Domain Decomposition Algorithms for Mortar discretizations Hyea - PowerPoint PPT Presentation

Domain Decomposition Algorithms for Mortar discretizations Hyea Hyun Kim Courant Institute (NYU) Email: hhk2@cims.nyu.edu July 4, 2006 1 / 50

Outline Outline Mortar 1. Mortar discretizations discretizations DD for mortar Nonmatching triangulations, discretizations Overlapping Schwarz Geometrically nonconforming partitions. BDDC and FETI–DP for mortar 2. Domain decomposition algorithms Analysis Additional Overlapping Schwarz algorithms, Applications Numerical Results FETI-DP, BDDC algorithms. Conclusions 3. Additional applications to Elasticity, Stokes, Inexact coarse problem. 4. Numerical results 5. Conclusions 2 / 50

Mortar element methods (by Bernardi, Maday, and Patera (1994)) Outline Coupling different approximations in N different subdomains Mortar discretizations DD for mortar discretizations Overlapping Schwarz BDDC and FETI–DP for mortar Analysis Additional � N Applications X = i =1 X i , finite element Numerical Results Conclusions space Glue ( v 1 , · · · , v N ) ∈ X across the interface F ij � ( v i − v j ) ψ ds = 0 , ∀ ψ ∈ M ( F ij ) (1) F ij We call (1) mortar matching condition . 3 / 50

Geometrically Nonconforming partitions Outline F ij Mortar Ω j F l discretizations Ω i DD for mortar discretizations F ik Overlapping Schwarz BDDC and Ω k FETI–DP for mortar Analysis Additional Applications Numerical Results Conclusions F ij = ∂ Ω i ∩ ∂ Ω j : interface { F l } : a collection of subdomain faces such that � � F l ∩ F k = ∅ F l = F ij , l ij F l : nonmortar side, { F ij , F ik } : mortar sides. 4 / 50

Lagrange multipliers spaces ✔ Outline M ( F ) on each nonmortar face F Mortar discretizations ✘ the same dimension as that of finite element functions DD for mortar discretizations supported in F Overlapping Schwarz BDDC and ✘ contains constant functions FETI–DP for mortar Analysis ✔ Examples, standard (left) and dual (right) Additional Applications Numerical Results Conclusions 1 1 F F 2 M (F) M(F) 1 1 F F −1 (by Wohlmuth) computationally more efficient, easy to implement 5 / 50

Mortar Discretization ✔ Outline Mortar finite element space Mortar discretizations X ⊂ X = � N DD for mortar � i =1 X i satisfying mortar discretizations Overlapping Schwarz matching condition BDDC and FETI–DP for mortar Analysis ✔ Error estimate for a mortar discretization: Additional Applications Numerical Results For elliptic problems with P 1 -finite elements in X i , Conclusions N N � � � u − u h � 2 h 2 i | log( h i ) | � u � 2 1 , Ω i ≤ 2 , Ω i . i =1 i =1 | log( h i ) | : from geometrical nonconformity 6 / 50

Previous DD algorithms for mortar discretization ✔ Substructuring methods Outline Mortar discretizations (by Achdou, Maday, and Widlund) DD for mortar discretizations Geometrically nonconforming partitions Overlapping Schwarz Condition number bound (1 + log H h ) 2 BDDC and FETI–DP for mortar Analysis Additional ✔ Overlapping Schwarz Applications (by Achdou and Maday) Numerical Results Conclusions ✘ Convergence analysis ✘ Additional coarse space ✘ Condition number bound (1 + ( H δ )) 7 / 50

New Results ✔ Extension to 3 D geometrically non-conforming Outline Mortar discretizations partitions DD for mortar discretizations ✔ Smaller coarse problems Overlapping Schwarz ✔ Simpler analysis BDDC and FETI–DP for mortar Analysis for Additional Applications Numerical Results ✔ Overlapping Schwarz methods Conclusions ✔ Dual–Primal FETI methods (by Farhat et al) ✔ BDDC methods (by Dohrmann) 8 / 50

Overlapping Schwarz algorithm for mortar discretization Outline (Joint work with Olof B. Widlund) Mortar discretizations ✔ Nonoverlapping subdomain partition { Ω i } i DD for mortar discretizations equipped with mortar discretization Overlapping Schwarz Overlapping subregion partition { � BDDC and ✔ Ω j } j FETI–DP for mortar ✔ Analysis Coarse triangulation { T k } k Additional Applications Numerical Results Conclusions coarse triangulation overlapping subregions (coarse problem) (local problems) 9 / 50

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Preconditioner (a coarse space contained in the mortar finite element space) Outline Mortar discretizations DD for mortar discretizations Overlapping Schwarz BDDC and FETI–DP for mortar Analysis Coarse finite element space X H Additional Applications Numerical Results I h ( v ) : X H → X defined by ✔ Conclusions I h ( v ) = ( I h 1 ( v ) , · · · , I h N ( v )) , I h i ( v ) : nodal interpolant to X i . Interpolant I m : X H → � X defined by modifying I h ( v ) on ✔ the nonmortar side to satisfy the mortar matching. 11 / 50

The coarse problem ✔ Coarse function space Outline Mortar discretizations DD for mortar V H = I m ( X H ) ⊂ � discretizations X. Overlapping Schwarz BDDC and FETI–DP for mortar Coarse problem Find T 0 u ∈ V H , ✔ Analysis Additional Applications ∀ v H ∈ V H . a ( T 0 u, v H ) = a ( u, v H ) , Numerical Results Conclusions 12 / 50

Condition number bound ✔ Condition number estimate Outline Mortar discretizations � � DD for mortar � N � H j )(1 + log H k discretizations κ ( T j ) ≤ C max (1 + ) Overlapping Schwarz δ j h k j,k BDDC and j =0 FETI–DP for mortar Analysis Additional Note: Additional log-factor from geometrically Applications non-conforming partitions. Numerical Results Conclusions � H j : subregion diameter δ j : overlapping width H k /h k : the num. of elements across a subdomain Ω k 13 / 50

BDDC and FETI–DP for the mortar discretization ✔ Outline Form two equivalent linear systems of mortar Mortar discretization discretizations DD for mortar discretizations 1. primal form Overlapping Schwarz BDDC and 2. dual form FETI–DP for mortar Analysis Additional Applications ✔ Develop BDDC and FETI–DP Numerical Results BDDC (primal formulation) Conclusions FETI–DP (dual formulation) ✔ Providing preconditioners as efficient as the ones in the conforming case κ ( B DDC ) , κ ( F DP ) ≤ C (1 + log ( H/h )) 2 14 / 50