Boundary Element Domain Decomposition Methods Challenges and - - PowerPoint PPT Presentation

▶
boundary element domain decomposition methods challenges
SMART_READER_LITE
LIVE PREVIEW

Boundary Element Domain Decomposition Methods Challenges and - - PowerPoint PPT Presentation

Aug 20, 2023 109 likes •840 views

Institut f ur Numerische Mathematik Boundary Element Domain Decomposition Methods Challenges and Applications Olaf Steinbach Institut f ur Numerische Mathematik Technische Universit at Graz SFB 404 Mehrfeldprobleme in der

slide-1
SLIDE 1

Institut f¨ ur Numerische Mathematik

Boundary Element Domain Decomposition Methods Challenges and Applications

Olaf Steinbach

Institut f¨ ur Numerische Mathematik Technische Universit¨ at Graz SFB 404 Mehrfeldprobleme in der Kontinuumsmechanik, Stuttgart in collaboration with

  • U. Langer, G. Of, W. L. Wendland, W. Zulehner
  • O. Steinbach

DD 17, 6.7.2006 1 / 23

slide-2
SLIDE 2

Institut f¨ ur Numerische Mathematik

◮ Coupling of Finite and Boundary Element Methods

[Bettess, Kelly, Zienkiewicz ’77, ’79; Brezzi, Johnson, Nedelec ’78; Brezzi, Johnson ’79; Johnson, Nedelec ’80; . . . ]

  • O. Steinbach

DD 17, 6.7.2006 2 / 23

slide-3
SLIDE 3

Institut f¨ ur Numerische Mathematik

◮ Coupling of Finite and Boundary Element Methods

[Bettess, Kelly, Zienkiewicz ’77, ’79; Brezzi, Johnson, Nedelec ’78; Brezzi, Johnson ’79; Johnson, Nedelec ’80; . . . ]

◮ Symmetric Coupling of Finite and Boundary Element Methods

[Costabel ’87; Langer ’94; . . . ]

  • O. Steinbach

DD 17, 6.7.2006 2 / 23

slide-4
SLIDE 4

Institut f¨ ur Numerische Mathematik

◮ Coupling of Finite and Boundary Element Methods

[Bettess, Kelly, Zienkiewicz ’77, ’79; Brezzi, Johnson, Nedelec ’78; Brezzi, Johnson ’79; Johnson, Nedelec ’80; . . . ]

◮ Symmetric Coupling of Finite and Boundary Element Methods

[Costabel ’87; Langer ’94; . . . ]

◮ Symmetric Boundary Element Domain Decomposition Methods

[Hsiao, Wendland ’90; Carstensen, Kuhn, Langer ’98; OS ’96; . . . ]

  • O. Steinbach

DD 17, 6.7.2006 2 / 23

slide-5
SLIDE 5

Institut f¨ ur Numerische Mathematik

◮ Coupling of Finite and Boundary Element Methods

[Bettess, Kelly, Zienkiewicz ’77, ’79; Brezzi, Johnson, Nedelec ’78; Brezzi, Johnson ’79; Johnson, Nedelec ’80; . . . ]

◮ Symmetric Coupling of Finite and Boundary Element Methods

[Costabel ’87; Langer ’94; . . . ]

◮ Symmetric Boundary Element Domain Decomposition Methods

[Hsiao, Wendland ’90; Carstensen, Kuhn, Langer ’98; OS ’96; . . . ]

◮ Steklov–Poincar´

e Operator Domain Decomposition Methods

[Agoshkov, Lebedev ’85; Hsiao, Wendland ’92; Hsiao, Schnack, Wendland ’99, ’00; OS ’03; . . . ]

  • O. Steinbach

DD 17, 6.7.2006 2 / 23

slide-6
SLIDE 6

Institut f¨ ur Numerische Mathematik

◮ Coupling of Finite and Boundary Element Methods

[Bettess, Kelly, Zienkiewicz ’77, ’79; Brezzi, Johnson, Nedelec ’78; Brezzi, Johnson ’79; Johnson, Nedelec ’80; . . . ]

◮ Symmetric Coupling of Finite and Boundary Element Methods

[Costabel ’87; Langer ’94; . . . ]

◮ Symmetric Boundary Element Domain Decomposition Methods

[Hsiao, Wendland ’90; Carstensen, Kuhn, Langer ’98; OS ’96; . . . ]

◮ Steklov–Poincar´

e Operator Domain Decomposition Methods

[Agoshkov, Lebedev ’85; Hsiao, Wendland ’92; Hsiao, Schnack, Wendland ’99, ’00; OS ’03; . . . ]

◮ Hybrid Domain Decomposition Methods (Mortar, Three Field, FETI)

[Agouzal, Thomas ’85; Bernardi, Maday, Patera ’85; Wohlmuth ’01; Brezzi, Marini ’04; Farhat, Roux ’91; Klawonn, Widlund ’01; Toselli, Widlund ’05; . . . ]

  • O. Steinbach

DD 17, 6.7.2006 2 / 23

slide-7
SLIDE 7

Institut f¨ ur Numerische Mathematik

◮ Coupling of Finite and Boundary Element Methods

[Bettess, Kelly, Zienkiewicz ’77, ’79; Brezzi, Johnson, Nedelec ’78; Brezzi, Johnson ’79; Johnson, Nedelec ’80; . . . ]

◮ Symmetric Coupling of Finite and Boundary Element Methods

[Costabel ’87; Langer ’94; . . . ]

◮ Symmetric Boundary Element Domain Decomposition Methods

[Hsiao, Wendland ’90; Carstensen, Kuhn, Langer ’98; OS ’96; . . . ]

◮ Steklov–Poincar´

e Operator Domain Decomposition Methods

[Agoshkov, Lebedev ’85; Hsiao, Wendland ’92; Hsiao, Schnack, Wendland ’99, ’00; OS ’03; . . . ]

◮ Hybrid Domain Decomposition Methods (Mortar, Three Field, FETI)

[Agouzal, Thomas ’85; Bernardi, Maday, Patera ’85; Wohlmuth ’01; Brezzi, Marini ’04; Farhat, Roux ’91; Klawonn, Widlund ’01; Toselli, Widlund ’05; . . . ]

◮ Sparse Boundary Element Tearing and Interconnecting Methods

[Langer, OS ’03; Langer, Of, OS, Zulehner ’05; Of ’05]

  • O. Steinbach

DD 17, 6.7.2006 2 / 23

slide-8
SLIDE 8

Institut f¨ ur Numerische Mathematik

Model Problem −div[α(x)∇u(x)] = f (x) for x ∈ Ω, u(x) = g(x) for x ∈ Γ = ∂Ω Nonoverlapping Domain Decomposition Ω =

p

  • i=1

Ωi, Ωi ∩ Ωj = ∅ for i = j, Γi = ∂Ωi

  • O. Steinbach

DD 17, 6.7.2006 3 / 23

slide-9
SLIDE 9

Institut f¨ ur Numerische Mathematik

Model Problem −div[α(x)∇u(x)] = f (x) for x ∈ Ω, u(x) = g(x) for x ∈ Γ = ∂Ω Nonoverlapping Domain Decomposition Ω =

p

  • i=1

Ωi, Ωi ∩ Ωj = ∅ for i = j, Γi = ∂Ωi Local Boundary Value Problems −αi∆ui(x) = fi(x) for x ∈ Ωi, ui(x) = g(x) for x ∈ Γi ∩ Γ Transmission Boundary Conditions ui(x) = uj(x), αi ∂ ∂ni ui(x) + αj ∂ ∂nj uj(x) = 0 for x ∈ Γij = Γi ∩ Γj

  • O. Steinbach

DD 17, 6.7.2006 3 / 23

slide-10
SLIDE 10

Institut f¨ ur Numerische Mathematik

Local Dirichlet Boundary Value Problem −αi∆ui(x) = fi(x) for x ∈ Ωi, ui(x) = gi(x) for x ∈ Γi = ∂Ωi

  • O. Steinbach

DD 17, 6.7.2006 4 / 23

slide-11
SLIDE 11

Institut f¨ ur Numerische Mathematik

Local Dirichlet Boundary Value Problem −αi∆ui(x) = fi(x) for x ∈ Ωi, ui(x) = gi(x) for x ∈ Γi = ∂Ωi Representation Formula for x ∈ Ωi ui(x) =

  • Γi

U∗(x, y)ti(y)dsy−

  • Γi

gi(y) ∂ ∂ny U∗(x, y)dsy+ 1 αi

  • Ωi

U∗(x, y)fi(y)dy Fundamental Solution U∗(x, y) = 1 4π 1 |x − y|, ti(y) = ∂ ∂ny ui(y), y ∈ Γi

  • O. Steinbach

DD 17, 6.7.2006 4 / 23

slide-12
SLIDE 12

Institut f¨ ur Numerische Mathematik

Local Dirichlet Boundary Value Problem −αi∆ui(x) = fi(x) for x ∈ Ωi, ui(x) = gi(x) for x ∈ Γi = ∂Ωi Representation Formula for x ∈ Ωi ui(x) =

  • Γi

U∗(x, y)ti(y)dsy−

  • Γi

gi(y) ∂ ∂ny U∗(x, y)dsy+ 1 αi

  • Ωi

U∗(x, y)fi(y)dy Fundamental Solution U∗(x, y) = 1 4π 1 |x − y|, ti(y) = ∂ ∂ny ui(y), y ∈ Γi Boundary Integral Equation for x ∈ Γi 1 4π

  • Γi

ti(y) |x − y|dsy = 1 2gi(x)+ 1 4π

  • Γi

gi(y) ∂ ∂ni(y) 1 |x − y|dsy− 1 αi 1 4π

  • Ωi

f (y) |x − y|dy

  • O. Steinbach

DD 17, 6.7.2006 4 / 23

slide-13
SLIDE 13

Institut f¨ ur Numerische Mathematik

Boundary Integral Equation for x ∈ Γi (Viti)(x) = 1 2gi(x) + (Kigi)(x) − 1 αi (N0,ifi)(x)

  • O. Steinbach

DD 17, 6.7.2006 5 / 23

slide-14
SLIDE 14

Institut f¨ ur Numerische Mathematik

Boundary Integral Equation for x ∈ Γi (Viti)(x) = 1 2gi(x) + (Kigi)(x) − 1 αi (N0,ifi)(x) Single Layer Potential Vi : H−1/2(Γi) → H1/2(Γi), Viwi, wiΓi ≥ cVi

1 wi2 H−1/2(Γi), n = 2 : diam Ωi < 1

  • O. Steinbach

DD 17, 6.7.2006 5 / 23

slide-15
SLIDE 15

Institut f¨ ur Numerische Mathematik

Boundary Integral Equation for x ∈ Γi (Viti)(x) = 1 2gi(x) + (Kigi)(x) − 1 αi (N0,ifi)(x) Single Layer Potential Vi : H−1/2(Γi) → H1/2(Γi), Viwi, wiΓi ≥ cVi

1 wi2 H−1/2(Γi), n = 2 : diam Ωi < 1

Dirichlet to Neumann Map ti = V −1

i

(1 2I + Ki)gi − 1 αi V −1

i

N0,ifi

  • O. Steinbach

DD 17, 6.7.2006 5 / 23

slide-16
SLIDE 16

Institut f¨ ur Numerische Mathematik

Boundary Integral Equation for x ∈ Γi (Viti)(x) = 1 2gi(x) + (Kigi)(x) − 1 αi (N0,ifi)(x) Single Layer Potential Vi : H−1/2(Γi) → H1/2(Γi), Viwi, wiΓi ≥ cVi

1 wi2 H−1/2(Γi), n = 2 : diam Ωi < 1

Dirichlet to Neumann Map ti = V −1

i

(1 2I + Ki)gi − 1 αi V −1

i

N0,ifi Steklov–Poincar´ e Operator Si = V −1

i

(1 2I + Ki)

  • O. Steinbach

DD 17, 6.7.2006 5 / 23

slide-17
SLIDE 17

Institut f¨ ur Numerische Mathematik

Representation Formula for x ∈ Ωi ui(x) =

  • Γi

U∗(x, y)ti(y)dsy−

  • Γi

gi(y) ∂ ∂ny U∗(x, y)dsy+ 1 αi

  • Ωi

U∗(x, y)fi(y)dy

  • O. Steinbach

DD 17, 6.7.2006 6 / 23

slide-18
SLIDE 18

Institut f¨ ur Numerische Mathematik

Representation Formula for x ∈ Ωi ui(x) =

  • Γi

U∗(x, y)ti(y)dsy−

  • Γi

gi(y) ∂ ∂ny U∗(x, y)dsy+ 1 αi

  • Ωi

U∗(x, y)fi(y)dy Computation of the Normal Derivative ti(x) = 1 2ti(x) +

  • Γi

∂ ∂nx U∗(x, y)ti(y)dsy − ∂ ∂nx

  • Γi

gi(y) ∂ ∂ny U∗(x, y)dsy + 1 αi ∂ ∂nx

  • Ωi

U∗(x, y)fi(y)dy

  • O. Steinbach

DD 17, 6.7.2006 6 / 23

slide-19
SLIDE 19

Institut f¨ ur Numerische Mathematik

Representation Formula for x ∈ Ωi ui(x) =

  • Γi

U∗(x, y)ti(y)dsy−

  • Γi

gi(y) ∂ ∂ny U∗(x, y)dsy+ 1 αi

  • Ωi

U∗(x, y)fi(y)dy Computation of the Normal Derivative ti(x) = 1 2ti(x) +

  • Γi

∂ ∂nx U∗(x, y)ti(y)dsy − ∂ ∂nx

  • Γi

gi(y) ∂ ∂ny U∗(x, y)dsy + 1 αi ∂ ∂nx

  • Ωi

U∗(x, y)fi(y)dy Hypersingular Boundary Integral Equation for x ∈ Γi ti(x) = 1 2ti(x) + (K ′

i ti)(x) + (Digi)(x) + 1

αi (N1,ifi)(x)

  • O. Steinbach

DD 17, 6.7.2006 6 / 23

slide-20
SLIDE 20

Institut f¨ ur Numerische Mathematik

Dirichlet to Neumann Map ti = Digi + (1 2I + K ′

i )ti + 1

αi N1,ifi

  • O. Steinbach

DD 17, 6.7.2006 7 / 23

slide-21
SLIDE 21

Institut f¨ ur Numerische Mathematik

Dirichlet to Neumann Map ti = Digi + (1 2I + K ′

i )ti + 1

αi N1,ifi = Digi + (1 2I + K ′

i )[V −1 i

(1 2I + Ki)gi − 1 αi V −1

i

N0,ifi] + 1 αi N1,ifi

  • O. Steinbach

DD 17, 6.7.2006 7 / 23

slide-22
SLIDE 22

Institut f¨ ur Numerische Mathematik

Dirichlet to Neumann Map ti = Digi + (1 2I + K ′

i )ti + 1

αi N1,ifi = Digi + (1 2I + K ′

i )[V −1 i

(1 2I + Ki)gi − 1 αi V −1

i

N0,ifi] + 1 αi N1,ifi = Sigi − 1 αi Nifi Steklov–Poincar´ e Operator Si = V −1

i

(1 2I + Ki) = Di + (1 2I + K ′

i )V −1 i

(1 2I + Ki) = . . .

  • O. Steinbach

DD 17, 6.7.2006 7 / 23

slide-23
SLIDE 23

Institut f¨ ur Numerische Mathematik

Dirichlet to Neumann Map ti = Digi + (1 2I + K ′

i )ti + 1

αi N1,ifi = Digi + (1 2I + K ′

i )[V −1 i

(1 2I + Ki)gi − 1 αi V −1

i

N0,ifi] + 1 αi N1,ifi = Sigi − 1 αi Nifi Steklov–Poincar´ e Operator Si = V −1

i

(1 2I + Ki) = Di + (1 2I + K ′

i )V −1 i

(1 2I + Ki) = . . .

◮ Mapping Properties of Si : H1/2(Γi) → H−1/2(Γi) ◮ Definition of Si via Domain Variational Formulation (FEM)

  • O. Steinbach

DD 17, 6.7.2006 7 / 23

slide-24
SLIDE 24

Institut f¨ ur Numerische Mathematik

Boundary Integral Equations (Calderon Projector) for f = 0

  • ui

ti

  • =
  • 1

2I − Ki

Vi Di

1 2I + K ′ i

ui ti

  • O. Steinbach

DD 17, 6.7.2006 8 / 23

slide-25
SLIDE 25

Institut f¨ ur Numerische Mathematik

Boundary Integral Equations (Calderon Projector) for f = 0

  • ui

ti

  • =
  • 1

2I − Ki

Vi Di

1 2I + K ′ i

ui ti

  • Corollary [Plemelj 1911]

KiVi = ViK ′

i ,

ViDi = 1 4I − K 2

i

  • O. Steinbach

DD 17, 6.7.2006 8 / 23

slide-26
SLIDE 26

Institut f¨ ur Numerische Mathematik

Boundary Integral Equations (Calderon Projector) for f = 0

  • ui

ti

  • =
  • 1

2I − Ki

Vi Di

1 2I + K ′ i

ui ti

  • Corollary [Plemelj 1911]

KiVi = ViK ′

i ,

ViDi = 1 4I − K 2

i

Theorem [OS, Wendland 2001] (1 2I + Ki)viV −1

i

≤ cK(Γi) viV −1

i

for all vi ∈ H1/2(Γi) with cK(Γi) = 1 2 +

  • 1

4 − cVi

1 cDi 1

< 1 shape sensitive

  • O. Steinbach

DD 17, 6.7.2006 8 / 23

slide-27
SLIDE 27

Institut f¨ ur Numerische Mathematik

Theorem SiviVi ≤ cK(Γi) viV −1

i

for all vi ∈ H1/2(Γi)

  • O. Steinbach

DD 17, 6.7.2006 9 / 23

slide-28
SLIDE 28

Institut f¨ ur Numerische Mathematik

Theorem SiviVi ≤ cK(Γi) viV −1

i

for all vi ∈ H1/2(Γi) Sivi, viΓi ≥ [1 − cK(Γi)] vi2

V −1

i

for all vi ∈ H1/2(Γi), vi⊥1

  • O. Steinbach

DD 17, 6.7.2006 9 / 23

slide-29
SLIDE 29

Institut f¨ ur Numerische Mathematik

Theorem SiviVi ≤ cK(Γi) viV −1

i

for all vi ∈ H1/2(Γi) Sivi, viΓi ≥ [1 − cK(Γi)] vi2

V −1

i

for all vi ∈ H1/2(Γi), vi⊥1 Define

  • Siui, viΓi = Siui, viΓi + βiui, weq,iΓivi, weq,iΓi,

Viweq,i = 1

  • O. Steinbach

DD 17, 6.7.2006 9 / 23

slide-30
SLIDE 30

Institut f¨ ur Numerische Mathematik

Theorem SiviVi ≤ cK(Γi) viV −1

i

for all vi ∈ H1/2(Γi) Sivi, viΓi ≥ [1 − cK(Γi)] vi2

V −1

i

for all vi ∈ H1/2(Γi), vi⊥1 Define

  • Siui, viΓi = Siui, viΓi + βiui, weq,iΓivi, weq,iΓi,

Viweq,i = 1 Theorem c

e Si 1 V −1 i

vi, viΓi ≤ Sivi, viΓi ≤ c

e Si 2 V −1 i

vi, viΓi with c

e Si 1 = min{1 − cK(Γi), βi1, weq,iΓi},

c

e Si 2 = max{cK(Γi), βi1, weq,iΓi}

  • O. Steinbach

DD 17, 6.7.2006 9 / 23

slide-31
SLIDE 31

Institut f¨ ur Numerische Mathematik

Theorem SiviVi ≤ cK(Γi) viV −1

i

for all vi ∈ H1/2(Γi) Sivi, viΓi ≥ [1 − cK(Γi)] vi2

V −1

i

for all vi ∈ H1/2(Γi), vi⊥1 Define

  • Siui, viΓi = Siui, viΓi + βiui, weq,iΓivi, weq,iΓi,

Viweq,i = 1 Theorem c

e Si 1 V −1 i

vi, viΓi ≤ Sivi, viΓi ≤ c

e Si 2 V −1 i

vi, viΓi with c

e Si 1 = min{1 − cK(Γi), βi1, weq,iΓi},

c

e Si 2 = max{cK(Γi), βi1, weq,iΓi}

Optimal Scaling βi = 1 21, weq,iΓi

  • O. Steinbach

DD 17, 6.7.2006 9 / 23

slide-32
SLIDE 32

Institut f¨ ur Numerische Mathematik

Dirichlet to Neumann Map (Partial Differential Equation) αiti(x) = αi(Siui)(x) − (Nifi)(x) for x ∈ Γi Dirichlet Boundary Condition ui(x) = g(x) for x ∈ Γi ∩ Γ Transmission Conditions ui(x) = uj(x), αiti(x) + αjtj(x) = 0 for x ∈ Γij

  • O. Steinbach

DD 17, 6.7.2006 10 / 23

slide-33
SLIDE 33

Institut f¨ ur Numerische Mathematik

Dirichlet to Neumann Map (Partial Differential Equation) αiti(x) = αi(Siui)(x) − (Nifi)(x) for x ∈ Γi Dirichlet Boundary Condition ui(x) = g(x) for x ∈ Γi ∩ Γ Transmission Conditions ui(x) = uj(x), αiti(x) + αjtj(x) = 0 for x ∈ Γij Dirichlet Domain Decomposition Approach Find u ∈ H1/2(ΓS) such that u(x) = g(x) for x ∈ Γ and αi(Siu|Γi)(x) + αj(Sju|Γj)(x) = (Nifi)(x) + (Njfj)(x) for x ∈ Γij

  • O. Steinbach

DD 17, 6.7.2006 10 / 23

slide-34
SLIDE 34

Institut f¨ ur Numerische Mathematik

Variational Problem Find u ∈ H1/2(ΓS) such that u(x) = g(x) for x ∈ Γ and

p

  • i=1

αi

  • Γi

(Siu|Γi)(x)v|Γi (x)dsx =

p

  • i=1
  • Γi

(Nifi)(x)v|Γi(x)dsx for all v ∈ H1/2(ΓS), v(x) = 0 for x ∈ Γ

  • O. Steinbach

DD 17, 6.7.2006 11 / 23

slide-35
SLIDE 35

Institut f¨ ur Numerische Mathematik

Variational Problem Find u ∈ H1/2(ΓS) such that u(x) = g(x) for x ∈ Γ and

p

  • i=1

αi

  • Γi

(Siu|Γi)(x)v|Γi (x)dsx =

p

  • i=1
  • Γi

(Nifi)(x)v|Γi(x)dsx for all v ∈ H1/2(ΓS), v(x) = 0 for x ∈ Γ Galerkin Variational Problem Find u0,h ∈ S1

h(ΓS) ⊂ H1/2

(ΓS) such that

p

  • i=1

αi

  • Γi

(Siu0,h|Γi)(x)vh|Γi (x)dsx =

p

  • i=1
  • Γi

[(Nifi)(x) − αi(Siug)(x)]v|Γi(x)dsx for all vh ∈ S1

h(ΓS), where ug is some bounded extension of g.

  • O. Steinbach

DD 17, 6.7.2006 11 / 23

slide-36
SLIDE 36

Institut f¨ ur Numerische Mathematik

Linear System

p

  • i=1

αiA⊤

i Si,hAiu = p

  • i=1

A⊤

i f i,

Si,h[ℓ, k] = Siϕi

k, ϕi ℓΓi

  • O. Steinbach

DD 17, 6.7.2006 12 / 23

slide-37
SLIDE 37

Institut f¨ ur Numerische Mathematik

Linear System

p

  • i=1

αiA⊤

i Si,hAiu = p

  • i=1

A⊤

i f i,

Si,h[ℓ, k] = Siϕi

k, ϕi ℓΓi

Non–Symmetric BEM Approximation Siui = V −1

i

(1 2I + Ki)ui = wi : Viwi, τiΓi = (1 2I + Ki)ui, τiΓi

  • O. Steinbach

DD 17, 6.7.2006 12 / 23

slide-38
SLIDE 38

Institut f¨ ur Numerische Mathematik

Linear System

p

  • i=1

αiA⊤

i Si,hAiu = p

  • i=1

A⊤

i f i,

Si,h[ℓ, k] = Siϕi

k, ϕi ℓΓi

Non–Symmetric BEM Approximation Siui = V −1

i

(1 2I + Ki)ui = wi : Viwi, τiΓi = (1 2I + Ki)ui, τiΓi Galerkin Approximation wi,h ∈ S0

h(Γi) : Viwi,h, τi,hΓi = (1

2I + Ki)ui, τi,hΓi

  • O. Steinbach

DD 17, 6.7.2006 12 / 23

slide-39
SLIDE 39

Institut f¨ ur Numerische Mathematik

Linear System

p

  • i=1

αiA⊤

i Si,hAiu = p

  • i=1

A⊤

i f i,

Si,h[ℓ, k] = Siϕi

k, ϕi ℓΓi

Non–Symmetric BEM Approximation Siui = V −1

i

(1 2I + Ki)ui = wi : Viwi, τiΓi = (1 2I + Ki)ui, τiΓi Galerkin Approximation wi,h ∈ S0

h(Γi) : Viwi,h, τi,hΓi = (1

2I + Ki)ui, τi,hΓi Approximation

  • Siui = wi,h,
  • Si,h = M⊤

i,hV −1 i,h (1

2Mi,h + Ki,h)

  • O. Steinbach

DD 17, 6.7.2006 12 / 23

slide-40
SLIDE 40

Institut f¨ ur Numerische Mathematik

Linear System

p

  • i=1

αiA⊤

i Si,hAiu = p

  • i=1

A⊤

i f i,

Si,h[ℓ, k] = Siϕi

k, ϕi ℓΓi

Non–Symmetric BEM Approximation Siui = V −1

i

(1 2I + Ki)ui = wi : Viwi, τiΓi = (1 2I + Ki)ui, τiΓi Galerkin Approximation wi,h ∈ S0

h(Γi) : Viwi,h, τi,hΓi = (1

2I + Ki)ui, τi,hΓi Approximation

  • Siui = wi,h,
  • Si,h = M⊤

i,hV −1 i,h (1

2Mi,h + Ki,h) Stability Condition cS vi,hH1/2(Γi) ≤ sup

0=τi,h∈S0

h(Γi)

vi,h, τi,hΓi τi,hH−1/2(Γi) for all vi,h ∈ S1

h(Γi)

  • O. Steinbach

DD 17, 6.7.2006 12 / 23

slide-41
SLIDE 41

Institut f¨ ur Numerische Mathematik

Symmetric BEM Approximation Siui = Diui + (1 2I + K ′

i )V −1 i

(1 2I + Ki)ui = Diui + (1 2I + K ′

i )wi

where Viwi, τiΓi = (1 2I + Ki)ui, τiΓi

  • O. Steinbach

DD 17, 6.7.2006 13 / 23

slide-42
SLIDE 42

Institut f¨ ur Numerische Mathematik

Symmetric BEM Approximation Siui = Diui + (1 2I + K ′

i )V −1 i

(1 2I + Ki)ui = Diui + (1 2I + K ′

i )wi

where Viwi, τiΓi = (1 2I + Ki)ui, τiΓi Galerkin Approximation wi,h ∈ S0

h(Γi) : Viwi,h, τi,hΓi = (1

2I + Ki)ui, τi,hΓi Approximation

  • Si = Diui + (1

2I + K ′

i )wi,h,

  • Si,h = Di,h + (1

2M⊤

i,h + K ⊤ i,h)V −1 i,h (1

2Mi,h + Ki,h)

  • O. Steinbach

DD 17, 6.7.2006 13 / 23

slide-43
SLIDE 43

Institut f¨ ur Numerische Mathematik

Symmetric BEM Approximation Siui = Diui + (1 2I + K ′

i )V −1 i

(1 2I + Ki)ui = Diui + (1 2I + K ′

i )wi

where Viwi, τiΓi = (1 2I + Ki)ui, τiΓi Galerkin Approximation wi,h ∈ S0

h(Γi) : Viwi,h, τi,hΓi = (1

2I + Ki)ui, τi,hΓi Approximation

  • Si = Diui + (1

2I + K ′

i )wi,h,

  • Si,h = Di,h + (1

2M⊤

i,h + K ⊤ i,h)V −1 i,h (1

2Mi,h + Ki,h) Stability

  • Sivi, viΓi ≥ Divi, viΓi ≥ cDi

1 |vi|2 H1/2(Γi)

  • O. Steinbach

DD 17, 6.7.2006 13 / 23

slide-44
SLIDE 44

Institut f¨ ur Numerische Mathematik

Symmetric BEM Approximation Siui = Diui + (1 2I + K ′

i )V −1 i

(1 2I + Ki)ui = Diui + (1 2I + K ′

i )wi

where Viwi, τiΓi = (1 2I + Ki)ui, τiΓi Galerkin Approximation wi,h ∈ S0

h(Γi) : Viwi,h, τi,hΓi = (1

2I + Ki)ui, τi,hΓi Approximation

  • Si = Diui + (1

2I + K ′

i )wi,h,

  • Si,h = Di,h + (1

2M⊤

i,h + K ⊤ i,h)V −1 i,h (1

2Mi,h + Ki,h) Stability

  • Sivi, viΓi ≥ Divi, viΓi ≥ cDi

1 |vi|2 H1/2(Γi)

FEM Approximation

  • Si,h = KCiCi − K ⊤

CiIiK −1 IiIi KCiIi

  • O. Steinbach

DD 17, 6.7.2006 13 / 23

slide-45
SLIDE 45

Institut f¨ ur Numerische Mathematik

Tearing and Interconnecting

Ω1 Ω2 Ω3 Ω4

  • O. Steinbach

DD 17, 6.7.2006 14 / 23

slide-46
SLIDE 46

Institut f¨ ur Numerische Mathematik

Tearing and Interconnecting

Ω1 Ω2 Ω3 Ω4

  • O. Steinbach

DD 17, 6.7.2006 14 / 23

slide-47
SLIDE 47

Institut f¨ ur Numerische Mathematik

Tearing and Interconnecting

Ω1 Ω2 Ω3 Ω4

All–Floating BETI [Of ’05] Total FETI [Dostal et. al. ’05]

  • O. Steinbach

DD 17, 6.7.2006 14 / 23

slide-48
SLIDE 48

Institut f¨ ur Numerische Mathematik

Linear System        α1 S1,h −B⊤

1

... . . . αp Sp,h −B⊤

p

B1 . . . Bp              u1 . . . up λ       =       f 1 . . . f p g      

  • O. Steinbach

DD 17, 6.7.2006 15 / 23

slide-49
SLIDE 49

Institut f¨ ur Numerische Mathematik

Linear System        α1 S1,h −B⊤

1

... . . . αp Sp,h −B⊤

p

B1 . . . Bp              u1 . . . up λ       =       f 1 . . . f p g       Local System αi Si,hui = f i + B⊤

i λ,

  • Si,h1 = 0
  • O. Steinbach

DD 17, 6.7.2006 15 / 23

slide-50
SLIDE 50

Institut f¨ ur Numerische Mathematik

Linear System        α1 S1,h −B⊤

1

... . . . αp Sp,h −B⊤

p

B1 . . . Bp              u1 . . . up λ       =       f 1 . . . f p g       Local System αi Si,hui = f i + B⊤

i λ,

  • Si,h1 = 0

Solvability Condition (f i + B⊤

i λ, 1) = 0

  • O. Steinbach

DD 17, 6.7.2006 15 / 23

slide-51
SLIDE 51

Institut f¨ ur Numerische Mathematik

Linear System        α1 S1,h −B⊤

1

... . . . αp Sp,h −B⊤

p

B1 . . . Bp              u1 . . . up λ       =       f 1 . . . f p g       Local System αi Si,hui = f i + B⊤

i λ,

  • Si,h1 = 0

Solvability Condition (f i + B⊤

i λ, 1) = 0

Extended Linear System αi Si,hui = αi[ Si,h + βiaia⊤

i ]ui = f i + B⊤ i λ,

ai,k = ϕi

k, weq,iΓi

  • O. Steinbach

DD 17, 6.7.2006 15 / 23

slide-52
SLIDE 52

Institut f¨ ur Numerische Mathematik

Linear System        α1 S1,h −B⊤

1

... . . . αp Sp,h −B⊤

p

B1 . . . Bp              u1 . . . up λ       =       f 1 . . . f p g       Local System αi Si,hui = f i + B⊤

i λ,

  • Si,h1 = 0

Solvability Condition (f i + B⊤

i λ, 1) = 0

Extended Linear System αi Si,hui = αi[ Si,h + βiaia⊤

i ]ui = f i + B⊤ i λ,

ai,k = ϕi

k, weq,iΓi

Solution ui = 1 αi

  • S−1

i,h [f i + B⊤ i λ] + γi1,

(f i + B⊤

i λ, 1) = 0

  • O. Steinbach

DD 17, 6.7.2006 15 / 23

slide-53
SLIDE 53

Institut f¨ ur Numerische Mathematik

Dual Problem

p

  • i=1

1 αi Bi S−1

i

B⊤

i λ + p

  • i=1

γiBi1 = −

p

  • i=1

1 αi Bi S−1

i,h f i,

(f i + B⊤

i λ, 1) = 0

  • O. Steinbach

DD 17, 6.7.2006 16 / 23

slide-54
SLIDE 54

Institut f¨ ur Numerische Mathematik

Dual Problem

p

  • i=1

1 αi Bi S−1

i

B⊤

i λ + p

  • i=1

γiBi1 = −

p

  • i=1

1 αi Bi S−1

i,h f i,

(f i + B⊤

i λ, 1) = 0

Fλ + Gγ = d, G ⊤λ = e

  • O. Steinbach

DD 17, 6.7.2006 16 / 23

slide-55
SLIDE 55

Institut f¨ ur Numerische Mathematik

Dual Problem

p

  • i=1

1 αi Bi S−1

i

B⊤

i λ + p

  • i=1

γiBi1 = −

p

  • i=1

1 αi Bi S−1

i,h f i,

(f i + B⊤

i λ, 1) = 0

Fλ + Gγ = d, G ⊤λ = e Projection P⊤ = I − G(G ⊤G)−1G ⊤, P⊤Gγ = 0

  • O. Steinbach

DD 17, 6.7.2006 16 / 23

slide-56
SLIDE 56

Institut f¨ ur Numerische Mathematik

Dual Problem

p

  • i=1

1 αi Bi S−1

i

B⊤

i λ + p

  • i=1

γiBi1 = −

p

  • i=1

1 αi Bi S−1

i,h f i,

(f i + B⊤

i λ, 1) = 0

Fλ + Gγ = d, G ⊤λ = e Projection P⊤ = I − G(G ⊤G)−1G ⊤, P⊤Gγ = 0 Linear System P⊤Fλ = P⊤d

  • O. Steinbach

DD 17, 6.7.2006 16 / 23

slide-57
SLIDE 57

Institut f¨ ur Numerische Mathematik

Dual Problem

p

  • i=1

1 αi Bi S−1

i

B⊤

i λ + p

  • i=1

γiBi1 = −

p

  • i=1

1 αi Bi S−1

i,h f i,

(f i + B⊤

i λ, 1) = 0

Fλ + Gγ = d, G ⊤λ = e Projection P⊤ = I − G(G ⊤G)−1G ⊤, P⊤Gγ = 0 Linear System P⊤Fλ = P⊤d Preconditioner CF ∼ F =

p

  • i=1

1 αi Bi S−1

i

B⊤

i

  • O. Steinbach

DD 17, 6.7.2006 16 / 23

slide-58
SLIDE 58

Institut f¨ ur Numerische Mathematik

Dual Problem

p

  • i=1

1 αi Bi S−1

i

B⊤

i λ + p

  • i=1

γiBi1 = −

p

  • i=1

1 αi Bi S−1

i,h f i,

(f i + B⊤

i λ, 1) = 0

Fλ + Gγ = d, G ⊤λ = e Projection P⊤ = I − G(G ⊤G)−1G ⊤, P⊤Gγ = 0 Linear System P⊤Fλ = P⊤d Preconditioner CF ∼ F =

p

  • i=1

1 αi Bi S−1

i

B⊤

i

Scaled Hypersingular BETI Preconditioner [Langer, OS ’03] C −1

F

= (BC −1

α B⊤)−1BC −1 α DhC −1 α B⊤(BC −1 α B⊤)−1

  • O. Steinbach

DD 17, 6.7.2006 16 / 23

slide-59
SLIDE 59

Institut f¨ ur Numerische Mathematik

Coupled Linear System                 α1V1,h −α1 K1,h ... ... αpVp,h −αp Kp,h α1 K ⊤

1,h

α1 D1,h −B⊤

1

... ... . . . αp K ⊤

p,h

αp Dp,h −B⊤

p

B1 . . . Bp                                t1 . . . tp u1 . . . up λ                =                f 0,1 . . . f 0,p f 1,1 . . . f 1,p g               

  • O. Steinbach

DD 17, 6.7.2006 17 / 23

slide-60
SLIDE 60

Institut f¨ ur Numerische Mathematik

Coupled Linear System                 α1V1,h −α1 K1,h ... ... αpVp,h −αp Kp,h α1 K ⊤

1,h

α1 D1,h −B⊤

1

... ... . . . αp K ⊤

p,h

αp Dp,h −B⊤

p

B1 . . . Bp                                t1 . . . tp u1 . . . up λ                =                f 0,1 . . . f 0,p f 1,1 . . . f 1,p g                Twofold Saddle Point Problem → Transformation into positive definite symmetric system

[Bramble, Pasciak ’88; Zulehner ’02; Langer, Of, OS, Zulehner ’05]

→ Scaling of Preconditioners is needed!

  • O. Steinbach

DD 17, 6.7.2006 17 / 23

slide-61
SLIDE 61

Institut f¨ ur Numerische Mathematik

Preconditioner for Discrete Steklov–Poincar´ e Operator: CSi ∼ Si,h

◮ Single Layer Potential [OS, Wendland ’95, ’98]

C −1

Si

= ¯ M−1

i,h ¯

Vi,h ¯ M−1

i,h ◮ Geometric/Algebraic Multigrid for discrete Hypersingular Integral

Operator Di,h ∼ Si,h

  • O. Steinbach

DD 17, 6.7.2006 18 / 23

slide-62
SLIDE 62

Institut f¨ ur Numerische Mathematik

Preconditioner for Discrete Steklov–Poincar´ e Operator: CSi ∼ Si,h

◮ Single Layer Potential [OS, Wendland ’95, ’98]

C −1

Si

= ¯ M−1

i,h ¯

Vi,h ¯ M−1

i,h ◮ Geometric/Algebraic Multigrid for discrete Hypersingular Integral

Operator Di,h ∼ Si,h Preconditioner for Discrete Single Layer Potential

◮ Geometric Multigrid/Multilevel [Maischak, Stephan, Tran,. . . ] ◮ Artificial Multilevel Preconditioning [OS ’03] ◮ Algebraic Multigrid for Sparse BEM [Langer, Pusch ’05; Of ’05]

  • O. Steinbach

DD 17, 6.7.2006 18 / 23

slide-63
SLIDE 63

Institut f¨ ur Numerische Mathematik

Galerkin Discretisation of Boundary Integral Operators

◮ non–local kernels → dense stiffness matrices ◮ singular fundamental solution → singular surface integrals ◮ integration by parts → weakly singular surface integrals only

  • O. Steinbach

DD 17, 6.7.2006 19 / 23

slide-64
SLIDE 64

Institut f¨ ur Numerische Mathematik

Galerkin Discretisation of Boundary Integral Operators

◮ non–local kernels → dense stiffness matrices ◮ singular fundamental solution → singular surface integrals ◮ integration by parts → weakly singular surface integrals only

Fast Boundary Element Methods

◮ Fast Multipole Algorithm [Greengard, Rokhlin ’87; . . . ] ◮ Panel Clustering [Hackbusch, Nowak ’89; . . . ] ◮ Adaptive Cross Approximation [Bebendorf, Rjasanow ’03; . . . ] ◮ Hierarchical Matrices [Hackbusch ’99; . . . ] ◮ Wavelets [Dahmen, Pr¨

  • ßdorf, Schneider ’93; . . . ]
  • O. Steinbach

DD 17, 6.7.2006 19 / 23

slide-65
SLIDE 65

Institut f¨ ur Numerische Mathematik

Galerkin Discretisation of Boundary Integral Operators

◮ non–local kernels → dense stiffness matrices ◮ singular fundamental solution → singular surface integrals ◮ integration by parts → weakly singular surface integrals only

Fast Boundary Element Methods

◮ Fast Multipole Algorithm [Greengard, Rokhlin ’87; . . . ] ◮ Panel Clustering [Hackbusch, Nowak ’89; . . . ] ◮ Adaptive Cross Approximation [Bebendorf, Rjasanow ’03; . . . ] ◮ Hierarchical Matrices [Hackbusch ’99; . . . ] ◮ Wavelets [Dahmen, Pr¨

  • ßdorf, Schneider ’93; . . . ]

Complexity O(Ni logq Ni) (Fast Multipole: q = 2)

  • O. Steinbach

DD 17, 6.7.2006 19 / 23

slide-66
SLIDE 66

Institut f¨ ur Numerische Mathematik

Theorem Requirement: algebraic multigrid preconditioner for Vi,h is optimal. Solving by Bramble Pasciak transformation and conjugate gradient method: Twofold saddle point problem of the standard BETI method:

◮ number of iterations O((1 + log(H/h))2) ◮ O((H/h)2(1 + log(H/h))4) arithmetical operations

  • O. Steinbach

DD 17, 6.7.2006 20 / 23

slide-67
SLIDE 67

Institut f¨ ur Numerische Mathematik

Theorem Requirement: algebraic multigrid preconditioner for Vi,h is optimal. Solving by Bramble Pasciak transformation and conjugate gradient method: Twofold saddle point problem of the standard BETI method:

◮ number of iterations O((1 + log(H/h))2) ◮ O((H/h)2(1 + log(H/h))4) arithmetical operations

Twofold saddle point problem of the all–floating BETI method:

◮ number of iterations O(1 + log(H/h)) ◮ O((H/h)2(1 + log(H/h))3) arithmetical operations

  • O. Steinbach

DD 17, 6.7.2006 20 / 23

slide-68
SLIDE 68

Institut f¨ ur Numerische Mathematik

Theorem Requirement: algebraic multigrid preconditioner for Vi,h is optimal. Solving by Bramble Pasciak transformation and conjugate gradient method: Twofold saddle point problem of the standard BETI method:

◮ number of iterations O((1 + log(H/h))2) ◮ O((H/h)2(1 + log(H/h))4) arithmetical operations

Twofold saddle point problem of the all–floating BETI method:

◮ number of iterations O(1 + log(H/h)) ◮ O((H/h)2(1 + log(H/h))3) arithmetical operations

The use of fast boundary element methods does not perturb the convergence rates of the approximation.

  • O. Steinbach

DD 17, 6.7.2006 20 / 23

slide-69
SLIDE 69

Institut f¨ ur Numerische Mathematik

Example: Linear elasticity (steel and concrete) 18 subdomains

BETI all–floating L t2 It. t2 It. 31 19( 21( 10)) 39 22( 17( 10)) 1 217 28( 33( 14)) 170 24( 27( 14)) 2 2129 35( 44( 14)) 1437 27( 33( 14)) 3 14149 42( 51( 14)) 9005 32( 36( 14)) 4 116404 47( 54( 14)) 77111 38( 38( 15))

  • O. Steinbach

DD 17, 6.7.2006 21 / 23

slide-70
SLIDE 70

Institut f¨ ur Numerische Mathematik

Example: Linear elasticity (steel and concrete) 18 subdomains

BETI all–floating L t2 It. t2 It. 31 19( 21( 10)) 39 22( 17( 10)) 1 217 28( 33( 14)) 170 24( 27( 14)) 2 2129 35( 44( 14)) 1437 27( 33( 14)) 3 14149 42( 51( 14)) 9005 32( 36( 14)) 4 116404 47( 54( 14)) 77111 38( 38( 15)) Dirichlet DD BETI all–floating L Ni t2 It. t2 It. t2 It. 24 7 53( 10) 7 78 8 65 1 96 25 110( 14) 19 100 19 82 2 384 181 130( 14) 112 114 115 85 3 1536 986 148( 14) 562 129 476 95 4 6144 6902 154( 14) 4352 153 3119 105 5 24576 59264 166( 16) 31645 172 23008 120

  • O. Steinbach

DD 17, 6.7.2006 21 / 23

slide-71
SLIDE 71

Institut f¨ ur Numerische Mathematik

5 10 15

  • 5

5 5 10 15

  • 5

5

[Courtesy to Z. Andjelic, H. Andr¨ a, J. Breuer, G. Of, S. Rjasanow]

  • O. Steinbach

DD 17, 6.7.2006 22 / 23

slide-72
SLIDE 72

Institut f¨ ur Numerische Mathematik

References

  • 1. O. Steinbach: Stability Estimates for Hybrid Coupled Domain

Decomposition Methods. Springer Lecture Notes in Mathematics, vol. 1809, 2003.

  • 2. U. Langer, O. Steinbach: Boundary element tearing and interconnecting
  • methods. Computing 71 (2003) 205–228.
  • 3. S. Rjasanow, O. Steinbach: The Fast Solution of Boundary Integral
  • Equations. Mathematical and Analytical Techniques with Applications to

Engineering, Springer, New York, 2006, in press.

  • 4. G. Of, O. Steinbach, W. L. Wendland: Boundary Element Tearing and

Interconnecting Domain Decomposition Methods. In: Multifield Problems in Fluid and Solid Mechanics (R. Helmig, A. Mielke,

  • B. I. Wohlmuth eds.). Lecture Notes in Applied and Computational

Mechanics, vol. 28, pp. 461–490, 2006, in press.

  • 5. U. Langer, G. Of, O. Steinbach, W. Zulehner: Inexact data–sparse

boundary element tearing and interconnecting methods. Berichte aus dem Institut f¨ ur Mathematik D, Bericht 2005/4, TU Graz, 2005.

  • O. Steinbach

DD 17, 6.7.2006 23 / 23