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

BDDC Domain Decomposition Algorithms Olof B. Widlund Courant Institute, New York University 75th

Domain Decomposition Algorithms for Mortar discretizations Hyea Hyun Kim Courant Institute (NYU)

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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

BDDC Domain Decomposition Algorithms Olof B. Widlund Courant Institute, New York University 75th

Domain Decomposition Algorithms for Mortar discretizations Hyea Hyun Kim Courant Institute (NYU)

BDDC Algorithms with Adaptive Choices of Primal Constraints Olof B. Widlund Courant Institute,

Higher order solution of ODEs arising from DG space semi-discretization of nonstationary

Multispace and Multilevel BDDC Jan Mandel University of Colorado at Denver and Health Sciences

Discretization and Solution of and Solution of Discretization Convection- -Diffusion Problems

Parallel Implementation of BDDC for Mixed-Hybrid Formulation of Flow in Porous Media Jakub

BLM AREA B, UNIT B-2A MORTAR INVESTIGATION FORMER FORT ORD, CALIFORNIA PROJECT DESCRIPTION

Powdered Mortar Plasticiser Improves workability of bricklaying and plastering mortars The

HIGH-LEVEL SEARCH Instead of directly solving a High-level Search Methods given probem

Adaptive Multilevel BDDC Jan Mandel Includes joint work with Clark Dohrmann, Bed rich Soused

Multilevel and Adaptive Iterative Substructuring Methods Jan Mandel University of Colorado Denver

Coast Rescue Equipment Line throwing from early 1800s to 1948 Equipment Heaving Line

Comparative Experiments E.g. Tension bond strength of mortar (kgf / cm 2 ) Measurements of

Cosserat Rod with rh- Adaptive Discretization Jiahao Wen 1 , Jiong Chen 1 , Nobuyuki Umetani 2 ,

3. Interpolation: Closing the Gaps of Discretization . . . 3. Interpolation: Closing the Gaps of

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All-in-One Marketing &amp; Communications Platform for Brick &amp; Mortar Businesses One

All-in-One Marketing & Communications Platform for Brick & Mortar Businesses One