Boundary Element Domain Decomposition Methods Challenges and Applications

Transcription

1 Boundary Element Doman Decomposton Methods Challenges and Applcatons Olaf Stenbach Insttut für Numersche Mathematk Technsche Unverstät Graz SFB 404 Mehrfeldprobleme n der Kontnuumsmechank, Stuttgart n collaboraton wth U. Langer, G. Of, W. L. Wendland, W. Zulehner 1 / 23

2 Couplng of Fnte and Boundary Element Methods [Bettess, Kelly, Zenkewcz 77, 79; Brezz, Johnson, Nedelec 78; Brezz, Johnson 79; Johnson, Nedelec 80;...] 2 / 23

3 Couplng of Fnte and Boundary Element Methods [Bettess, Kelly, Zenkewcz 77, 79; Brezz, Johnson, Nedelec 78; Brezz, Johnson 79; Johnson, Nedelec 80;...] Symmetrc Couplng of Fnte and Boundary Element Methods [Costabel 87; Langer 94;...] 2 / 23

4 Couplng of Fnte and Boundary Element Methods [Bettess, Kelly, Zenkewcz 77, 79; Brezz, Johnson, Nedelec 78; Brezz, Johnson 79; Johnson, Nedelec 80;...] Symmetrc Couplng of Fnte and Boundary Element Methods [Costabel 87; Langer 94;...] Symmetrc Boundary Element Doman Decomposton Methods [Hsao, Wendland 90; Carstensen, Kuhn, Langer 98; OS 96;...] 2 / 23

5 Couplng of Fnte and Boundary Element Methods [Bettess, Kelly, Zenkewcz 77, 79; Brezz, Johnson, Nedelec 78; Brezz, Johnson 79; Johnson, Nedelec 80;...] Symmetrc Couplng of Fnte and Boundary Element Methods [Costabel 87; Langer 94;...] Symmetrc Boundary Element Doman Decomposton Methods [Hsao, Wendland 90; Carstensen, Kuhn, Langer 98; OS 96;...] Steklov Poncaré Operator Doman Decomposton Methods [Agoshkov, Lebedev 85; Hsao, Wendland 92; Hsao, Schnack, Wendland 99, 00; OS 03;...] 2 / 23

6 Couplng of Fnte and Boundary Element Methods [Bettess, Kelly, Zenkewcz 77, 79; Brezz, Johnson, Nedelec 78; Brezz, Johnson 79; Johnson, Nedelec 80;...] Symmetrc Couplng of Fnte and Boundary Element Methods [Costabel 87; Langer 94;...] Symmetrc Boundary Element Doman Decomposton Methods [Hsao, Wendland 90; Carstensen, Kuhn, Langer 98; OS 96;...] Steklov Poncaré Operator Doman Decomposton Methods [Agoshkov, Lebedev 85; Hsao, Wendland 92; Hsao, Schnack, Wendland 99, 00; OS 03;...] Hybrd Doman Decomposton Methods (Mortar, Three Feld, FETI) [Agouzal, Thomas 85; Bernard, Maday, Patera 85; Wohlmuth 01; Brezz, Marn 04; Farhat, Roux 91; Klawonn, Wdlund 01; Tosell, Wdlund 05;...] 2 / 23

7 Couplng of Fnte and Boundary Element Methods [Bettess, Kelly, Zenkewcz 77, 79; Brezz, Johnson, Nedelec 78; Brezz, Johnson 79; Johnson, Nedelec 80;...] Symmetrc Couplng of Fnte and Boundary Element Methods [Costabel 87; Langer 94;...] Symmetrc Boundary Element Doman Decomposton Methods [Hsao, Wendland 90; Carstensen, Kuhn, Langer 98; OS 96;...] Steklov Poncaré Operator Doman Decomposton Methods [Agoshkov, Lebedev 85; Hsao, Wendland 92; Hsao, Schnack, Wendland 99, 00; OS 03;...] Hybrd Doman Decomposton Methods (Mortar, Three Feld, FETI) [Agouzal, Thomas 85; Bernard, Maday, Patera 85; Wohlmuth 01; Brezz, Marn 04; Farhat, Roux 91; Klawonn, Wdlund 01; Tosell, Wdlund 05;...] Sparse Boundary Element Tearng and Interconnectng Methods [Langer, OS 03; Langer, Of, OS, Zulehner 05; Of 05] 2 / 23

8 Model Problem dv[α(x) u(x)] = f (x) for x Ω, u(x) = g(x) for x Γ = Ω Nonoverlappng Doman Decomposton Ω = p Ω, Ω Ω j = for j, Γ = Ω =1 3 / 23

9 Model Problem dv[α(x) u(x)] = f (x) for x Ω, u(x) = g(x) for x Γ = Ω Nonoverlappng Doman Decomposton Ω = p Ω, Ω Ω j = for j, Γ = Ω =1 Local Boundary Value Problems α u (x) = f (x) for x Ω, u (x) = g(x) for x Γ Γ Transmsson Boundary Condtons u (x) = u j (x), α n u (x) + α j n j u j (x) = 0 for x Γ j = Γ Γ j 3 / 23

10 Local Drchlet Boundary Value Problem α u (x) = f (x) for x Ω, u (x) = g (x) for x Γ = Ω 4 / 23

11 Local Drchlet Boundary Value Problem α u (x) = f (x) for x Ω, u (x) = g (x) for x Γ = Ω Representaton Formula for x Ω u (x) = U (x,y)t (y)ds y g (y) U (x,y)ds y + 1 n y α Γ Γ Ω U (x,y)f (y)dy Fundamental Soluton U (x,y) = 1 1 4π x y, t (y) = u (y), y Γ n y 4 / 23

12 Local Drchlet Boundary Value Problem α u (x) = f (x) for x Ω, u (x) = g (x) for x Γ = Ω Representaton Formula for x Ω u (x) = U (x,y)t (y)ds y g (y) U (x,y)ds y + 1 n y α Γ Γ Ω U (x,y)f (y)dy Fundamental Soluton U (x,y) = 1 1 4π x y, t (y) = u (y), y Γ n y Boundary Integral Equaton for x Γ 1 t (y) 4π x y ds y = 1 2 g (x)+ 1 1 g (y) 4π n (y) x y ds y 1 1 α 4π Γ Γ Ω f (y) x y dy 4 / 23

13 Boundary Integral Equaton for x Γ (V t )(x) = 1 2 g (x) + (K g )(x) 1 α (N 0, f )(x) 5 / 23

14 Boundary Integral Equaton for x Γ (V t )(x) = 1 2 g (x) + (K g )(x) 1 α (N 0, f )(x) Sngle Layer Potental V : H 1/2 (Γ ) H 1/2 (Γ ), V w,w Γ c V 1 w 2 H 1/2 (Γ ), n = 2 : damω < 1 5 / 23

15 Boundary Integral Equaton for x Γ (V t )(x) = 1 2 g (x) + (K g )(x) 1 α (N 0, f )(x) Sngle Layer Potental V : H 1/2 (Γ ) H 1/2 (Γ ), V w,w Γ c V 1 w 2 H 1/2 (Γ ), n = 2 : damω < 1 Drchlet to Neumann Map t = V 1 ( 1 2 I + K )g 1 V 1 N 0, f α 5 / 23

16 Boundary Integral Equaton for x Γ (V t )(x) = 1 2 g (x) + (K g )(x) 1 α (N 0, f )(x) Sngle Layer Potental V : H 1/2 (Γ ) H 1/2 (Γ ), V w,w Γ c V 1 w 2 H 1/2 (Γ ), n = 2 : damω < 1 Drchlet to Neumann Map t = V 1 Steklov Poncaré Operator ( 1 2 I + K )g 1 V 1 N 0, f α S = V 1 ( 1 2 I + K ) 5 / 23

17 Representaton Formula for x Ω u (x) = U (x,y)t (y)ds y g (y) U (x,y)ds y + 1 n y α Γ Γ Ω U (x,y)f (y)dy 6 / 23

18 Representaton Formula for x Ω u (x) = U (x,y)t (y)ds y g (y) U (x,y)ds y + 1 n y α Γ Γ Ω U (x,y)f (y)dy Computaton of the Normal Dervatve t (x) = 1 2 t (x) + U (x,y)t (y)ds y n x n x Γ Γ + 1 U (x,y)f (y)dy α n x Ω g (y) U (x,y)ds y n y 6 / 23

19 Representaton Formula for x Ω u (x) = U (x,y)t (y)ds y g (y) U (x,y)ds y + 1 n y α Γ Γ Ω U (x,y)f (y)dy Computaton of the Normal Dervatve t (x) = 1 2 t (x) + U (x,y)t (y)ds y n x n x Γ Γ + 1 U (x,y)f (y)dy α n x Ω g (y) U (x,y)ds y n y Hypersngular Boundary Integral Equaton for x Γ t (x) = 1 2 t (x) + (K t )(x) + (D g )(x) + 1 α (N 1, f )(x) 6 / 23

20 Drchlet to Neumann Map t = D g + ( 1 2 I + K )t + 1 α N 1, f 7 / 23

21 Drchlet to Neumann Map t = D g + ( 1 2 I + K )t + 1 α N 1, f = D g + ( 1 2 I + K )[V 1 ( 1 2 I + K )g 1 V 1 N 0, f ] + 1 N 1, f α α 7 / 23

22 Drchlet to Neumann Map t = D g + ( 1 2 I + K )t + 1 α N 1, f = D g + ( 1 2 I + K )[V 1 ( 1 2 I + K )g 1 V 1 N 0, f ] + 1 N 1, f α α = S g 1 α N f Steklov Poncaré Operator S = V 1 ( 1 2 I + K ) = D + ( 1 2 I + K )V 1 ( 1 2 I + K ) =... 7 / 23

23 Drchlet to Neumann Map t = D g + ( 1 2 I + K )t + 1 α N 1, f = D g + ( 1 2 I + K )[V 1 ( 1 2 I + K )g 1 V 1 N 0, f ] + 1 N 1, f α α = S g 1 α N f Steklov Poncaré Operator S = V 1 ( 1 2 I + K ) = D + ( 1 2 I + K )V 1 ( 1 2 I + K ) =... Mappng Propertes of S : H 1/2 (Γ ) H 1/2 (Γ ) Defnton of S va Doman Varatonal Formulaton (FEM) 7 / 23

24 Boundary Integral Equatons (Calderon Projector) for f = 0 ( ) ( 1 u 2 = I K ) ( ) V u 1 t D 2 I + K t 8 / 23

25 Boundary Integral Equatons (Calderon Projector) for f = 0 ( ) ( 1 u 2 = I K ) ( ) V u 1 t D 2 I + K t Corollary [Plemelj 1911] K V = V K, V D = 1 4 I K2 8 / 23

26 Boundary Integral Equatons (Calderon Projector) for f = 0 ( ) ( 1 u 2 = I K ) ( ) V u 1 t D 2 I + K t Corollary [Plemelj 1911] K V = V K, V D = 1 4 I K2 Theorem [OS, Wendland 2001] ( 1 2 I + K )v V 1 c K (Γ ) v V 1 for all v H 1/2 (Γ ) wth c K (Γ ) = cv 1 cd 1 < 1 shape senstve 8 / 23

27 Theorem S v V c K (Γ ) v V 1 for all v H 1/2 (Γ ) 9 / 23

28 Theorem S v V c K (Γ ) v V 1 for all v H 1/2 (Γ ) S v,v Γ [1 c K (Γ )] v 2 V 1 for all v H 1/2 (Γ ),v 1 9 / 23

29 Theorem S v V c K (Γ ) v V 1 for all v H 1/2 (Γ ) Defne S v,v Γ [1 c K (Γ )] v 2 V 1 for all v H 1/2 (Γ ),v 1 S u,v Γ = S u,v Γ + β u,w eq, Γ v,w eq, Γ, V w eq, = 1 9 / 23

30 Theorem S v V c K (Γ ) v V 1 for all v H 1/2 (Γ ) Defne S v,v Γ [1 c K (Γ )] v 2 V 1 for all v H 1/2 (Γ ),v 1 S u,v Γ = S u,v Γ + β u,w eq, Γ v,w eq, Γ, V w eq, = 1 Theorem wth c e S 1 1 S V v,v Γ S v,v Γ c e 2 1 V v,v Γ c e S 1 = mn{1 c K(Γ ),β 1,w eq, Γ }, c e S 2 = max{c K(Γ ),β 1,w eq, Γ } 9 / 23

31 Theorem S v V c K (Γ ) v V 1 for all v H 1/2 (Γ ) Defne S v,v Γ [1 c K (Γ )] v 2 V 1 for all v H 1/2 (Γ ),v 1 S u,v Γ = S u,v Γ + β u,w eq, Γ v,w eq, Γ, V w eq, = 1 Theorem wth c e S 1 1 S V v,v Γ S v,v Γ c e 2 1 V v,v Γ c e S 1 = mn{1 c K(Γ ),β 1,w eq, Γ }, c e S 2 = max{c K(Γ ),β 1,w eq, Γ } Optmal Scalng 1 β = 2 1,w eq, Γ 9 / 23

32 Drchlet to Neumann Map (Partal Dfferental Equaton) α t (x) = α (S u )(x) (N f )(x) for x Γ Drchlet Boundary Condton u (x) = g(x) for x Γ Γ Transmsson Condtons u (x) = u j (x), α t (x) + α j t j (x) = 0 for x Γ j 10 / 23

33 Drchlet to Neumann Map (Partal Dfferental Equaton) α t (x) = α (S u )(x) (N f )(x) for x Γ Drchlet Boundary Condton u (x) = g(x) for x Γ Γ Transmsson Condtons u (x) = u j (x), α t (x) + α j t j (x) = 0 for x Γ j Drchlet Doman Decomposton Approach Fnd u H 1/2 (Γ S ) such that u(x) = g(x) for x Γ and α (S u Γ )(x) + α j (S j u Γj )(x) = (N f )(x) + (N j f j )(x) for x Γ j 10 / 23

34 Varatonal Problem Fnd u H 1/2 (Γ S ) such that u(x) = g(x) for x Γ and p α =1 Γ (S u Γ )(x)v Γ (x)ds x = p =1 Γ (N f )(x)v Γ (x)ds x for all v H 1/2 (Γ S ), v(x) = 0 for x Γ 11 / 23

35 Varatonal Problem Fnd u H 1/2 (Γ S ) such that u(x) = g(x) for x Γ and p α =1 Γ (S u Γ )(x)v Γ (x)ds x = p =1 Γ (N f )(x)v Γ (x)ds x for all v H 1/2 (Γ S ), v(x) = 0 for x Γ Galerkn Varatonal Problem Fnd u 0,h S 1 h (Γ S) H 1/2 0 (Γ S ) such that p α =1 Γ (S u 0,h Γ )(x)v h Γ (x)ds x = p =1 Γ [(N f )(x) α (S u g )(x)]v Γ (x)ds x for all v h S 1 h (Γ S), where u g s some bounded extenson of g. 11 / 23

36 Lnear System p α A S,h A u = =1 p A f, =1 S,h [l,k] = S ϕ k,ϕ l Γ 12 / 23

37 Lnear System p p α A S,h A u = A f, =1 =1 Non Symmetrc BEM Approxmaton S,h [l,k] = S ϕ k,ϕ l Γ S u = V 1 ( 1 2 I + K )u = w : V w,τ Γ = ( 1 2 I + K )u,τ Γ 12 / 23

38 Lnear System p p α A S,h A u = A f, =1 =1 Non Symmetrc BEM Approxmaton S,h [l,k] = S ϕ k,ϕ l Γ S u = V 1 ( 1 2 I + K )u = w : V w,τ Γ = ( 1 2 I + K )u,τ Γ Galerkn Approxmaton w,h S 0 h(γ ) : V w,h,τ,h Γ = ( 1 2 I + K )u,τ,h Γ 12 / 23

39 Lnear System p p α A S,h A u = A f, =1 =1 Non Symmetrc BEM Approxmaton S,h [l,k] = S ϕ k,ϕ l Γ S u = V 1 ( 1 2 I + K )u = w : V w,τ Γ = ( 1 2 I + K )u,τ Γ Galerkn Approxmaton Approxmaton w,h S 0 h(γ ) : V w,h,τ,h Γ = ( 1 2 I + K )u,τ,h Γ Ŝ u = w,h, Ŝ,h = M,hV 1,h (1 2 M,h + K,h ) 12 / 23

40 Lnear System p p α A S,h A u = A f, =1 =1 Non Symmetrc BEM Approxmaton S,h [l,k] = S ϕ k,ϕ l Γ S u = V 1 ( 1 2 I + K )u = w : V w,τ Γ = ( 1 2 I + K )u,τ Γ Galerkn Approxmaton Approxmaton w,h S 0 h(γ ) : V w,h,τ,h Γ = ( 1 2 I + K )u,τ,h Γ Stablty Condton Ŝ u = w,h, Ŝ,h = M,hV 1,h (1 2 M,h + K,h ) c S v,h H 1/2 (Γ ) v,h,τ,h Γ sup 0 τ,h Sh 0(Γ ) τ,h H 1/2 (Γ ) for all v,h S 1 h(γ ) 12 / 23

41 Symmetrc BEM Approxmaton where S u = D u + ( 1 2 I + K )V 1 ( 1 2 I + K )u = D u + ( 1 2 I + K )w V w,τ Γ = ( 1 2 I + K )u,τ Γ 13 / 23

42 Symmetrc BEM Approxmaton where S u = D u + ( 1 2 I + K )V 1 ( 1 2 I + K )u = D u + ( 1 2 I + K )w Galerkn Approxmaton V w,τ Γ = ( 1 2 I + K )u,τ Γ Approxmaton w,h S 0 h(γ ) : V w,h,τ,h Γ = ( 1 2 I + K )u,τ,h Γ Ŝ = D u + ( 1 2 I + K )w,h, Ŝ,h = D,h + ( 1 2 M,h + K,h)V 1,h (1 2 M,h + K,h ) 13 / 23

43 Symmetrc BEM Approxmaton where S u = D u + ( 1 2 I + K )V 1 ( 1 2 I + K )u = D u + ( 1 2 I + K )w Galerkn Approxmaton V w,τ Γ = ( 1 2 I + K )u,τ Γ Approxmaton w,h S 0 h(γ ) : V w,h,τ,h Γ = ( 1 2 I + K )u,τ,h Γ Ŝ = D u + ( 1 2 I + K )w,h, Ŝ,h = D,h + ( 1 2 M,h + K,h)V 1,h (1 2 M,h + K,h ) Stablty Ŝv,v Γ D v,v Γ c D 1 v 2 H 1/2 (Γ ) 13 / 23

44 Symmetrc BEM Approxmaton where S u = D u + ( 1 2 I + K )V 1 ( 1 2 I + K )u = D u + ( 1 2 I + K )w Galerkn Approxmaton V w,τ Γ = ( 1 2 I + K )u,τ Γ Approxmaton w,h S 0 h(γ ) : V w,h,τ,h Γ = ( 1 2 I + K )u,τ,h Γ Ŝ = D u + ( 1 2 I + K )w,h, Ŝ,h = D,h + ( 1 2 M,h + K,h)V 1,h (1 2 M,h + K,h ) Stablty Ŝv,v Γ D v,v Γ c D 1 v 2 H 1/2 (Γ ) FEM Approxmaton Ŝ,h = K C C K C I K 1 I I K C I 13 / 23

45 Tearng and Interconnectng Ω 1 Ω 2 Ω 3 Ω 4 14 / 23

46 Tearng and Interconnectng Ω 1 Ω 2 Ω 3 Ω 4 14 / 23

47 Tearng and Interconnectng Ω 1 Ω 2 Ω 3 Ω 4 All Floatng BETI [Of 05] Total FETI [Dostal et. al. 05] 14 / 23

48 Lnear System α 1 Ŝ 1,h B α p Ŝ p,h Bp B 1... B p u 1. u p λ = f 1. f p g 15 / 23

49 Lnear System α 1 Ŝ 1,h B α p Ŝ p,h Bp B 1... B p u 1. u p λ = f 1. f p g Local System α Ŝ,h u = f + B λ, S,h 1 = 0 15 / 23

50 Lnear System α 1 Ŝ 1,h B α p Ŝ p,h Bp B 1... B p u 1. u p λ = f 1. f p g Local System α Ŝ,h u = f + B λ, S,h 1 = 0 Solvablty Condton (f + B λ,1) = 0 15 / 23

51 Lnear System α 1 Ŝ 1,h B α p Ŝ p,h Bp B 1... B p u 1. u p λ = f 1. f p g Local System α Ŝ,h u = f + B λ, S,h 1 = 0 Solvablty Condton (f + B λ,1) = 0 Extended Lnear System α S,h u = α [Ŝ,h + β a a ]u = f + B λ, a,k = ϕ k,w eq, Γ 15 / 23

52 Lnear System α 1 Ŝ 1,h B α p Ŝ p,h Bp B 1... B p u 1. u p λ = f 1. f p g Local System α Ŝ,h u = f + B λ, S,h 1 = 0 Solvablty Condton (f + B λ,1) = 0 Extended Lnear System α S,h u = α [Ŝ,h + β a a ]u = f + B λ, a,k = ϕ k,w eq, Γ Soluton u = 1 α S 1,h [f + B λ] + γ 1, (f + B λ,1) = 0 15 / 23

53 Dual Problem p =1 1 α B S 1 B λ + p γ B 1 = =1 p =1 1 α B S 1,h f, (f + B λ,1) = 0 16 / 23

54 Dual Problem p =1 1 α B S 1 B λ + p γ B 1 = =1 p =1 1 α B S 1,h f, (f + B λ,1) = 0 Fλ + Gγ = d, G λ = e 16 / 23

55 Dual Problem p =1 1 α B S 1 B λ + p γ B 1 = =1 p =1 1 α B S 1,h f, (f + B λ,1) = 0 Fλ + Gγ = d, G λ = e Projecton P = I G(G G) 1 G, P Gγ = 0 16 / 23

56 Dual Problem p =1 1 α B S 1 B λ + p γ B 1 = =1 p =1 1 α B S 1,h f, (f + B λ,1) = 0 Fλ + Gγ = d, G λ = e Projecton Lnear System P = I G(G G) 1 G, P Gγ = 0 P Fλ = P d 16 / 23

57 Dual Problem p =1 1 α B S 1 B λ + p γ B 1 = =1 p =1 1 α B S 1,h f, (f + B λ,1) = 0 Projecton Lnear System Precondtoner Fλ + Gγ = d, G λ = e P = I G(G G) 1 G, P Gγ = 0 P Fλ = P d p 1 1 C F F = B S α =1 B 16 / 23

58 Dual Problem p =1 1 α B S 1 B λ + p γ B 1 = =1 p =1 1 α B S 1,h f, (f + B λ,1) = 0 Projecton Lnear System Precondtoner Fλ + Gγ = d, G λ = e P = I G(G G) 1 G, P Gγ = 0 C F F = P Fλ = P d p 1 1 B S α =1 Scaled Hypersngular BETI Precondtoner [Langer, OS 03] C 1 F = (BC 1 α B ) 1 BC 1 α D h C 1 α B (BC 1 α B ) 1 B 16 / 23

59 Coupled Lnear System α 1 V 1,h α 1 K1,h α p V p,h α p Kp,h α 1 K 1,h α 1 D1,h B α p K p,h α p Dp,h B p B 1... B p t 1... t p u 1... u p λ = f 0,1.. f 0,p f 1,1... f 1,p g 17 / 23

60 Coupled Lnear System α 1 V 1,h α 1 K1,h α p V p,h α p Kp,h α 1 K 1,h α 1 D1,h B α p K p,h α p Dp,h B p B 1... B p t 1... t p u 1... u p λ = f 0,1.. f 0,p f 1,1... f 1,p g Twofold Saddle Pont Problem Transformaton nto postve defnte symmetrc system [Bramble, Pascak 88; Zulehner 02; Langer, Of, OS, Zulehner 05] Scalng of Precondtoners s needed! 17 / 23

61 Precondtoner for Dscrete Steklov Poncaré Operator: C S S,h Sngle Layer Potental [OS, Wendland 95, 98] C 1 S = M 1 V,h,h M 1,h Geometrc/Algebrac Multgrd for dscrete Hypersngular Integral Operator D,h S,h 18 / 23

62 Precondtoner for Dscrete Steklov Poncaré Operator: C S S,h Sngle Layer Potental [OS, Wendland 95, 98] C 1 S = M 1 V,h,h M 1,h Geometrc/Algebrac Multgrd for dscrete Hypersngular Integral Operator D,h S,h Precondtoner for Dscrete Sngle Layer Potental Geometrc Multgrd/Multlevel [Maschak, Stephan, Tran,...] Artfcal Multlevel Precondtonng [OS 03] Algebrac Multgrd for Sparse BEM [Langer, Pusch 05; Of 05] 18 / 23

63 Galerkn Dscretsaton of Boundary Integral Operators non local kernels dense stffness matrces sngular fundamental soluton sngular surface ntegrals ntegraton by parts weakly sngular surface ntegrals only 19 / 23

64 Galerkn Dscretsaton of Boundary Integral Operators non local kernels dense stffness matrces sngular fundamental soluton sngular surface ntegrals ntegraton by parts weakly sngular surface ntegrals only Fast Boundary Element Methods Fast Multpole Algorthm [Greengard, Rokhln 87;...] Panel Clusterng [Hackbusch, Nowak 89;...] Adaptve Cross Approxmaton [Bebendorf, Rjasanow 03;...] Herarchcal Matrces [Hackbusch 99;...] Wavelets [Dahmen, Prößdorf, Schneder 93;...] 19 / 23

65 Galerkn Dscretsaton of Boundary Integral Operators non local kernels dense stffness matrces sngular fundamental soluton sngular surface ntegrals ntegraton by parts weakly sngular surface ntegrals only Fast Boundary Element Methods Fast Multpole Algorthm [Greengard, Rokhln 87;...] Panel Clusterng [Hackbusch, Nowak 89;...] Adaptve Cross Approxmaton [Bebendorf, Rjasanow 03;...] Herarchcal Matrces [Hackbusch 99;...] Wavelets [Dahmen, Prößdorf, Schneder 93;...] Complexty O(N log q N ) (Fast Multpole: q = 2) 19 / 23

66 Theorem Requrement: algebrac multgrd precondtoner for V,h s optmal. Solvng by Bramble Pascak transformaton and conjugate gradent method: Twofold saddle pont problem of the standard BETI method: number of teratons O((1 + log(h/h)) 2 ) O((H/h) 2 (1 + log(h/h)) 4 ) arthmetcal operatons 20 / 23

67 Theorem Requrement: algebrac multgrd precondtoner for V,h s optmal. Solvng by Bramble Pascak transformaton and conjugate gradent method: Twofold saddle pont problem of the standard BETI method: number of teratons O((1 + log(h/h)) 2 ) O((H/h) 2 (1 + log(h/h)) 4 ) arthmetcal operatons Twofold saddle pont problem of the all floatng BETI method: number of teratons O(1 + log(h/h)) O((H/h) 2 (1 + log(h/h)) 3 ) arthmetcal operatons 20 / 23

68 Theorem Requrement: algebrac multgrd precondtoner for V,h s optmal. Solvng by Bramble Pascak transformaton and conjugate gradent method: Twofold saddle pont problem of the standard BETI method: number of teratons O((1 + log(h/h)) 2 ) O((H/h) 2 (1 + log(h/h)) 4 ) arthmetcal operatons Twofold saddle pont problem of the all floatng BETI method: number of teratons O(1 + log(h/h)) O((H/h) 2 (1 + log(h/h)) 3 ) arthmetcal operatons The use of fast boundary element methods does not perturb the convergence rates of the approxmaton. 20 / 23

69 Example: Lnear elastcty (steel and concrete) 18 subdomans BETI all floatng L t 2 It. t 2 It ( 21( 10)) 39 22( 17( 10)) ( 33( 14)) ( 27( 14)) ( 44( 14)) ( 33( 14)) ( 51( 14)) ( 36( 14)) ( 54( 14)) ( 38( 15)) 21 / 23

70 Example: Lnear elastcty (steel and concrete) 18 subdomans BETI all floatng L t 2 It. t 2 It ( 21( 10)) 39 22( 17( 10)) ( 33( 14)) ( 27( 14)) ( 44( 14)) ( 33( 14)) ( 51( 14)) ( 36( 14)) ( 54( 14)) ( 38( 15)) Drchlet DD BETI all floatng L N t 2 It. t 2 It. t 2 It ( 10) ( 14) ( 14) ( 14) ( 14) ( 16) / 23

71 [Courtesy to Z. Andjelc, H. Andrä, J. Breuer, G. Of, S. Rjasanow] 22 / 23

72 References 1. O. Stenbach: Stablty Estmates for Hybrd Coupled Doman Decomposton Methods. Sprnger Lecture Notes n Mathematcs, vol. 1809, U. Langer, O. Stenbach: Boundary element tearng and nterconnectng methods. Computng 71 (2003) S. Rjasanow, O. Stenbach: The Fast Soluton of Boundary Integral Equatons. Mathematcal and Analytcal Technques wth Applcatons to Engneerng, Sprnger, New York, 2006, n press. 4. G. Of, O. Stenbach, W. L. Wendland: Boundary Element Tearng and Interconnectng Doman Decomposton Methods. In: Multfeld Problems n Flud and Sold Mechancs (R. Helmg, A. Melke, B. I. Wohlmuth eds.). Lecture Notes n Appled and Computatonal Mechancs, vol. 28, pp , 2006, n press. 5. U. Langer, G. Of, O. Stenbach, W. Zulehner: Inexact data sparse boundary element tearng and nterconnectng methods. Berchte aus dem Insttut für Mathematk D, Bercht 2005/4, TU Graz, / 23