A three-level BDDC algorithm for mortar discretization Hyea Hyun - PowerPoint PPT Presentation

A three-level BDDC algorithm for mortar discretization Hyea Hyun Kim National Institute for Mathematical Sciences Email: hhk@nims.re.kr Joint project with Xuemin Tu ( University of California, Berkeley ) November 21, 2006 1 / 18

Outline Outline BDDC algorithm 1. Three-level BDDC algorithms Inexact coarse problem 2. Extension to Mortar discretizations Mortar discretization Three-Level BDDC 3. Condition number analysis for mortar 4. Numerical results Numerical Result 2 / 18

BDDC algorithm We consider a discrete problem of the elliptic problem using a Outline BDDC algorithm finite element space � X in Ω , Inexact coarse problem Mortar discretization Three-Level BDDC for mortar Numerical Result −∇ · ( ρ ( x ) ∇ u ( x )) = f ( x ) , x ∈ Ω , u ( x ) = 0 , x ∈ ∂ Ω . � Au = f ( ∗ ) . When � A is large and sparse, iterative methods with a preconditioner are much more efficient than the direct method (Gaussian elimination). Goal: Find a good preconditioner for (*), easily adaptable to parallel processors with a good scalability. 3 / 18

BDDC algorithm Outline Domain Decomposition (nonoverlapping) BDDC algorithm Inexact coarse problem N � Mortar discretization Ω = Ω i . Three-Level BDDC for mortar i =1 Numerical Result X i : Finite element space in Ω i Local problem : A i , 1 ≤ i ≤ N A coarse problem : A 0 from a special coarse finite element space defined in Ω , its dimension is comparable to N , the number of subdomains. � N Preconditioner M − 1 = i D i A − 1 i =0 R t i D i R i , with a good scalability κ ( M − 1 � A ) ≤ C (1 + log ( H/h )) 2 . H/h : local problem size. 4 / 18

Construction of local and coarse problems Finite Element Spaces Outline BDDC algorithm Inexact coarse problem �� Mortar discretization �� Three-Level BDDC �� for mortar �� Numerical Result �� N �� X X X = i =1 X i Selection of dual and primal unknowns u ∆ : dual, u Π : primal Enforcing strong continuity at the primal unknowns on X gives the space � X . We subassemble local problem matrices at the primal unknowns and obtain a discrete problem � A in � X . 5 / 18

Subassembly We order the local problem matrix into dual and primal, Outline BDDC algorithm Inexact coarse � � problem A ( i ) A ( i ) Mortar discretization ∆∆ ∆Π A i = Three-Level BDDC A ( i ) A ( i ) for mortar Π∆ ΠΠ Numerical Result We obtain a subassembled matrix,   A (1) A (1) ∆Π R (1) ∆∆ Π   ...     � A = ,   A ( N ) A ( N ) ∆Π R ( N )     ∆∆ Π � t A (1) t A ( N ) t A ( i ) R (1) R ( N ) i R ( i ) ΠΠ R ( i ) · · · Π Π∆ Π Π∆ Π Π R ( i ) Π : X Π → X ( i ) Π : restriction. 6 / 18

Construction of local and coarse problems We simply denote it by Outline BDDC algorithm Inexact coarse � � problem A ∆∆ A ∆Π � Mortar discretization A = . Three-Level BDDC A t A ΠΠ for mortar ∆Π Numerical Result Using a block Cholesky factorization, we obtain � � � � � � A − 1 I 0 A ∆∆ 0 I ∆∆ A ∆Π � A = , ∆Π A − 1 A t I 0 S ΠΠ 0 I ∆∆ ∆Π A − 1 S ΠΠ = A ΠΠ − A t ∆∆ A ∆Π . In addition, we have � � � � � � − A − 1 A − 1 I ∆∆ A ∆Π 0 I 0 A − 1 = � ∆∆ , S − 1 ∆Π A − 1 − A t 0 I 0 I ΠΠ ∆∆ 7 / 18

BDDC preconditioner Outline Discrete problem using � X , BDDC algorithm Inexact coarse problem � Au = f. Mortar discretization Three-Level BDDC for mortar Using the discrete problem � A of � Numerical Result X , we build a preconditioner M − 1 = � R t D � A − 1 D � R, where � R : � X → � X (extension), D is a certain weight. � � − A − 1 ∆∆ A ∆Π Let Φ = . I �� A − 1 0 M − 1 = � D � R t D + Φ S − 1 ΠΠ Φ t ∆∆ R. 0 0 8 / 18

To improve scalability Scalability of M − 1 ( H/h : local problem size ) Outline BDDC algorithm Inexact coarse problem κ ( M − 1 A ) ≤ C (1 + log ( H/h )) 2 . Mortar discretization Three-Level BDDC for mortar ✄ The number of iterations does not depend on the number Numerical Result of subdomains (processors). Using N times larger number of subdomains with a fixed local problem size, we can solve N times larger problem in a similar wall clock time. ✄ Note that, at each iteration we solve the local problems A − 1 ∆∆ and the coarse problem S − 1 ΠΠ exactly . 9 / 18

To improve scalability ✄ However, as the increase of the number of subdomains the Outline BDDC algorithm coarse problem becomes the bottleneck so that the scalability Inexact coarse problem can be expected only up to a limited number of subdomains Mortar discretization Three-Level BDDC (processors) (up to 1,000). The largest one with 100,000 for mortar Numerical Result processors is available. To improve the scalability, we want to solve the coarse problem cheaply. �� A − 1 0 M − 1 = � D � R t D + Φ S − 1 ΠΠ Φ t ∆∆ R. 0 0 ✄ We will replace S − 1 ΠΠ with M − 1 S , a BDDC preconditioner for S ΠΠ , by introducing another level, subregion partition, where each subregion is a union of subdomains. 10 / 18

Mortar discretization ✔ Mortar matching condition Outline BDDC algorithm Inexact coarse problem Mortar discretization Three-Level BDDC for mortar Numerical Result � ( v i − v j ) ψ ds = 0 , ∀ ψ ∈ M ij Γ ij � N Mortar finite element space is a subspace of i =1 X i that satisfies the mortar matching condition. Mortar discretization is to approximate the solution in the mortar finite element space . 11 / 18

Three-level algorithm The mortar discretization gives Outline BDDC algorithm Inexact coarse problem Au = f. Mortar discretization Three-Level BDDC for mortar We build a BDDC preconditioner by selecting appropriate Numerical Result primal unknowns, � ( v i − v j ) ψ ds = 0 , ψ = 1 , F ij � � v Π = v i ds = v j ds. F ij F ij We then transform unknowns v i into ( v ∆ , v Π ) and obtain partially assembled matrix � A to build a BDDC preconditioner. 12 / 18

A three-level BDDC algorithm for mortar discretization Hyea Hyun - PowerPoint PPT Presentation

A three-level BDDC algorithm for mortar discretization Hyea Hyun Kim National Institute for Mathematical Sciences Email: hhk@nims.re.kr Joint project with Xuemin Tu ( University of California, Berkeley ) November 21, 2006 1 / 18 Outline

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