Boundary Element Domain Decomposition Methods Challenges and - PowerPoint PPT Presentation

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

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

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

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

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

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

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

Institut f¨ ur Numerische Mathematik Model Problem − div[ α ( x ) ∇ u ( x )] = f ( x ) for x ∈ Ω , for x ∈ Γ = ∂ Ω u ( x ) = g ( x ) Nonoverlapping Domain Decomposition p � Ω = Ω i , Ω i ∩ Ω j = ∅ for i � = j , Γ i = ∂ Ω i i =1 O. Steinbach DD 17, 6.7.2006 3 / 23

Institut f¨ ur Numerische Mathematik Model Problem − div[ α ( x ) ∇ u ( x )] = f ( x ) for x ∈ Ω , for x ∈ Γ = ∂ Ω u ( x ) = g ( x ) Nonoverlapping Domain Decomposition p � Ω = Ω i , Ω i ∩ Ω j = ∅ for i � = j , Γ i = ∂ Ω i i =1 Local Boundary Value Problems − α i ∆ u i ( x ) = f i ( x ) for x ∈ Ω i , for x ∈ Γ i ∩ Γ u i ( x ) = g ( x ) Transmission Boundary Conditions ∂ ∂ u i ( x ) = u j ( x ) , α i u i ( x ) + α j u j ( x ) = 0 for x ∈ Γ ij = Γ i ∩ Γ j ∂ n i ∂ n j O. Steinbach DD 17, 6.7.2006 3 / 23

Institut f¨ ur Numerische Mathematik Local Dirichlet Boundary Value Problem − α i ∆ u i ( x ) = f i ( x ) for x ∈ Ω i , u i ( x ) = g i ( x ) for x ∈ Γ i = ∂ Ω i O. Steinbach DD 17, 6.7.2006 4 / 23

Institut f¨ ur Numerische Mathematik Local Dirichlet Boundary Value Problem − α i ∆ u i ( x ) = f i ( x ) for x ∈ Ω i , u i ( x ) = g i ( x ) for x ∈ Γ i = ∂ Ω i Representation Formula for x ∈ Ω i � � � g i ( y ) ∂ U ∗ ( x , y ) ds y + 1 U ∗ ( x , y ) t i ( y ) ds y − U ∗ ( x , y ) f i ( y ) dy u i ( x ) = ∂ n y α i Γ i Γ i Ω i Fundamental Solution 1 1 ∂ U ∗ ( x , y ) = | x − y | , t i ( y ) = u i ( y ) , y ∈ Γ i 4 π ∂ n y O. Steinbach DD 17, 6.7.2006 4 / 23

Institut f¨ ur Numerische Mathematik Local Dirichlet Boundary Value Problem − α i ∆ u i ( x ) = f i ( x ) for x ∈ Ω i , u i ( x ) = g i ( x ) for x ∈ Γ i = ∂ Ω i Representation Formula for x ∈ Ω i � � � g i ( y ) ∂ U ∗ ( x , y ) ds y + 1 U ∗ ( x , y ) t i ( y ) ds y − U ∗ ( x , y ) f i ( y ) dy u i ( x ) = ∂ n y α i Γ i Γ i Ω i Fundamental Solution 1 1 ∂ U ∗ ( x , y ) = | x − y | , t i ( y ) = u i ( y ) , y ∈ Γ i 4 π ∂ n y Boundary Integral Equation for x ∈ Γ i � � � 1 | x − y | ds y = 1 t i ( y ) 2 g i ( x )+ 1 ∂ | x − y | ds y − 1 1 1 f ( y ) g i ( y ) | x − y | dy 4 π 4 π ∂ n i ( y ) α i 4 π Γ i Γ i Ω i O. Steinbach DD 17, 6.7.2006 4 / 23

Institut f¨ ur Numerische Mathematik Boundary Integral Equation for x ∈ Γ i ( V i t i )( x ) = 1 2 g i ( x ) + ( K i g i )( x ) − 1 ( N 0 , i f i )( x ) α i O. Steinbach DD 17, 6.7.2006 5 / 23

Institut f¨ ur Numerische Mathematik Boundary Integral Equation for x ∈ Γ i ( V i t i )( x ) = 1 2 g i ( x ) + ( K i g i )( x ) − 1 ( N 0 , i f i )( x ) α i Single Layer Potential V i : H − 1 / 2 (Γ i ) → H 1 / 2 (Γ i ) , � V i w i , w i � Γ i ≥ c V i 1 � w i � 2 H − 1 / 2 (Γ i ) , n = 2 : diam Ω i < 1 O. Steinbach DD 17, 6.7.2006 5 / 23

Institut f¨ ur Numerische Mathematik Boundary Integral Equation for x ∈ Γ i ( V i t i )( x ) = 1 2 g i ( x ) + ( K i g i )( x ) − 1 ( N 0 , i f i )( x ) α i Single Layer Potential V i : H − 1 / 2 (Γ i ) → H 1 / 2 (Γ i ) , � V i w i , w i � Γ i ≥ c V i 1 � w i � 2 H − 1 / 2 (Γ i ) , n = 2 : diam Ω i < 1 Dirichlet to Neumann Map (1 2 I + K i ) g i − 1 t i = V − 1 V − 1 N 0 , i f i i i α i O. Steinbach DD 17, 6.7.2006 5 / 23

Institut f¨ ur Numerische Mathematik Boundary Integral Equation for x ∈ Γ i ( V i t i )( x ) = 1 2 g i ( x ) + ( K i g i )( x ) − 1 ( N 0 , i f i )( x ) α i Single Layer Potential V i : H − 1 / 2 (Γ i ) → H 1 / 2 (Γ i ) , � V i w i , w i � Γ i ≥ c V i 1 � w i � 2 H − 1 / 2 (Γ i ) , n = 2 : diam Ω i < 1 Dirichlet to Neumann Map (1 2 I + K i ) g i − 1 t i = V − 1 V − 1 N 0 , i f i i i α i Steklov–Poincar´ e Operator (1 S i = V − 1 2 I + K i ) i O. Steinbach DD 17, 6.7.2006 5 / 23

Institut f¨ ur Numerische Mathematik Representation Formula for x ∈ Ω i � � � g i ( y ) ∂ U ∗ ( x , y ) ds y + 1 U ∗ ( x , y ) t i ( y ) ds y − U ∗ ( x , y ) f i ( y ) dy u i ( x ) = ∂ n y α i Γ i Γ i Ω i O. Steinbach DD 17, 6.7.2006 6 / 23

Institut f¨ ur Numerische Mathematik Representation Formula for x ∈ Ω i � � � g i ( y ) ∂ U ∗ ( x , y ) ds y + 1 U ∗ ( x , y ) t i ( y ) ds y − U ∗ ( x , y ) f i ( y ) dy u i ( x ) = ∂ n y α i Γ i Γ i Ω i Computation of the Normal Derivative � � 1 ∂ ∂ g i ( y ) ∂ U ∗ ( x , y ) t i ( y ) ds y − U ∗ ( x , y ) ds y t i ( x ) = 2 t i ( x ) + ∂ n x ∂ n x ∂ n y Γ i Γ i � + 1 ∂ U ∗ ( x , y ) f i ( y ) dy α i ∂ n x Ω i O. Steinbach DD 17, 6.7.2006 6 / 23

Institut f¨ ur Numerische Mathematik Representation Formula for x ∈ Ω i � � � g i ( y ) ∂ U ∗ ( x , y ) ds y + 1 U ∗ ( x , y ) t i ( y ) ds y − U ∗ ( x , y ) f i ( y ) dy u i ( x ) = ∂ n y α i Γ i Γ i Ω i Computation of the Normal Derivative � � 1 ∂ ∂ g i ( y ) ∂ U ∗ ( x , y ) t i ( y ) ds y − U ∗ ( x , y ) ds y t i ( x ) = 2 t i ( x ) + ∂ n x ∂ n x ∂ n y Γ i Γ i � + 1 ∂ U ∗ ( x , y ) f i ( y ) dy α i ∂ n x Ω i Hypersingular Boundary Integral Equation for x ∈ Γ i t i ( x ) = 1 i t i )( x ) + ( D i g i )( x ) + 1 2 t i ( x ) + ( K ′ ( N 1 , i f i )( x ) α i O. Steinbach DD 17, 6.7.2006 6 / 23