A piecewise homogeneous world of waves Scattering of transient waves AR & FJS & TQ by piecewise homogeneous obstacles Notation Problem Approximation Alexander Rieder & Francisco–Javier Sayas Theory & Tianyu Qiu Estimates TDBIE Technical University of Vienna Full Department of Mathematical Sciences, University of Delaware discretization Rice University RICAM Workshop November 9, 2016
The team (different stages of 2016) A piecewise homogeneous world of waves AR & FJS & TQ Notation Problem Approximation Theory Estimates TDBIE Full discretization Work with Alex Rieder (visiting from TUW), strongly based on Tim Qiu’s with add-ons by Tom Brown
Contents, disclaimers, warnings A piecewise homogeneous world of waves Work in progress AR & FJS & TQ Acoustic waves on piecewise homogeneous materials — Galerkin BEM in space — RKCQ in time Notation Problem Don’t bother with the elastic case (trivial, trivial) Approximation Maxwell (I’m pretty, pretty, pretty sure it’s quite similar) Theory Our numerical experiments are just convergence lines, Estimates showing not entirely expected results TDBIE Full I’ll talk about analysis mainly discretization Apologies for the very mathy slides
Analysis? Really? A piecewise homogeneous world of waves AR & FJS & TQ Notation Problem Approximation Theory Estimates TDBIE Full discretization
Literature A piecewise homogeneous world of Single trace formulations (FD): Von Petersdorff, 1989 waves AR & FJS & Single and multi-trace formulations (FD): Claeys & TQ Hiptmair, 2013. Additional work by them and Notation Jerez-Hanckes Problem Time domain tools from theory of evolutionary Approximation equations (say you own a copy of Pazy’s book and you Theory understand what’s in it) Estimates TDBIE Results on RK for abstract evolutionary equations: Full Alonso-Mallo & Palencia, 2003. With sequels discretization RKCQ theory by Banjai, Lubich, & Melenk, 2011 The formulation can be used with other discretization methods (full Galerkin, etc)
A piecewise homogeneous world of waves AR & FJS & TQ Notation Problem NOTATION Approximation Theory AND SCATTERING PROBLEM Estimates TDBIE Full discretization
Geometric setting A piecewise Ω ℓ ( ℓ = 1 , . . . , L ) mutually disjoint bounded Lipschitz homogeneous world of domains (with possibly intersecting boundaries) waves AR & FJS & Surrounding domain (air or something) TQ L Notation � Ω 0 := R d \ Ω ℓ Problem Approximation ℓ = 1 Theory Skeleton Estimates L � TDBIE Γ := ∂ Ω ℓ Full discretization ℓ = 1 Piecewise constant physical parameters: κ, c : R d → ( 0 , ∞ ) κ | Ω ℓ ≡ κ ℓ > 0 , c | Ω ℓ ≡ c ℓ > 0 , ℓ = 0 , . . . , L
Scattering problem (scattered wave) Incident wave. u inc : R → H 1 A piecewise loc (Ω 0 ) (Extended by zero to homogeneous u inc .) the interior domains as � world of waves AR & FJS & c − 2 0 u inc ( t ) = κ 0 ∆ u inc ( t ) ∀ t ∈ R , TQ supp u inc ( t ) ⊂ Ω 0 ∀ t ≤ 0 . Notation Problem Scattered wave-field u : [ 0 , ∞ ) → H 1 ( R d \ Γ) . For alll t ≥ 0 Approximation Theory in R d \ Γ c − 2 ¨ Estimates u ( t ) = ∇ · ( κ ∇ u )( t ) TDBIE u inc ( t ) ∈ H 1 ( B ) u ( t ) − � Full discretization u inc ( t )) ∈ H ( div , B ) κ ∇ ( u ( t ) − � ˙ u ( 0 ) = 0 , u ( 0 ) = 0 (What’s B )
Traces on one boundary A piecewise homogeneous world of waves Interior, exterior, jump, average... AR & FJS & TQ } : H 1 ( R d \ ∂ Ω ℓ ) → H 1 / 2 ( ∂ Ω ℓ ) γ int ℓ , γ ext ℓ , � γ ℓ · � , { { γ ℓ ·} Notation Problem � γ ℓ u � := γ int ℓ u − γ ext } := 1 2 ( γ int ℓ u + γ ext ℓ u , { { γ ℓ u } ℓ u ) . Approximation Theory Normal traces... Estimates } : H ( div , R d \ ∂ Ω ℓ ) → H − 1 / 2 ( ∂ Ω ℓ ) γ int ν,ℓ , γ ext TDBIE ν,ℓ , � γ ν,ℓ · � , { { γ ν,ℓ ·} Full discretization All normals point out
Packing up spaces and traces A piecewise homogeneous Collecting fields in product spaces world of waves L L � � AR & FJS & H div := H ( div , R d \ ∂ Ω ℓ ) , H 1 ( R d \ ∂ Ω ℓ ) , TQ H := Notation ℓ = 0 ℓ = 0 Problem L L � � H − 1 / 2 := H − 1 / 2 ( ∂ Ω ℓ ) , H 1 / 2 := H 1 / 2 ( ∂ Ω ℓ ) , Approximation Theory ℓ = 0 ℓ = 0 Estimates TDBIE γ int , γ ext , � γ · � , { } : H → H 1 / 2 , { γ ·} Full discretization } : H div → H − 1 / 2 γ int ν , γ ext ν , � γ ν · � , { { γ ν ·} The field inside Ω ℓ will be extended by zero outside. Why?
A piecewise homogeneous world of waves AR & FJS & TQ Notation Problem ANOTHER WAY Approximation Theory OF WRITING THE PROBLEM Estimates TDBIE Full discretization
Enforcing interface continuity A piecewise homogeneous world of waves Y : = { ( γ int ℓ u ) L ℓ = 0 : u ∈ H 1 ( R d ) } AR & FJS & TQ = { φ ∈ H 1 / 2 : ∃ u ∈ H 1 ( R d ) , φ = γ int u } , Notation X : = { ( γ int ν,ℓ v ) L ℓ = 0 : v ∈ H ( div , R d ) } Problem = { λ ∈ H − 1 / 2 : ∃ v ∈ H ( div , R d ) , λ = γ int Approximation ν v } . Theory Estimates Lemma (Restricting and gluing — odd conditions) TDBIE Full Let U ∈ H satisfy � γ U � ∈ Y and γ ext U ∈ Y and let discretization u : R d → R be defined by u | Ω ℓ := u ℓ | Ω ℓ . Then u ∈ H 1 ( R d ) . Similarly, let V ∈ H div satisfy � γ ν V � ∈ X and γ ext ν V ∈ X and let v : R d → R d be defined by v | Ω ℓ := v ℓ | Ω ℓ . Then v ∈ H ( div , R d ) .
A trivial example: disjoint homogeneous obstacles A piecewise homogeneous world of waves L � H ± 1 / 2 ( ∂ Ω 0 ) ≡ H ± 1 / 2 ( ∂ Ω ℓ ) AR & FJS & TQ ℓ = 1 Notation Problem L � Approximation H 1 / 2 ( ∂ Ω ℓ ) ∋ ( φ 1 , . . . , φ L ) Theory ℓ = 1 Estimates ← → (( φ 1 , . . . , φ L ) , φ 1 , . . . , φ L ) ∈ Y TDBIE Full L � discretization H − 1 / 2 ( ∂ Ω ℓ ) ∋ ( λ 1 , . . . , λ L ) ℓ = 1 ← → (( − λ 1 , . . . , − λ L ) , λ 1 , . . . , λ L ) ∈ X
An observation and more notation A piecewise homogeneous world of waves The spaces X and Y are respective polar to each other, i.e., AR & FJS & TQ Y ◦ = { µ ∈ H − 1 / 2 : � φ , µ � Γ = 0 ∀ φ ∈ Y} = X Notation Problem and Approximation Theory X ◦ = { φ ∈ H 1 / 2 : � φ , µ � Γ = 0 ∀ µ ∈ X} = Y Estimates TDBIE Full discretization I call’em polar sets because I cannot pronounce annihilator
The reformulation: with pieces Data A piecewise homogeneous world of β 0 := ( γ int β 1 := ( κ 0 γ int ν, 0 ∇ ∂ − 1 0 u inc , 0 , . . . , 0 ) , u inc , 0 , . . . , 0 ) waves t AR & FJS & TQ Unknowns Notation U = ( u ℓ ) L V = ( v ℓ ) L ℓ = 0 : [ 0 , ∞ ) → H div ℓ = 0 : [ 0 , ∞ ) → H , Problem Approximation First order system ( ℓ = 0 , . . . , L ) Theory Estimates u ℓ ( t ) = c 2 ˙ ˙ ℓ ∇ · v ℓ ( t ) , v ℓ ( t ) = κ ℓ ∇ u ℓ ( t ) , t ≥ 0 TDBIE Full Transmission conditions ( t ≥ 0) discretization � γ U � ( t ) + β 0 ( t ) ∈ Y � γ ν V � ( t ) + β 1 ( t ) ∈ X γ ext U ( t ) ∈ X ◦ = Y γ ext ν V ( t ) ∈ Y ◦ Vanishing initial conditions
The formulation: all together and why it works Data: U inc := ( � u inc , 0 , . . . , 0 ) A piecewise homogeneous Unknowns world of waves AR & FJS & V : [ 0 , ∞ ) → H div U : [ 0 , ∞ ) → H , TQ Notation First order system Problem Approximation ˙ ˙ U ( t ) = T c 2 ∇ · V ( t ) , V ( t ) = T κ ∇ U ( t ) , t ≥ 0 Theory Estimates Transmission conditions TDBIE Full � γ U + U inc � ( t ) ∈ Y discretization γ ext ( U + U inc )( t ) ∈ Y � γ ν T κ ∇ ( U + U inc ) � ( t ) ∈ X γ ext ν T κ ∇ ( U + U inc )( t ) ∈ X
A piecewise homogeneous world of waves AR & FJS & TQ Notation Problem AN APPROXIMATION Approximation Theory TO THE PROBLEM Estimates TDBIE Full discretization
Discrete spaces A piecewise homogeneous world of Choose two closed (finite dimensional) subspaces waves AR & FJS & TQ X h ⊂ X Y h ⊂ Y Notation Look at their polar sets Problem Approximation h := { φ ∈ H 1 / 2 : � φ , µ � Γ = 0 X ◦ ∀ µ ∈ X h } , Theory Estimates h := { λ ∈ H − 1 / 2 : � ψ , λ � Γ = 0 Y ◦ ∀ ψ ∈ Y h } . TDBIE Full Note that discretization X h ⊂ X = Y ◦ ⊂ Y ◦ Y h ⊂ Y = X ◦ ⊂ X 0 h . h
Continuous vs semidiscrete (1) A piecewise homogeneous world of Unknowns waves AR & FJS & V : [ 0 , ∞ ) → H div U : [ 0 , ∞ ) → H , TQ Notation First order system Problem Approximation ˙ ˙ U ( t ) = T c 2 ∇ · V ( t ) , V ( t ) = T κ ∇ U ( t ) , ∀ t , t ≥ 0 Theory Estimates Transmission conditions TDBIE Full � γ U � ( t ) + β 0 ( t ) ∈ Y � γ ν V � ( t ) + β 1 ( t ) ∈ X discretization γ ext U ( t ) ∈ X ◦ ν V ( t ) ∈ Y ◦ γ ext Vanishing initial conditions
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