Meshfree Adaptative Aitken-Schwarz DTD Domain Decomposition for - PowerPoint PPT Presentation

AS DDM Meshfree Adaptative Aitken-Schwarz DTD Domain Decomposition for Darcy flow Outline DtoN map The GSAM Aitken- D.Tromeur-Dervout Schwarz Adaptive CDCSP/ICJ-UMR5208 Universit´ e Lyon 1, Aitken- Schwarz 15 Bd Latarjet, 69622 Villeurbanne, France. Aitken meshfree Dedicated to Alain Bourgeat’s 60 th birthday Scaling Up and Modeling for Transport and Flow in Porous Media Dubrovnik, Croatia, 13-16 October 2008 Partially founded by : GDR MOMAS, ANR-TL-07 LIBRAERO, ANR-CIS-07 MICAS

AS DDM DTD Outline DtoN map Objectives : make a Schwarz DDM that has : The GSAM scalable properties Aitken- Schwarz Artificial condition independant of the parameter Adaptive Aitken- (even make convergent a divergent Schwarz method) Schwarz can be used as ”black box”, no direct impact on the Aitken meshfree implementation of local solver.

Outline AS DDM DTD Outline The Dirichlet-Neumann Map 1 DtoN map The GSAM Aitken- The Generalized Schwarz Alternating Method 2 Schwarz Adaptive Aitken- Schwarz The Aitken-Schwarz Method 3 Aitken meshfree Non separable operator , non regular mesh, adaptive 4 Aitken-Schwarz Aitken meshfree acceleration 5

The Dirichlet to Neumann map AS DDM DTD Outline Let Ω ⊂ R n a bounded domain with Γ := ∂ Ω Lipschitz. DtoN map The GSAM The trace operator : γ 0 Aitken- Schwarz ∀ u ∈ H 1 (Ω) , ∃ γ 0 u ∈ H 1 / 2 (Γ) satisfying Adaptive Aitken- Schwarz || γ 0 u || H 1 / 2 (Γ) ≤ c T . || u || H 1 (Ω) . (1) Aitken meshfree vice versa the bounded extension operator : ε ∀ v ∈ H 1 / 2 (Γ) , ∃ ε v ∈ H 1 (Ω) satisfying γ 0 ε v = v and || ε v || H 1 (Ω) ≤ c IT . || v || H 1 / 2 (Γ) . (2)

The Dirichlet to Neumann map AS DDM DTD Outline Let Ω ⊂ R n a bounded domain with Γ := ∂ Ω Lipschitz. DtoN map The GSAM The trace operator : γ 0 Aitken- Schwarz ∀ u ∈ H 1 (Ω) , ∃ γ 0 u ∈ H 1 / 2 (Γ) satisfying Adaptive Aitken- Schwarz || γ 0 u || H 1 / 2 (Γ) ≤ c T . || u || H 1 (Ω) . (1) Aitken meshfree vice versa the bounded extension operator : ε ∀ v ∈ H 1 / 2 (Γ) , ∃ ε v ∈ H 1 (Ω) satisfying γ 0 ε v = v and || ε v || H 1 (Ω) ≤ c IT . || v || H 1 / 2 (Γ) . (2)

∂ [ a ji ( x ) ∂ Set L ( x ) u ( x ) = − Σ n u ( x )] , a ji ∈ L ∞ (Ω) (3) AS DDM i , j = 1 ∂ x j ∂ x i DTD Outline L ( . ) is assumed to be uniformly elliptic, DtoN map The GSAM i , j = 1 a ji ( x ) ξ j ξ l ≥ c 0 . | ξ | 2 , ∀ ξ ∈ R n , ∀ x ∈ Ω Σ n Aitken- Schwarz Adaptive The conormal derivative γ 1 is given by Aitken- Schwarz i , j = 1 n j ( x )[ a ji ( x ) ∂ Aitken γ 1 u ( x ) := Σ n u ( x )] , ∀ x ∈ Γ meshfree ∂ x i where n ( x ) is the exterior unit normal vector. n ∂ v ( x ) a ji ( x ) ∂ � � a ( u , v ) = u ( x ) ∂ x j ∂ x i Ω i , j = 1 � � = Lu ( x ) v ( x ) dx + γ 1 u ( x ) γ 0 v ( x ) dS x Ω Γ

∂ [ a ji ( x ) ∂ Set L ( x ) u ( x ) = − Σ n u ( x )] , a ji ∈ L ∞ (Ω) (3) AS DDM i , j = 1 ∂ x j ∂ x i DTD Outline L ( . ) is assumed to be uniformly elliptic, DtoN map The GSAM i , j = 1 a ji ( x ) ξ j ξ l ≥ c 0 . | ξ | 2 , ∀ ξ ∈ R n , ∀ x ∈ Ω Σ n Aitken- Schwarz Adaptive The conormal derivative γ 1 is given by Aitken- Schwarz i , j = 1 n j ( x )[ a ji ( x ) ∂ Aitken γ 1 u ( x ) := Σ n u ( x )] , ∀ x ∈ Γ meshfree ∂ x i where n ( x ) is the exterior unit normal vector. n ∂ v ( x ) a ji ( x ) ∂ � � a ( u , v ) = u ( x ) ∂ x j ∂ x i Ω i , j = 1 � � = Lu ( x ) v ( x ) dx + γ 1 u ( x ) γ 0 v ( x ) dS x Ω Γ

∂ [ a ji ( x ) ∂ Set L ( x ) u ( x ) = − Σ n u ( x )] , a ji ∈ L ∞ (Ω) (3) AS DDM i , j = 1 ∂ x j ∂ x i DTD Outline L ( . ) is assumed to be uniformly elliptic, DtoN map The GSAM i , j = 1 a ji ( x ) ξ j ξ l ≥ c 0 . | ξ | 2 , ∀ ξ ∈ R n , ∀ x ∈ Ω Σ n Aitken- Schwarz Adaptive The conormal derivative γ 1 is given by Aitken- Schwarz i , j = 1 n j ( x )[ a ji ( x ) ∂ Aitken γ 1 u ( x ) := Σ n u ( x )] , ∀ x ∈ Γ meshfree ∂ x i where n ( x ) is the exterior unit normal vector. n ∂ v ( x ) a ji ( x ) ∂ � � a ( u , v ) = u ( x ) ∂ x j ∂ x i Ω i , j = 1 � � = Lu ( x ) v ( x ) dx + γ 1 u ( x ) γ 0 v ( x ) dS x Ω Γ

Necas Lem. ⇒ ∃ ! u = u 0 + ε g ∈ H 1 (Ω) sol. of Dirichlet Pb AS DDM L ( x ) u ( x ) = f ( x ) , for x ∈ Ω , γ 0 u ( x ) = g ( x ) , for x ∈ Γ (4) DTD Outline Then defining the linear application ∀ w ∈ H 1 / 2 (Γ) DtoN map The GSAM � Aitken- l ( w ) = a ( u , ε w ) − f ( x ) ε w ( c ) dx . Schwarz Ω Adaptive Aitken- Riez thm : ∃ λ ∈ H − 1 / 2 (Γ) : � λ, w � L 2 (Γ) = l ( w ) ∀ w ∈ H 1 / 2 (Γ) . Schwarz Aitken meshfree Hence, the conormal derivative λ ∈ H − 1 / 2 (Γ) satisfies � � f ε w dx ∀ w ∈ H 1 / 2 (Γ) . λ w ds x = a ( u 0 + ε g , ε w ) − Γ Ω ⇒ f fixed, we have a DtoN map : g = γ 0 u �→ λ := γ 1 u γ 1 u ( x ) = Sg ( x ) − Nf ( x ) , ∀ w ∈ Γ (5)

Necas Lem. ⇒ ∃ ! u = u 0 + ε g ∈ H 1 (Ω) sol. of Dirichlet Pb AS DDM L ( x ) u ( x ) = f ( x ) , for x ∈ Ω , γ 0 u ( x ) = g ( x ) , for x ∈ Γ (4) DTD Outline Then defining the linear application ∀ w ∈ H 1 / 2 (Γ) DtoN map The GSAM � Aitken- l ( w ) = a ( u , ε w ) − f ( x ) ε w ( c ) dx . Schwarz Ω Adaptive Aitken- Riez thm : ∃ λ ∈ H − 1 / 2 (Γ) : � λ, w � L 2 (Γ) = l ( w ) ∀ w ∈ H 1 / 2 (Γ) . Schwarz Aitken meshfree Hence, the conormal derivative λ ∈ H − 1 / 2 (Γ) satisfies � � f ε w dx ∀ w ∈ H 1 / 2 (Γ) . λ w ds x = a ( u 0 + ε g , ε w ) − Γ Ω ⇒ f fixed, we have a DtoN map : g = γ 0 u �→ λ := γ 1 u γ 1 u ( x ) = Sg ( x ) − Nf ( x ) , ∀ w ∈ Γ (5)

Necas Lem. ⇒ ∃ ! u = u 0 + ε g ∈ H 1 (Ω) sol. of Dirichlet Pb AS DDM L ( x ) u ( x ) = f ( x ) , for x ∈ Ω , γ 0 u ( x ) = g ( x ) , for x ∈ Γ (4) DTD Outline Then defining the linear application ∀ w ∈ H 1 / 2 (Γ) DtoN map The GSAM � Aitken- l ( w ) = a ( u , ε w ) − f ( x ) ε w ( c ) dx . Schwarz Ω Adaptive Aitken- Riez thm : ∃ λ ∈ H − 1 / 2 (Γ) : � λ, w � L 2 (Γ) = l ( w ) ∀ w ∈ H 1 / 2 (Γ) . Schwarz Aitken meshfree Hence, the conormal derivative λ ∈ H − 1 / 2 (Γ) satisfies � � f ε w dx ∀ w ∈ H 1 / 2 (Γ) . λ w ds x = a ( u 0 + ε g , ε w ) − Γ Ω ⇒ f fixed, we have a DtoN map : g = γ 0 u �→ λ := γ 1 u γ 1 u ( x ) = Sg ( x ) − Nf ( x ) , ∀ w ∈ Γ (5)

Necas Lem. ⇒ ∃ ! u = u 0 + ε g ∈ H 1 (Ω) sol. of Dirichlet Pb AS DDM L ( x ) u ( x ) = f ( x ) , for x ∈ Ω , γ 0 u ( x ) = g ( x ) , for x ∈ Γ (4) DTD Outline Then defining the linear application ∀ w ∈ H 1 / 2 (Γ) DtoN map The GSAM � Aitken- l ( w ) = a ( u , ε w ) − f ( x ) ε w ( c ) dx . Schwarz Ω Adaptive Aitken- Riez thm : ∃ λ ∈ H − 1 / 2 (Γ) : � λ, w � L 2 (Γ) = l ( w ) ∀ w ∈ H 1 / 2 (Γ) . Schwarz Aitken meshfree Hence, the conormal derivative λ ∈ H − 1 / 2 (Γ) satisfies � � f ε w dx ∀ w ∈ H 1 / 2 (Γ) . λ w ds x = a ( u 0 + ε g , ε w ) − Γ Ω ⇒ f fixed, we have a DtoN map : g = γ 0 u �→ λ := γ 1 u γ 1 u ( x ) = Sg ( x ) − Nf ( x ) , ∀ w ∈ Γ (5)

Outline AS DDM DTD Outline The Dirichlet-Neumann Map 1 DtoN map The GSAM Aitken- The Generalized Schwarz Alternating Method 2 Schwarz Adaptive Aitken- Schwarz The Aitken-Schwarz Method 3 Aitken meshfree Non separable operator , non regular mesh, adaptive 4 Aitken-Schwarz Aitken meshfree acceleration 5

The Generalized Schwarz Alternating Method (GSAM) B. Engquist and H.-K. Zhao, Appl. Numer. Math. 27 (1998), no. 4, 341–365. AS DDM DTD Consider Ω = Ω 1 ∪ Ω 2 with the two artificial boundaries Γ 1 , Γ 2 Outline intersecting ∂ Ω . DtoN map The GSAM Algorithm Aitken- Schwarz L ( x ) u 2 n + 1 f ( x ) , ∀ x ∈ Ω 1 , u 2 n + 1 Adaptive ( x ) = ( x ) = g ( x ) , ∀ x ∈ ∂ Ω 1 \ Γ 1 , Aitken- 1 1 Schwarz ∂ u 2 n + 1 ∂ u 2 n ( x ) 2 ( x ) Λ 1 u 2 n + 1 1 = Λ 1 u 2 n + λ 1 2 + λ 1 , ∀ x ∈ Γ 1 Aitken 1 ∂ n 1 ∂ n 1 meshfree L ( x ) u 2 n + 2 f ( x ) , ∀ x ∈ Ω 2 , u 2 n + 2 ( x ) = ( x ) = g ( x ) , ∀ x ∈ ∂ Ω 2 \ Γ 2 , 2 2 ∂ u 2 n + 2 ∂ u 2 n + 1 ( x ) ( x ) Λ 2 u 2 n + 2 2 = Λ 2 u 2 n + 1 1 + λ 2 + λ 2 , ∀ x ∈ Γ 2 . 2 1 ∂ n 2 ∂ n 2 where Λ i ’s are some operators and λ i ’s are constants. (Λ 1 = I , λ 1 = 0 , Λ 2 = 0 , λ 2 = 1 ) Schwarz Neumann-Dirichlet Algorithm

AS DDM If λ 1 = 1 and Λ 1 is the DtoN operator at Γ 1 associated to the DTD homogeneous PDE in Ω 2 with homogeneous boundary Outline condition on ∂ Ω 2 ∩ ∂ Ω then GSAM converge in two steps. DtoN map proof Let e n i = u − u n , i = 1 , 2 , , then The GSAM Aitken- L ( x ) e 1 0 , ∀ x ∈ Ω 1 , e 1 Schwarz 1 ( x ) = 1 ( x ) = 0 , ∀ x ∈ ∂ Ω 1 \ Γ 1 , Adaptive ∂ e 1 2 + ∂ e 0 1 ( x ) 2 ( x ) Aitken- Λ 1 e 1 = Λ 1 e 0 + , ∀ x ∈ Γ 1 Schwarz 1 ∂ n 1 ∂ n 1 Aitken meshfree since Λ 1 is the DtoN operator at Γ 1 in Ω 2 ∂ e 0 − ∂ e 0 + ∂ e 0 2 + Λ 1 e 0 2 2 = 0 , ⇒ e 1 = 1 = 0 in Ω 1 2 ∂ n 1 ∂ n 2 ∂ n 2 Hence we get the exact solution in two steps []