Hardy space infinite elements for exterior Maxwell problems L. - - PowerPoint PPT Presentation

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

Hardy space infinite elements for exterior Maxwell problems

• L. Nannen, T. Hohage, A. Schädle, J. Schöberl

Linz, November 2011

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

electromagnetic scattering problem

• Ω

curl u · curl v − εκ2 u · vdx = g(v) κ = 2.7684 κ = 2.8

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

electromagnetic resonance problem

Definition (resonance problem)

Let (κ2, u) ∈ C × Hloc(curl, Ω) \ {0} with ℜ(κ) > 0 be a solution to the eigenvalue problem

• Ω

curl u · curl v dx = κ2

• Ω

ε u · v dx + BC + RC. Then we call κ a resonance, ℜ(κ) > 0 the resonance frequency and Q :=

ℜ(κ) 2|ℑ(κ)| the quality factor.

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

• utline

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

first order ABC

Silver-Müller radiation condition: lim

|x|→∞ |x|

• curl u × x

|x| − iκu

• = 0

first order ABC: curl u × ν − iκu = 0

• n Γ := ∂Ba(0)

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

first order ABC

Silver-Müller radiation condition: lim

|x|→∞ |x|

• curl u × x

|x| − iκu

• = 0

first order ABC: curl u × ν − iκu = 0

• n Γ := ∂Ba(0)

pros: nothing to implement, no additional dofs cons: poor accuracy, wrong solutions for resonance problems

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

first order ABC

Silver-Müller radiation condition: lim

|x|→∞ |x|

• curl u × x

|x| − iκu

• = 0

first order ABC: curl u × ν − iκu = 0

• n Γ := ∂Ba(0)

pros: nothing to implement, no additional dofs cons: poor accuracy, wrong solutions for resonance problems reason: factor exp(±iκ|x|) in the asymptotic behaviour of u

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

BEM

Stratton-Chu formula: u(x) = curl

• Γ

ν(y) × u(y)Φ(x, y)ds(y) − 1 κ2 curl curl

• Γ

ν(y) × curl u(y)Φ(x, y)ds(y), x ∈ R3 \ Ba(0) with Φ(x, y) = 1 4π exp(iκ|x − y|) |x − y|

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

BEM

Stratton-Chu formula: u(x) = curl

• Γ

ν(y) × u(y)Φ(x, y)ds(y) − 1 κ2 curl curl

• Γ

ν(y) × curl u(y)Φ(x, y)ds(y), x ∈ R3 \ Ba(0) with Φ(x, y) = 1 4π exp(iκ|x − y|) |x − y| pros: boundary integrals, fast convergence, non-convex Γ cons: dependence on κ, Green’s function needed

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

classical infinite elements

tensor product ansatz: u(|x|, ˆ x) =

N

• l=0

ψl(|x|) el(ˆ x) +

N

• l=1

αl(|x|) gl(ˆ x)ˆ x, |x| ≥ 1, with

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

classical infinite elements

tensor product ansatz: u(|x|, ˆ x) =

N

• l=0

ψl(|x|) el(ˆ x) +

N

• l=1

αl(|x|) gl(ˆ x)ˆ x, |x| ≥ 1, with ψ0(r) :=1 r exp(iκ(r − 1)), ψl(r) := 1 r l+1 − 1 r

• exp(iκ(r − 1)),

α(r) := 1 r l+1 exp(iκ(r − 1)).

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

classical infinite elements

tensor product ansatz: u(|x|, ˆ x) =

N

• l=0

ψl(|x|) el(ˆ x) +

N

• l=1

αl(|x|) gl(ˆ x)ˆ x, |x| ≥ 1, with ψ0(r) :=1 r exp(iκ(r − 1)), ψl(r) := 1 r l+1 − 1 r

• exp(iκ(r − 1)),

α(r) := 1 r l+1 exp(iκ(r − 1)). pros: fast convergence in |x| cons: dependence on κ, complicated theory/ implementation

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

classical infinite elements

tensor product ansatz: u(|x|, ˆ x) =

N

• l=0

ψl(|x|) el(ˆ x) +

N

• l=1

αl(|x|) gl(ˆ x)ˆ x, |x| ≥ 1, with ψ0(r) :=1 r exp(iκ(r − 1)), ψl(r) := 1 r l+1 − 1 r

• exp(iκ(r − 1)),

α(r) := 1 r l+1 exp(iκ(r − 1)). pros: fast convergence in |x| cons: dependence on κ, complicated theory/ implementation Demkowicz & Pal, An infinite element for Maxwell’s equations , Computer Methods in Applied Mechanics and Engineering, 1998.

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

complex scaling

unisotropic damping: exp(iκ|x|) − → exp(iκσ|x|) Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Physics Reports, 1998. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J.Comput.Phy.,1994.

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

complex scaling

unisotropic damping: exp(iκ|x|) − → exp(iκσ|x|) Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Physics Reports, 1998. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J.Comput.Phy.,1994. pros: simple to implement, generalized linear eigenvalue problem, well known cons: many parameters to chose, artificial resonances

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

Helmholtz equation in 1d

Helmholtz equation: −u′′(r) − κ2u(r) = 0

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

Helmholtz equation in 1d

Helmholtz equation: −u′′(r) − κ2u(r) = 0 Laplace transformation: ˆ u(s) := (Lu)(s) = ∞ e−sru(r)dr , ℜ(s) > 0 u(r) = C1e+iκr + C2e−iκr L

• ˆ

u(s) = C1 s − iκ + C2 s + iκ.

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

Helmholtz equation in 1d

Helmholtz equation: −u′′(r) − κ2u(r) = 0 Laplace transformation: ˆ u(s) := (Lu)(s) = ∞ e−sru(r)dr , ℜ(s) > 0 u(r) = C1e+iκr + C2e−iκr L

• ˆ

u(s) = C1 s − iκ + C2 s + iκ. ℜ(s) ℑ(s) iκ −iκ incoming

• utgoing

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

Möbius transformation

ϕ(z) = iκ0 z+1

z−1 iκ0

Definition (Hardy space H+(S1))

Let S1 := {z | |z| = 1}. Then F ∈ H+(S1) iff

• F ∈ L2(S1) and
• L2-boundary value of a holomorphic function in

D := {z | |z| < 1}.

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

pole condition

Definition

u is outgoing, iff MLu ∈ H+(S1).

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

pole condition

Definition

u is outgoing, iff MLu ∈ H+(S1). Schmidt & Deuflhard, Discrete Transparent Boundary Conditions for the Numerical Solution of Fresnel’s Equation, Computers Math. Applic., 1995. Hohage & Schmidt & Zschiedrich, Solving time-harmonic scattering problems based on the pole condition. I. Theory, SIAM J. Math. Anal., 2003. Hohage & Nannen, Hardy space infinite elements for scattering and resonance problems, SIAM J. Numer. Anal., 2009.

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

Hardy space method in 1d

classical formulation: −u′′(r) − κ2u(r) = 0 in [0, ∞) u′(0) = 1, MLu ∈ H+(S1).

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

Hardy space method in 1d

classical formulation: −u′′(r) − κ2u(r) = 0 in [0, ∞) u′(0) = 1, MLu ∈ H+(S1). variational formulation: ∞

• u′v′ − κ2uv
• dr = v(0),

MLu ∈ H+(S1).

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

Hardy space method in 1d

classical formulation: −u′′(r) − κ2u(r) = 0 in [0, ∞) u′(0) = 1, MLu ∈ H+(S1). variational formulation: ∞

• u′v′ − κ2uv
• dr = v(0),

MLu ∈ H+(S1). transformation in the Hardy space H+(S1): −iκ0 π

• S1
• MLu′

(z)

• MLv′

(z)|dz| + κ2iκ0 π

• S1 (MLu) (z) (MLv) (z)|dz| = v(0).

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

basis functions

basis functions in H+(S1): (MLu)(z) ≈ 1 2iκ0  u0 + (z − 1)

N

• j=0

αjzj  

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

basis functions

basis functions in H+(S1): (MLu)(z) ≈ 1 2iκ0  u0 + (z − 1)

N

• j=0

αjzj   basis functions in space: u(r) ≈ eiκ0r  u0 +

N

• j=0

αj

j

• k=0

j k (2iκ0r)k+1 (k + 1)!  

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

basis functions

basis functions in H+(S1): (MLu)(z) ≈ 1 2iκ0  u0 + (z − 1)

N

• j=0

αjzj   basis functions in space: u(r) ≈ eiκ0r  u0 +

N

• j=0

αj

j

• k=0

j k (2iκ0r)k+1 (k + 1)!   derivative: (MLu′)(z) ≈ 1 2  u0 + (z + 1)

N

• j=0

αjzj  

sequence

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

system of linear equations:

variational formulation for MLu ∈ H+(S1): −iκ0 π

• S1
• MLu′

(z)

• MLv′

(z)|dz| + κ2iκ0 π

• S1 (MLu) (z) (MLv) (z)|dz| = v(0),

MLv ∈ H+(S1). system of linear equations:

             −2iκ0        1 1 1 2 1 ... ... ... 1 2 1 1 2        + κ2 2i

κ0

       1 −1 −1 2 −1 ... ... ... −1 2 −1 −1 2                            u0 α0 α1 . . . αN        =      1 . . .     

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

deRham diagram

H1(Ω)

∇

− → H(curl, Ω)

curl

− → H(div, Ω)

div

− → L2(Ω) πW  

• πV

 

• πQ

 

• πX

 

• Wh

∇

− → Vh

curl

− → Qh

div

− → Xh

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

deRham diagram

H1(Ω)

∇

− → H(curl, Ω)

curl

− → H(div, Ω)

div

− → L2(Ω) πW  

• πV

 

• πQ

 

• πX

 

• Wh

∇

− → Vh

curl

− → Qh

div

− → Xh motivation: curl ∇u = 0 ⇒ (0, ∇u) is eigenpair

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

deRham diagram

H1(Ω)

∇

− → H(curl, Ω)

curl

− → H(div, Ω)

div

− → L2(Ω) πW  

• πV

 

• πQ

 

• πX

 

• Wh

∇

− → Vh

curl

− → Qh

div

− → Xh motivation: curl ∇u = 0 ⇒ (0, ∇u) is eigenpair For uh = πWu it holds curl ∇uh = curl ∇πWu = πQ curl ∇u = 0

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

deRham diagram

H1(Ω)

∇

− → H(curl, Ω)

curl

− → H(div, Ω)

div

− → L2(Ω) πW  

• πV

 

• πQ

 

• πX

 

• Wh

∇

− → Vh

curl

− → Qh

div

− → Xh motivation: curl ∇u = 0 ⇒ (0, ∇u) is eigenpair For uh = πWu it holds curl ∇uh = curl ∇πWu = πQ curl ∇u = 0 ⇒ (0, ∇uh) is discrete eigenpair

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

generalized radial coordinates

• domain splitting:

Ω = Ωint ∪ Ωext

• tetrahedral finite elements for Ωint

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

generalized radial coordinates

• domain splitting:

Ω = Ωint ∪ Ωext

• tetrahedral finite elements for Ωint
• generalized radial coordinates for Ωext:

y z x ˆ x ξ

F V1 V3 V2 V2 V3 V1 V0

F(ξ, ˆ x) := ˆ x + ξ(ˆ x − V0)

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

sequences on the prism

surface sequence on the surface triangle T: H1(T)

∇ˆ

x

− → H(curl, T)

ν×∇ˆ

x·

− → L2(T) πWT  

• πVT

 

• πXT

 

• WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT

ˆ x ξ

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

sequences on the prism

surface sequence on the surface triangle T: H1(T)

∇ˆ

x

− → H(curl, T)

ν×∇ˆ

x·

− → L2(T) πWT  

• πVT

 

• πXT

 

• WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT radial sequence: H1(R+)

∂ξ

− → L2(R+)

ˆ x ξ

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

sequences on the prism

surface sequence on the surface triangle T: H1(T)

∇ˆ

x

− → H(curl, T)

ν×∇ˆ

x·

− → L2(T) πWT  

• πVT

 

• πXT

 

• WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT Hardy space sequence: H+(S1)

ˆ ∂ξ

− → H+(S1) πWξ  

• πW ′

ξ

 

• Wξ

ˆ ∂ξ

− → W ′

ξ

ˆ x ξ

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

sequences on the prism

surface sequence on the surface triangle T: H1(T)

∇ˆ

x

− → H(curl, T)

ν×∇ˆ

x·

− → L2(T) πWT  

• πVT

 

• πXT

 

• WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT Hardy space sequence: H+(S1)

ˆ ∂ξ

− → H+(S1) πWξ  

• πW ′

ξ

 

• Wξ := span{Ψ−1, ..., ΨN}

ˆ ∂ξ

− → W ′

ξ := span{ψ−1, ..., ψN }

ˆ x ξ

Ψ−1(z) :=

1 2iκ0 , Ψj(z) := z−1 2iκ0 zj, ψ−1(z) := 1 2, ψj(z) := z+1 2 zj

1D

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

tensor product elements

R+\T : WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT Wξ Wξ ⊗ WT − → Wξ ⊗ VT − → Wξ ⊗ XT ˆ ∂ξ ↓ ↓ ↓ ↓ W ′

ξ

W ′

ξ ⊗ WT

− → W ′

ξ ⊗ VT

− → W ′

ξ ⊗ XT

ˆ x ξ

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

tensor product elements

R+\T : WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT Wξ Wξ ⊗ WT − → Wξ ⊗ VT − → Wξ ⊗ XT ˆ ∂ξ ↓ ↓ ↓ ↓ W ′

ξ

W ′

ξ ⊗ WT

− → W ′

ξ ⊗ VT

− → W ′

ξ ⊗ XT

ˆ x ξ

tensor product sequence: Wξ ⊗ WT

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

tensor product elements

R+\T : WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT Wξ Wξ ⊗ WT − → Wξ ⊗ VT − → Wξ ⊗ XT ˆ ∂ξ ↓ ↓ ↓ ↓ W ′

ξ

W ′

ξ ⊗ WT

− → W ′

ξ ⊗ VT

− → W ′

ξ ⊗ XT

ˆ x ξ

tensor product sequence: Wξ ⊗ WT → Wξ ⊗ VT ⊕ W ′

ξ ⊗ WT

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

tensor product elements

R+\T : WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT Wξ Wξ ⊗ WT − → Wξ ⊗ VT − → Wξ ⊗ XT ˆ ∂ξ ↓ ↓ ↓ ↓ W ′

ξ

W ′

ξ ⊗ WT

− → W ′

ξ ⊗ VT

− → W ′

ξ ⊗ XT

ˆ x ξ

tensor product sequence: Wξ ⊗ WT → Wξ ⊗ VT ⊕ W ′

ξ ⊗ WT → Wξ ⊗ XT ⊕ W ′ ξ ⊗ VT

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

tensor product elements

R+\T : WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT Wξ Wξ ⊗ WT − → Wξ ⊗ VT − → Wξ ⊗ XT ˆ ∂ξ ↓ ↓ ↓ ↓ W ′

ξ

W ′

ξ ⊗ WT

− → W ′

ξ ⊗ VT

− → W ′

ξ ⊗ XT

ˆ x ξ

tensor product sequence: Wξ ⊗ WT → Wξ ⊗ VT ⊕ W ′

ξ ⊗ WT → Wξ ⊗ XT ⊕ W ′ ξ ⊗ VT → W ′ ξ ⊗ XT

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

tensor product elements

R+\T : WT

∇ˆ

x

− → VT

ν×∇ˆ

x·

− → XT Wξ Wξ ⊗ WT − → Wξ ⊗ VT − → Wξ ⊗ XT ˆ ∂ξ ↓ ↓ ↓ ↓ W ′

ξ

W ′

ξ ⊗ WT

− → W ′

ξ ⊗ VT

− → W ′

ξ ⊗ XT

ˆ x ξ

tensor product sequence: Wξ ⊗ WT → Wξ ⊗ VT ⊕ W ′

ξ ⊗ WT → Wξ ⊗ XT ⊕ W ′ ξ ⊗ VT → W ′ ξ ⊗ XT

Nannen, Hohage, Schädle & Schöberl, High order Curl-conforming Hardy space infinite elements for exterior Maxwell problems, arXiv:1103.2288v1.

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

basis functions in WK

WK := Wξ ⊗ WT

=

Ψ−1 Ψj wV2 wE23

l

⊗

wT

l

Ψ−1 ⊗ wV2 Ψ−1 ⊗ wE23

l

Ψj ⊗ wV2 Ψ−1 ⊗ wT

l

Ψj ⊗ wT

l

Ψj ⊗ wE23

l

wV3 wV1 19

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

basis functions in VK

VK := Wξ ⊗ VT ⊕ W ′

ξ ⊗ WT

=

⊗

Wξ ⊗ VT

=

⊗

W ′

ξ ⊗ WT

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

basis functions in VK

VK := Wξ ⊗ VT ⊕ W ′

ξ ⊗ WT

=

⊗

Wξ ⊗ VT

=

⊗

W ′

ξ ⊗ WT

conclusion: combinations of standard 2-dimensional elements with non-standard infinite elements

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

convergence of Hardy space method

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

electromagnetic resonance problem

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

electromagnetic scattering problem

• Ω

curl u · curl v − εκ2 u · vdx = g(v) κ = 2.7684 κ = 2.8

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

GaAs cavity

2 4 6 8 10 12 14 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

N

• rel. error
• rder 2
• rder 3
• rder 4
• rder 5
• rder 6

Karl et al., Reversed pyramids as novel optical micro-cavities , Superlattices and Microstructures, 2010.

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

summary

numerical results:

• acustics, electromagnetics, elastics
• inhomogeneous exterior domains
• arbitrary boundaries
• exponential convergence
• time-depending problems (A.Schädle)

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motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics

summary

numerical results:

• acustics, electromagnetics, elastics
• inhomogeneous exterior domains
• arbitrary boundaries
• exponential convergence
• time-depending problems (A.Schädle)

theoretical results:

• acoustics, (electromagnetics)
• homogeneous exterior domains
• spherical boundaries
• exponential/ super-algebraic convergence in 1D/Bessel

equation

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