 ) } restriction H s ( R n − 1 ) � H s (Γ) := C ∞ 0 (Γ) closure H s F := { u ∈ H s ( R n − 1 ) : supp u ⊂ F } support When Γ is Lipschitz it holds that For general open Γ = ( H − s (Γ)) ∗ with equal norms H s (Γ) ∼ ◮ � ◮ � � H s (Γ) ∼ � ◮ s ∈ N ⇒ � u � 2 Γ | ∂ α u | 2 L IPSCHITZ ◮ × | α |≤ s ◮ � ( ∼ H s (Γ) = H s = H s 00 (Γ) , s ≥ 0 ) ◮ × IS Γ ◮ H ± 1 / 2 = { 0 } L UXURY ! ◮ × ∂ Γ ◮ { H s (Γ) } s ∈ R and { � H s (Γ) } s ∈ R ◮ × are interpolation scales. 8

Sobolev spaces on rough subsets of R n − 1 We need Sobolev spaces on Γ ⊂ R n − 1 . For s ∈ R let � � � u ( ξ ) | 2 d ξ < ∞ H s ( R n − 1 )= u ∈ S ∗ ( R n − 1 ) : � u � 2 R n − 1 ( 1 + | ξ | 2 ) s | ˆ H s ( R n − 1 ) := For Γ ⊂ R n − 1 open and F ⊂ R n − 1 closed define [M C L EAN ] H s (Γ) := { u | Γ : u ∈ H s ( R n − 1 ) } restriction H s ( R n − 1 ) � H s (Γ) := C ∞ 0 (Γ) closure H s F := { u ∈ H s ( R n − 1 ) : supp u ⊂ F } support When Γ is Lipschitz it holds that For general open Γ = ( H − s (Γ)) ∗ with equal norms H s (Γ) ∼ ◮ � ◮ � � H s (Γ) ∼ � ◮ s ∈ N ⇒ � u � 2 Γ | ∂ α u | 2 L IPSCHITZ ◮ × | α |≤ s ◮ � ( ∼ H s (Γ) = H s = H s 00 (Γ) , s ≥ 0 ) ◮ × IS Γ ◮ H ± 1 / 2 = { 0 } L UXURY ! ◮ × ∂ Γ ◮ { H s (Γ) } s ∈ R and { � H s (Γ) } s ∈ R ◮ × are interpolation scales. 8

Sobolev spaces on rough subsets of R n − 1 We need Sobolev spaces on Γ ⊂ R n − 1 . For s ∈ R let � � � u ( ξ ) | 2 d ξ < ∞ H s ( R n − 1 )= u ∈ S ∗ ( R n − 1 ) : � u � 2 R n − 1 ( 1 + | ξ | 2 ) s | ˆ H s ( R n − 1 ) := For Γ ⊂ R n − 1 open and F ⊂ R n − 1 closed define [M C L EAN ] H s (Γ) := { u | Γ : u ∈ H s ( R n − 1 ) } restriction H s ( R n − 1 ) � H s (Γ) := C ∞ 0 (Γ) closure H s F := { u ∈ H s ( R n − 1 ) : supp u ⊂ F } support When Γ is Lipschitz it holds that For general open Γ = ( H − s (Γ)) ∗ with equal norms H s (Γ) ∼ ◮ � ◮ � � H s (Γ) ∼ � ◮ s ∈ N ⇒ � u � 2 Γ | ∂ α u | 2 L IPSCHITZ ◮ × | α |≤ s ◮ � ( ∼ H s (Γ) = H s = H s 00 (Γ) , s ≥ 0 ) ◮ × IS Γ ◮ H ± 1 / 2 = { 0 } L UXURY ! ◮ × ∂ Γ ◮ { H s (Γ) } s ∈ R and { � H s (Γ) } s ∈ R ◮ × are interpolation scales. 8

BVPs for open and compact screens BVP D op (Γ) for open screens Let Γ ⊂ Γ ∞ be bounded & open. Given g ∈ H 1 / 2 (Γ) (for instance, g = − ( γ ± u i ) | Γ ), find u ∈ C 2 ( D ) ∩ W 1 , loc ( D ) satisfying ∆ u + k 2 u = 0 in D , ( γ ± u ) | Γ = g , Sommerfeld RC . γ ± = traces : W 1 ( R n ± ) → H 1 / 2 (Γ ∞ ) 9