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

Hardy space infinite elements for exteriorMaxwell problems

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

Linz, November 2011

1

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 = 2.82

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 <(κ) > 0 be a solutionto the eigenvalue problem∫

Ωcurl u · curl v dx = κ2

∫Ωεu · v dx + BC + RC.

Then we call κ a resonance, <(κ) > 0 the resonance frequencyand Q := <(κ)

2|=(κ)| the quality factor.

3

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

outline

motivation

transparent boundary conditions

Hardy space infinite elements in 1d

sequence of infinite elements

numerics

4

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 on Γ := ∂Ba(0)

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

reason: factor exp(±iκ|x |) in the asymptotic behaviour of u

5

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 on Γ := ∂Ba(0)

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

reason: factor exp(±iκ|x |) in the asymptotic behaviour of u

5

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 on Γ := ∂Ba(0)

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

reason: factor exp(±iκ|x |) in the asymptotic behaviour of u

5

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

BEM

Stratton-Chu formula:

u(x) = curl∫

Γν(y)× u(y)Φ(x , y)ds(y)

− 1κ2 curl curl

∫Γν(y)× curl u(y)Φ(x , y)ds(y), x ∈ R3 \ Ba(0)

withΦ(x , y) =

14π

exp(iκ|x − y |)|x − y |

pros: boundary integrals, fast convergence, non-convex Γcons: dependence on κ, Green’s function needed

6

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

BEM

Stratton-Chu formula:

u(x) = curl∫

Γν(y)× u(y)Φ(x , y)ds(y)

− 1κ2 curl curl

∫Γν(y)× curl u(y)Φ(x , y)ds(y), x ∈ R3 \ Ba(0)

withΦ(x , y) =

14π

exp(iκ|x − y |)|x − y |

pros: boundary integrals, fast convergence, non-convex Γcons: dependence on κ, Green’s function needed

6

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

classical infinite elementstensor product ansatz:

u(|x |, x) =N∑

l=0

ψl(|x |) el(x) +N∑

l=1

αl(|x |) gl(x)x , |x | ≥ 1, with

ψ0(r) :=1r

exp(iκ(r − 1)),

ψl(r) :=

(1

r l+1 −1r

)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’sequations , Computer Methods in Applied Mechanics andEngineering, 1998.

7

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

classical infinite elementstensor product ansatz:

u(|x |, x) =N∑

l=0

ψl(|x |) el(x) +N∑

l=1

αl(|x |) gl(x)x , |x | ≥ 1, with

ψ0(r) :=1r

exp(iκ(r − 1)),

ψl(r) :=

(1

r l+1 −1r

)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’sequations , Computer Methods in Applied Mechanics andEngineering, 1998.

7

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

classical infinite elementstensor product ansatz:

u(|x |, x) =N∑

l=0

ψl(|x |) el(x) +N∑

l=1

αl(|x |) gl(x)x , |x | ≥ 1, with

ψ0(r) :=1r

exp(iκ(r − 1)),

ψl(r) :=

(1

r l+1 −1r

)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’sequations , Computer Methods in Applied Mechanics andEngineering, 1998.

7

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

classical infinite elementstensor product ansatz:

u(|x |, x) =N∑

l=0

ψl(|x |) el(x) +N∑

l=1

αl(|x |) gl(x)x , |x | ≥ 1, with

ψ0(r) :=1r

exp(iκ(r − 1)),

ψl(r) :=

(1

r l+1 −1r

)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’sequations , Computer Methods in Applied Mechanics andEngineering, 1998.

7

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: calculatingenergies, widths and cross-sections by complex scaling,Physics Reports, 1998.

Berenger, A perfectly matched layer for the absorption ofelectromagnetic waves, J.Comput.Phy.,1994.

pros: simple to implement, generalized linear eigenvalueproblem, well knowncons: many parameters to chose, artificial resonances

8

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: calculatingenergies, widths and cross-sections by complex scaling,Physics Reports, 1998.

Berenger, A perfectly matched layer for the absorption ofelectromagnetic waves, J.Comput.Phy.,1994.

pros: simple to implement, generalized linear eigenvalueproblem, well knowncons: many parameters to chose, artificial resonances

8

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

Helmholtz equation in 1dHelmholtz equation:

−u′′(r)− κ2u(r) = 0

Laplace transformation:

u(s) := (Lu)(s) =

∫ ∞0

e−sr u(r)dr , <(s) > 0

u(r) = C1e+iκr + C2e−iκr

Lww

u(s) =C1

s − iκ+

C2

s + iκ.

<(s)

=(s)

iκ

−iκ incoming

outgoing

9

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

Helmholtz equation in 1dHelmholtz equation:

−u′′(r)− κ2u(r) = 0

Laplace transformation:

u(s) := (Lu)(s) =

∫ ∞0

e−sr u(r)dr , <(s) > 0

u(r) = C1e+iκr + C2e−iκr

Lww

u(s) =C1

s − iκ+

C2

s + iκ.

<(s)

=(s)

iκ

−iκ incoming

outgoing

9

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

Helmholtz equation in 1dHelmholtz equation:

−u′′(r)− κ2u(r) = 0

Laplace transformation:

u(s) := (Lu)(s) =

∫ ∞0

e−sr u(r)dr , <(s) > 0

u(r) = C1e+iκr + C2e−iκr

Lww

u(s) =C1

s − iκ+

C2

s + iκ.

<(s)

=(s)

iκ

−iκ incoming

outgoing

9

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

Möbius transformation

ϕ(z) = iκ0z+1z−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| < 1.10

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

pole condition

Definitionu is outgoing, iff

MLu ∈ H+(S1).

Schmidt & Deuflhard, Discrete Transparent BoundaryConditions for the Numerical Solution of Fresnel’s Equation,Computers Math. Applic., 1995.

Hohage & Schmidt & Zschiedrich, Solving time-harmonicscattering problems based on the pole condition. I. Theory ,SIAM J. Math. Anal., 2003.

Hohage & Nannen, Hardy space infinite elements forscattering and resonance problems, SIAM J. Numer. Anal.,2009.

11

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

pole condition

Definitionu is outgoing, iff

MLu ∈ H+(S1).

Schmidt & Deuflhard, Discrete Transparent BoundaryConditions for the Numerical Solution of Fresnel’s Equation,Computers Math. Applic., 1995.

Hohage & Schmidt & Zschiedrich, Solving time-harmonicscattering problems based on the pole condition. I. Theory ,SIAM J. Math. Anal., 2003.

Hohage & Nannen, Hardy space infinite elements forscattering and resonance problems, SIAM J. Numer. Anal.,2009.

11

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

Hardy space method in 1dclassical formulation:

−u′′(r)− κ2u(r) = 0 in [0,∞)

u′(0) = 1, MLu ∈ H+(S1).

variational formulation:∫ ∞0

(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).

12

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

Hardy space method in 1dclassical formulation:

−u′′(r)− κ2u(r) = 0 in [0,∞)

u′(0) = 1, MLu ∈ H+(S1).

variational formulation:∫ ∞0

(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).

12

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

Hardy space method in 1dclassical formulation:

−u′′(r)− κ2u(r) = 0 in [0,∞)

u′(0) = 1, MLu ∈ H+(S1).

variational formulation:∫ ∞0

(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).

12

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

basis functionsbasis functions in H+(S1):

(MLu)(z) ≈ 12iκ0

u0 + (z − 1)N∑

j=0

αjz j

basis functions in space:

u(r) ≈ eiκ0r

u0 +N∑

j=0

αj

j∑k=0

(jk

)(2iκ0r)k+1

(k + 1)!

derivative:

(MLu′)(z) ≈ 12

u0 + (z + 1)N∑

j=0

αjz j

sequence

13

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

basis functionsbasis functions in H+(S1):

(MLu)(z) ≈ 12iκ0

u0 + (z − 1)N∑

j=0

αjz j

basis functions in space:

u(r) ≈ eiκ0r

u0 +N∑

j=0

αj

j∑k=0

(jk

)(2iκ0r)k+1

(k + 1)!

derivative:

(MLu′)(z) ≈ 12

u0 + (z + 1)N∑

j=0

αjz j

sequence

13

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

basis functionsbasis functions in H+(S1):

(MLu)(z) ≈ 12iκ0

u0 + (z − 1)N∑

j=0

αjz j

basis functions in space:

u(r) ≈ eiκ0r

u0 +N∑

j=0

αj

j∑k=0

(jk

)(2iκ0r)k+1

(k + 1)!

derivative:

(MLu′)(z) ≈ 12

u0 + (z + 1)N∑

j=0

αjz j

sequence

13

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

system of linear equations:

variational formulation forMLu ∈ 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 11 2 1

. . . . . . . . .1 2 1

1 2

+ κ2 2iκ0

1 −1−1 2 −1

. . . . . . . . .−1 2 −1

−1 2

u0α0α1...αN

=

10...0

14

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

y πV

y πQ

y πX

yWh

∇−→ Vhcurl−→ Qh

div−→ Xh

motivation:

curl∇u = 0 ⇒ (0,∇u) is eigenpair

For uh = πW u it holds

curl∇uh = curl∇πW u = πQ curl∇u = 0

⇒ (0,∇uh) is discrete eigenpair

15

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

y πV

y πQ

y πX

yWh

∇−→ Vhcurl−→ Qh

div−→ Xh

motivation:

curl∇u = 0 ⇒ (0,∇u) is eigenpair

For uh = πW u it holds

curl∇uh = curl∇πW u = πQ curl∇u = 0

⇒ (0,∇uh) is discrete eigenpair

15

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

y πV

y πQ

y πX

yWh

∇−→ Vhcurl−→ Qh

div−→ Xh

motivation:

curl∇u = 0 ⇒ (0,∇u) is eigenpair

For uh = πW u it holds

curl∇uh = curl∇πW u = πQ curl∇u = 0

⇒ (0,∇uh) is discrete eigenpair

15

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

y πV

y πQ

y πX

yWh

∇−→ Vhcurl−→ Qh

div−→ Xh

motivation:

curl∇u = 0 ⇒ (0,∇u) is eigenpair

For uh = πW u it holds

curl∇uh = curl∇πW u = πQ curl∇u = 0

⇒ (0,∇uh) is discrete eigenpair

15

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:

yz

xx

ξ

F

V1

V3

V2 V2 V3

V1

V0

F (ξ, x) := x + ξ(x − V0)

16

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:

yz

xx

ξ

F

V1

V3

V2 V2 V3

V1

V0

F (ξ, x) := x + ξ(x − V0)

16

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

sequences on the prismsurface sequence on the surface triangle T :

H1(T )∇x−→ H(curl,T )

ν×∇x ·−→ L2(T )

πWT

y πVT

y πXT

yWT

∇x−→ VTν×∇x ·−→ XT

Hardy space sequence:

H+(S1)∂ξ−→ H+(S1)

πWξ

y πW ′ξ

y

Wξ := spanΨ−1, ...,ΨN∂ξ−→ W ′

ξ := spanψ−1, ..., ψN

xξ

Ψ−1(z) := 12iκ0

, Ψj(z) := z−12iκ0

z j , ψ−1(z) := 12 , ψj(z) := z+1

2 z j

1D

17

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

sequences on the prismsurface sequence on the surface triangle T :

H1(T )∇x−→ H(curl,T )

ν×∇x ·−→ L2(T )

πWT

y πVT

y πXT

yWT

∇x−→ VTν×∇x ·−→ XT

radial sequence:

H1(R+)∂ξ−→ L2(R+)

Hardy space sequence:

H+(S1)∂ξ−→ H+(S1)

πWξ

y πW ′ξ

y

Wξ := spanΨ−1, ...,ΨN∂ξ−→ W ′

ξ := spanψ−1, ..., ψN

xξ

Ψ−1(z) := 12iκ0

, Ψj(z) := z−12iκ0

z j , ψ−1(z) := 12 , ψj(z) := z+1

2 z j

1D

17

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

sequences on the prismsurface sequence on the surface triangle T :

H1(T )∇x−→ H(curl,T )

ν×∇x ·−→ L2(T )

πWT

y πVT

y πXT

yWT

∇x−→ VTν×∇x ·−→ XT

Hardy space sequence:

H+(S1)∂ξ−→ H+(S1)

πWξ

y πW ′ξ

yWξ

∂ξ−→ W ′ξ

Wξ := spanΨ−1, ...,ΨN∂ξ−→ W ′

ξ := spanψ−1, ..., ψN

xξ

Ψ−1(z) := 12iκ0

, Ψj(z) := z−12iκ0

z j , ψ−1(z) := 12 , ψj(z) := z+1

2 z j

1D

17

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

sequences on the prismsurface sequence on the surface triangle T :

H1(T )∇x−→ H(curl,T )

ν×∇x ·−→ L2(T )

πWT

y πVT

y πXT

yWT

∇x−→ VTν×∇x ·−→ XT

Hardy space sequence:

H+(S1)∂ξ−→ H+(S1)

πWξ

y πW ′ξ

yWξ := spanΨ−1, ...,ΨN

∂ξ−→ W ′ξ := spanψ−1, ..., ψN

xξ

Ψ−1(z) := 12iκ0

, Ψj(z) := z−12iκ0

z j , ψ−1(z) := 12 , ψj(z) := z+1

2 z j

1D

17

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

tensor product elements

R+\T : WT∇x−→ VT

ν×∇x ·−→ XT

Wξ Wξ ⊗WT −→ Wξ ⊗ VT −→ Wξ ⊗ XT

∂ξ ↓ ↓ ↓ ↓

W ′ξ W ′

ξ ⊗WT −→ W ′ξ ⊗ VT −→ W ′

ξ ⊗ XTx

ξ

tensor product sequence:

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

ξ ⊗ VT →W ′ξ ⊗ XT

Nannen, Hohage, Schädle & Schöberl, High orderCurl-conforming Hardy space infinite elements for exteriorMaxwell problems, arXiv:1103.2288v1.

18

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

tensor product elements

R+\T : WT∇x−→ VT

ν×∇x ·−→ XT

Wξ Wξ ⊗WT −→ Wξ ⊗ VT −→ Wξ ⊗ XT

∂ξ ↓ ↓ ↓ ↓

W ′ξ W ′

ξ ⊗WT −→ W ′ξ ⊗ VT −→ W ′

ξ ⊗ XTx

ξ

tensor product sequence:

Wξ ⊗WT

→Wξ ⊗ VT ⊕W ′ξ ⊗WT →Wξ ⊗ XT ⊕W ′

ξ ⊗ VT →W ′ξ ⊗ XT

Nannen, Hohage, Schädle & Schöberl, High orderCurl-conforming Hardy space infinite elements for exteriorMaxwell problems, arXiv:1103.2288v1.

18

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

tensor product elements

R+\T : WT∇x−→ VT

ν×∇x ·−→ XT

Wξ Wξ ⊗WT −→ Wξ ⊗ VT −→ Wξ ⊗ XT

∂ξ ↓ ↓ ↓ ↓

W ′ξ W ′

ξ ⊗WT −→ W ′ξ ⊗ VT −→ W ′

ξ ⊗ XTx

ξ

tensor product sequence:

Wξ ⊗WT →Wξ ⊗ VT ⊕W ′ξ ⊗WT

→Wξ ⊗ XT ⊕W ′ξ ⊗ VT →W ′

ξ ⊗ XT

Nannen, Hohage, Schädle & Schöberl, High orderCurl-conforming Hardy space infinite elements for exteriorMaxwell problems, arXiv:1103.2288v1.

18

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

tensor product elements

R+\T : WT∇x−→ VT

ν×∇x ·−→ XT

Wξ Wξ ⊗WT −→ Wξ ⊗ VT −→ Wξ ⊗ XT

∂ξ ↓ ↓ ↓ ↓

W ′ξ W ′

ξ ⊗WT −→ W ′ξ ⊗ VT −→ W ′

ξ ⊗ XTx

ξ

tensor product sequence:

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

ξ ⊗ VT

→W ′ξ ⊗ XT

Nannen, Hohage, Schädle & Schöberl, High orderCurl-conforming Hardy space infinite elements for exteriorMaxwell problems, arXiv:1103.2288v1.

18

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

tensor product elements

R+\T : WT∇x−→ VT

ν×∇x ·−→ XT

Wξ Wξ ⊗WT −→ Wξ ⊗ VT −→ Wξ ⊗ XT

∂ξ ↓ ↓ ↓ ↓

W ′ξ W ′

ξ ⊗WT −→ W ′ξ ⊗ VT −→ W ′

ξ ⊗ XTx

ξ

tensor product sequence:

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

ξ ⊗ VT →W ′ξ ⊗ XT

Nannen, Hohage, Schädle & Schöberl, High orderCurl-conforming Hardy space infinite elements for exteriorMaxwell problems, arXiv:1103.2288v1.

18

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

tensor product elements

R+\T : WT∇x−→ VT

ν×∇x ·−→ XT

Wξ Wξ ⊗WT −→ Wξ ⊗ VT −→ Wξ ⊗ XT

∂ξ ↓ ↓ ↓ ↓

W ′ξ W ′

ξ ⊗WT −→ W ′ξ ⊗ VT −→ W ′

ξ ⊗ XTx

ξ

tensor product sequence:

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

ξ ⊗ VT →W ′ξ ⊗ XT

Nannen, Hohage, Schädle & Schöberl, High orderCurl-conforming Hardy space infinite elements for exteriorMaxwell problems, arXiv:1103.2288v1. 18

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

wE23l⊗ wTl

Ψ−1 ⊗ wV2

Ψ−1 ⊗ wE23l

Ψj ⊗ wV2

Ψ−1 ⊗ wTl

Ψj ⊗ wTl

Ψj ⊗ wE23l

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

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

20

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 elementswith non-standard infinite elements

20

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

convergence of Hardy space method

21

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

electromagnetic resonance problem

22

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 = 2.823

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

GaAs cavity

0 2 4 6 8 10 12 1410

−6

10−5

10−4

10−3

10−2

10−1

Nre

l. er

ror

order 2order 3order 4order 5order 6

Karl et al., Reversed pyramids as novel opticalmicro-cavities , Superlattices and Microstructures, 2010. 24

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

25

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

25