Preconditioning of Elliptic Saddle Point Systems by Substructuring and a Penalty Approach

Transcription

1 Preconditioning of Ellitic Saddle Point Systems by Sbstrctring and a Penalty roach 6 th International onference on Domain Decomosition Methods Janary 2-5, 2005 lark R. Dohrmann Strctral Dynamics Research Deartment Sandia National Laboratories lbqerqe, New Mexico Sandia is a mltirogram laboratory oerated by Sandia ororation, a Lockheed Martin omany, for the United States Deartment of Energy s National Nclear Secrity dministration nder contract DE-04-94L85000.

2 hanks DD6 scientific & organizing committees Sandia Rich Lehocq Pavel ochev Kendall Pierson Garth Reese University Jan Mandel

3 Overview Saddle Point Preconditioner related enalized roblem ositive real eigenvales conjgate gradients DD Preconditioner overview modified constraints Examles H(grad), H(div), H(crl)

4 Motivation original system: enalized (reglarized) system: xelsson (979) g f ~ ~ ~ 0, ~ > g f > 0 on kernel of, 0,, fll rank

5 r r z z ~ ) ~ ( ) ( ~ r r S z r z z + S ~ + exact soltion of enalized system rimal rather than dal Schr comlement considered is ) ~ ( ~ S +

6 + 0 0 f ρ f + ) ( ρ Some onnections 2 ) / ( ) ( 2 / f G + ρ enalty method for 0 and g 0: ρ / ~ I matrix same as S for G L + agmented Lagrangian:

7 r r z z ~ ) ( ~ r z z reglarized constraint reconditioning: enzi srvey (2005) reglarized constraint eqation satisfied exactly

8 Penalty Preconditioner ) ( ~ ) ~ ( r z z r r S z + S ~ + recall exact soltion of enalized system reconditioner: ) ( ~ ) ~ ( r z z r r M z + M reconditioner for S

9 r r I I M I I z z 0 ~ ~ 0 0 ~ 0 M matrix reresentation: M qestion: what abot sectrm of

10 eigenroblem: z λmz introdce: ramble & Pasciak (988), Klawonn (998) S M 0 H ~ 0 eigenvales same as those for HM z λhz symmetric can show H > 0 HM - > 0

11 G onnection (&P too) original linear system: w b eqivalent linear system: HM - w HM - b solve sing cg with H as reconditioner eigenvales of reconditioned system same as those of M - H not available, bt not a roblem H S M ~ 0 0

12 k k k k k k k k k k k k k k k k r r z z r r w w HM M ~ ~ + α α α α + ) ( ~ ) ~ ( a d a a M d d M a + d a d a a d S a ) ~ ( HM reqired calclations: recrrences:

13 heory heorem ( 0): Given α >, γ > 0, and α γ M M S α M 2 ~ γ M 2 R R n m eigenvales satisfy where 2 δ δ min δ max M ) and σ > 0, σ 2 > 0, σ + σ 2 σ ( δ { σ ( α / α ), σ /( α γ )} { 2α σ,(2 σ / α ) / γ }

14 recall with α > α M γ M Simlification for S S α M 2 ~ γ M ~ as ε 0 2 ~ ε 0 R R n m γ bonded below by /α 2 and γ 2 bonded above by /α ( α α )/ 2 σ ( M ) 3 / 2 / α 2 2

15 two goals:. Ensre α > (i.e. S M > 0) 2. Minimize α 2 /α (M good reconditioner for S ) otential isses:. Scaling reconditioner M to satisfy Goal 2. S becomes very oorly conditioned as ε 0 conclsion: effective reconditioner for S is essential

16 Reca of Penalty Preconditioner ased on aroximate soltion of related enalized roblem onjgate gradients can be sed to solve saddle oint system heory rovides conditions for scalability Effectiveness hinges on reconditioner for rimal Schr comlement S Ref:. R. Dohrmann and R.. Lehocq, rimal based enalty reconditioner for ellitic saddle oint systems, Sandia National Laboratories, echnical reort SND J.

17 DD Overview Primal Sbstrctring Preconditioner no coarse trianglation needed recent theory by Mandel et al mltilevel extensions straightforward M7 Nmerical Proerties (+log(h/h)) 2 condition nmber bonds reconditioned eigenvales all (woo hoo!) eigenvales identical to FEI-DP local and global roblems only reqire sarse solver for definite systems

18 FE discretization of de elements artitioned into sbstrctres solve for nknowns on sb bondaries (Schr comlement)

19 ilding locks dditive Schwarz: oarse grid correction v Sbstrctre correction v 2 Static condensation correction v 3 M r v + v2 + v 3 FEI-DP conterarts: v 2 N s s s s K s r rr r k λ v v 3 F I rc K * cc F I rc k λ Dirichlet reconditioner

20 oarse Grid and Sbstrctre Problems Λ Φ I Q Q K i i i i i 0 0 i i i ci K K Φ Φ K ci coarse element matrix assemble K ci K c K c ositive definite i i i i i i r v Q Q K λ sbstrctre roblem: coarse roblem:

21 Mixed Formlation of Linear Elasticity f 0 ( + ) f + - has same sarsity as (discontinos ressre ) DD reconditioner with enriched coarse sace investigated by Goldfeld (2003) related work for incomressible roblems in Pavarino and Widlnd (2002) and Li (200), see also M7

22 M λ µ Let µ 0 and (/λ)m where ) 2 )( ( ) ( 2 ν ν ν λ ν µ + + E E recall f + ) ( notice as ν ½ that λ condition nmber

23 2D Plane Strain Examle condition nmber estimates for 4 sbstrctre roblem with H/h 8 ν κ e2 6.2e3 of reconditioned system κ /( 2ν) for ν near ½ Q: why does κ as ν ½for DD reconditioner?

24 Exlanation Movement of sbstctre bondary nodes is weighted average of coarse grid and sbstrctre corrections (v and v 2 ) from neighboring sbs If sbstrctre volme changes, then strain energy of sbstrctre as ν ½ cg ste length α 0 since cg minimizes energy Soltion Modify standard DD constraint eqations to enable enforcement of zero volme change Same nmber of constraints in 2D slightly more in 3D

25 condition nmber estimates for 4 sbstrctre roblem with H/h 8 ν original e2 6.2e3 κ modified

26 Reca of DD Preconditioner Performance sensitive to vales of ν near ½ if standard constraints sed Sensitivity cased by sbstrctre volme changes for nearly incomressible materials Simle modification of constraints can effectively accommodate roblems with ν near ½ Ref:. R. Dohrmann, sbstrctring reconditioner for nearly incomressible elasticity roblems, Sandia National Laboratories, echnical reort SND J.

27 Examles Problem yes H(grad): incomressible elasticity H(div): Darcy s roblem H(crl): magnetostatics Preconditioning roaches Penalty/DD for H(grad) & H(div) DD for stabilized H(crl) roblem

28 2D Strctred Meshes H(grad): Q 2 -P H(grad): P 2+ -P H(div) & H(crl): R 0 & Ned

29 2D Unstrctred Mesh 2937 elements

30 3D Unstrctred Mesh 094 elements

31 Incomressible Elasticity mixed variational formlation: 2µ ε ( ) : ε ( v) dx + wdx f wdx Ω Ω Ω q dx 0 Ω w q where ε i,j () ( i,j + j,i )/2,,w H (Ω),,q L 2 (Ω), µ recall: 0 enalty reconditioner: ~ (2,2) block of for ν < ½

32 2D Incomressible Plane Strain Examle iterations for rtol 0-6 for 6 sbstrctre roblem with H/h 8 ν GMRES PG (6) (7.2) (3.0) 0 (2.7) Note: ν is Poisson ratio sed to define ~ in enalty reconditioner (2.7) (2.6)

33 Other Saddle Point Preconditioners block diagonal: Fortin, Silvester, Wathen, Klawonn, D M M 0 0 M M M 0 M M : DD reconditioner for, M : ressre mass matrix block trianglar: Elman, Silvester, Klawonn, MINRES GMRES

34 2D Strctred Mesh omarison iterations for rtol 0-6 for N sbstrctre roblem with H/h 4 N M M D M GMRES PG GMRES GMRES (.8) (2.) (2.6) (2.9) (3.0) (3.) (3.) (3.) Note: ν for enalty reconditioner

35 Unstrctred Mesh omarisons dim M M D M GMRES PG GMRES GMRES 2D (3.2) D (37) Note: ν in 2D and ν in 3D for enalty reconditioner

36 comatibility condition: Darcy s Problem governing eqations: K in Ω f in Ω n 0 on Ω Ω fdx 0 discretization: lowest-order R- simlicial elements enalty reconditioner: ~ chosen as εd s and DD for S

37 2D Darcy s Problem Examle iterations for rtol 0-6 for 6 sbstrctre roblem with H/h 8, K I ε GMRES PG 9 00 (.9e3) (2.e2) 20 (34) 0 (5.8) ~ Note: εd reconditioner s in enalty (.9) (.5)

38 Other Saddle Point Preconditioners block diagonal : norm eqivalence (Klawonn, 995) MD M 0 div 0 M (, w) wdx + ( )( w) dx div Ω Ω block diagonal 2: MD 2 M 0 0 M S S some others: balancing Nemann-Nemann (DD), overlaing methods (SPD redction for div free sace)

39 2D Strctred Mesh omarison iterations for rtol 0-6 for N sbstrctre roblem with H/h 4, K I N M M D GMRES PG GMRES (.0) (.2) 6 (.6) 8 0 ε (.7) (.8) 7 (.9) 0 0 similar reslts for 3D (.9) (2.0) 0

40 2D Strctred Mesh (H/h deendence) iterations for rtol 0-6 for 9 sbstrctre roblem with K I H/h PG 4 (.24) 5 (.53) 5 (.74) 5 (.92) 6 (2.09) similar reslts for 3D and K constant

41 Unstrctred Mesh omarisons dimen M M D GMRES PG GMRES 2D 5 5 (.3) 2 3D 8 8 (2.5) 7 Note: ε 0-5 for enalty reconditioner

42 Magnetostatics ( ) mixed variational form with olomb gage ( 0): ( / µ )( ) ( w) dx + w dx J wdx w H ( crl) 0 Ω Ω qdx Ω Note: was integrated by arts. mmer, bt J 0 J 0 and w 0 crl x b with crl 0, bt system consistent and niqe Ω 0 q H 0

43 Q: How to solve crl x b with crl 0? One otion: Solve in sace restricted to 0 need basis for this sace or back to saddle oint system nother otion: Solve crl x b sing G with reconditioner for crl + div dvantage: can aly reconditioners for H(grad) roblems! Ref:. R. Dohrmann, Preconditioning of crl-crl eqations by a enalty aroach, Sandia National Laboratories, echnical reort, in rearation.

44 Examle 28 elements, 208 edges, 8 nodes crl 80 zevals before 49 zevals after 2 zevals before 0 zevals after

45 Some Other Otions. recondition saddle oint system: lenary L3 (Zo) 2. introdce reglarized roblem: Rietzinger & Schöberl (2002) (/ µ )( ) ( w) dx Ω + σ wdx J wdx Ω Ω where σ > 0 ( crl + σ mass )x b, σ 0 eqivalance, and recondition: mltigrid (Hitmair, rnold, Falk, Winther, R&S, ) overlaing (oselli, Hitmair) sbstrctring (oselli, Widlnd, Wohlmth, H, Zo) FEI-DP: lenary L (oselli)

46 Recall crl + div. ly s and consider α x x x x crl x R( ) crl Note: max α is smallest nonzero eig of crl w λ w h /4 /8 /2 /6 /20 /24 α Note: PG convergence deends on /α for exact solves w/

47 2D Strctred Mesh DD reconditioner for with H/h 4, rtol 0-0 h α / / / / / / Similar reslts for 3D roblems, OS too iter cond

48 Reca of alk Penalty Preconditioner aroximate soltion of related enalized roblem symmetric indefinite, bt σ(m - ) real and ositive G for saddle oint systems DD Preconditioner well sited for se with enalty reconditioner constraint modification for near incomressibility alicable to roblems in H(grad), H(div), and H(crl) Divergence Stabilization ermits se of H(grad) reconditioners for crl-crl