Substructuring for multiscale problems

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1 Substructuring for multiscale problems Clemens Pechstein Johannes Kepler University Linz (A) jointly with Rob Scheichl Marcus Sarkis Clark Dohrmann DD 21, Rennes, June 2012

2 Outline 1 Introduction 2 Weighted Poincaré Inequalities (WPI) 3 FETI Basics 4 Multiscale Toolkit 5 TFETI 6 FETI-DP Clemens Pechstein Substructuring for multiscale problems 2 / 38

3 Outline 1 Introduction 2 Weighted Poincaré Inequalities (WPI) 3 FETI Basics 4 Multiscale Toolkit 5 TFETI 6 FETI-DP Clemens Pechstein Substructuring for multiscale problems 3 / 38

Model Problem Find u H 1 (Ω), u ΓD = 0 Ω R d d = 2, 3 Ω α u v dx = f (v) v H 1

Conditioning of FE system: ( α(x) ) sup x,y Ω α(y) }{{} O(10 3 ) O(10 10 ) h 2

flows oil reservoir simulation elasticitiy with heterogeneous material

4 Model Problem Find u H 1 (Ω), u ΓD = 0 Ω R d d = 2, 3 Ω α u v dx = f (v) v H 1 (Ω), v ΓD = 0 strongly varying coefficient α L + (Ω) (uniformly positive) Conditioning of FE system: ( α(x) ) sup x,y Ω α(y) }{{} O(10 3 ) O(10 10 ) h 2 Goal: iterative solvers, robust in h and α towards applications: porous media flows oil reservoir simulation elasticitiy with heterogeneous material nonlinear magnetostatics Clemens Pechstein Substructuring for multiscale problems 4 / 38

5 Spectral properties I spectrum square.dat -- H/h=32 B=I unit square uniform mesh h = 1/32 pure Neumann problem 1e σ(k h,α ) 1e-06 1e-08 1e-10 1e spectrum many_large_islands.dat -- H/h=32 B=I spectrum many_large_islands_qm.dat -- H/h=32 B=I 1e+06 1e e-06 1e-06 1e-08 1e-08 1e-10 1e-10 1e e Clemens Pechstein Substructuring for multiscale problems 5 / 38

6 Spectral properties II spectrum square.dat -- H/h=32 B=Kdiag unit square uniform mesh h = 1/32 pure Neumann problem σ(diag(k h,α ) 1 K h,α ) 1e e-06 1e-08 1e-10 1e spectrum many_large_islands.dat -- H/h=32 B=Kdiag spectrum many_large_islands_qm.dat -- H/h=32 B=Kdiag 1e+06 1e e-06 1e-06 1e-08 1e-08 1e-10 1e-10 1e e Clemens Pechstein Substructuring for multiscale problems 5 / 38

Spectral properties III unit square uniform mesh h = 1/32 pure Neumann problem σ(m 1 h,α K h,α) 1e+06 10000 100 1 0.01 0.

dat -- H/h=32 B=Malpha 1e+06 10000 100 1 0.01 0.0001 1e-06 1e-08 1e-10 1e+06 10000 100 1 0.01 0.0001 1e-06 1e-08 1e-10 spectrum many_large_islands_qm.

7 Spectral properties III unit square uniform mesh h = 1/32 pure Neumann problem σ(m 1 h,α K h,α) 1e e-06 1e-08 1e-10 1e-12 spectrum square.dat -- H/h=32 B=Malpha spectrum many_large_islands.dat -- H/h=32 B=Malpha 1e e-06 1e-08 1e-10 1e e-06 1e-08 1e-10 spectrum many_large_islands_qm.dat -- H/h=32 B=Malpha 1e e Clemens Pechstein Substructuring for multiscale problems 5 / 38

8 Weighted Poincaré Inequality (WPI) domain D, coefficient α L + (D) Definition C P,α (D) smallest constant such that u H 1 (D): inf c R u c L 2 (D),α C P,α (D) diam(d) u }{{} H 1 (D),α λ 2 (M 1 h,α K h,α) 1/2 where [Chua] [Chua & Wheeden] [Veeser & Verfürth] [Zikhov]... v 2 L 2 (D),α = α v 2 dx, v 2 H 1 (D),α = α v 2 dx D D infimum attained at weighted average: c = u D,α = D α u dx D α dx For quasi-monotone coefficients (details soon): C P,α (D) C independently of contrast [P. & Scheichl 11, 12*] early results by [Galvis & Efendiev 10] Clemens Pechstein Substructuring for multiscale problems 6 / 38

9 Weighted Poincaré Inequality (WPI) domain D, coefficient α L + (D) Definition C P,α (D) smallest constant such that u H 1 (D): inf c R u c L 2 (D),α C P,α (D) diam(d) u }{{} H 1 (D),α λ 2 (M 1 h,α K h,α) 1/2 where [Chua] [Chua & Wheeden] [Veeser & Verfürth] [Zikhov]... v 2 L 2 (D),α = α v 2 dx, v 2 H 1 (D),α = α v 2 dx D D infimum attained at weighted average: c = u D,α = D α u dx D α dx For quasi-monotone coefficients (details soon): C P,α (D) C independently of contrast [P. & Scheichl 11, 12*] early results by [Galvis & Efendiev 10] Clemens Pechstein Substructuring for multiscale problems 6 / 38

10 Preconditioners for multiscale problems Overlapping Schwarz Graham Lechner Scheichl 07 Scheichl Vainikko 07 Multilevel Methods (GMG AMG AMLI ρamge) Scheichl Vassilevski Zikatanov 12 Galvis Efendiev 10 Scheichl Vassilevski Zikatanov 12 eigensolves Willems 12* Galvis 12* Spillane Dolean Hauret Nataf P. Scheichl 12* * submitted/to appear Galvis Efendiev 10 Galvis Efendiev Willems Lazarov 12 heterogeneities NOT resolved by coarse(st) mesh or subdomains Iterative Substructuring (FETI, FETI DP) BDD, BDDC WPI P. & Scheichl 09, 12* & more P. & Scheichl 08, 09, 11 Dryja Sarkis 11 P. 12* Gippert Klawonn Rheinbach 12* Minisymposium M13 (Tu 16.30, We 10.30) links to upscaling Clemens Pechstein Substructuring for multiscale problems 7 / 38

11 Outline 1 Introduction 2 Weighted Poincaré Inequalities (WPI) 3 FETI Basics 4 Multiscale Toolkit 5 TFETI 6 FETI-DP Clemens Pechstein Substructuring for multiscale problems 8 / 38

12 Quasi-monotone coefficients Let α L + (D) be piecewise constant w.r.t. partition {Y k } n k=1 (Y k connected Lipschitz) α k := α Yk = const Y L... subregion with largest coefficient Definition α is quasi-monotone on D iff for each k there exists a path P kl from Y k to Y L : subsequent subregions share common (d 1)-dim. facet coefficient does not decrease along the path 3 4 (a) η η min min (b) (c) (d) Clemens Pechstein Substructuring for multiscale problems 9 / 38

13 path inequalities weighted inequality α quasi-monotone P kl corresp. path from Y k to Y L Y k X* YL X Y L Definition c kl best constant: u u X 2 L 2 (Y k ) c kl diam(d) 2 u 2 H 1 (P kl ) u H 1 (P kl ) Lemma u u X 2 L 2 (D),α ( n ) c kl diam(d) 2 u 2 H 1 (D),α u H1 (D) = C P,α (D) 2 k=1 ( n ) c kl k=1 independent of constrast! Clemens Pechstein Substructuring for multiscale problems 10 / 38

14 Explicit dependence on geometric scales P Y 1 L Y L 1 X* X 1 X 2 X 3 Lemma P kl... path from Y k to Y L c kl 2 Y l P kl X i Y l Y k Y l X i... interfaces or X diam(y l ) 2 diam(d) 2 C P(Y l, X i ) Lemma α quasi-monotone on D {Y l } form shape regular partition, not extremely long paths C P,α (D) 2 meas(d) ( diam(d) ) d 1 s max diam(d) 2 η d 2 η min min Clemens Pechstein Substructuring for multiscale problems 11 / 38

15 WPI for FE functions Definition α type-m quasi-monotone on D iff for each k there exists a path P kl from Y k to Y L : subsequent subregions share common m-dim. facet weights within the path do not decrease (a) η (b) (c) (d) Analogous statements hold true on FE space V h (D), but constants depend on h: m = d 2: 1 + log(η/h) More details: m = d 3: η/h [P. & Scheichl: IMAJNA 2012 (to appear)] Clemens Pechstein Substructuring for multiscale problems 12 / 38

16 Outline 1 Introduction 2 Weighted Poincaré Inequalities (WPI) 3 FETI Basics 4 Multiscale Toolkit 5 TFETI 6 FETI-DP Clemens Pechstein Substructuring for multiscale problems 13 / 38

17 FETI algorithms I Finite Element Tearing and Interconnecting FETI: [Farhat & Roux] FETI-DP: [Farhat, Lesoinne, Le Tallec, Pierson, Rixen] TFETI: [Dostál, Horák, Kučera], [Of, Steinbach] Dirichlet B.C. Neumann B.C. K 1 0 B 1 u K N BN u N = B 1 B N 0 λ [ ] [ ] K B u compactly: = B 0 λ f 1. f N 0 [ ] f 0 Clemens Pechstein Substructuring for multiscale problems 14 / 38

18 FETI algorithms II TFETI Dirichlet B.C. FETI-DP Dirichlet B.C. Neumann B.C. coarse DOF: p.w. constants ("rigid body modes") Neumann B.C. primal DOF: vertex values (weighted) edge averages??? solve P B K B λ = d precond. P B D S BD P = I Q G (G Q G) 1 G }{{} coarse solve B D, Q contain scalings solve B K 1 B λ = d precond. B D S BD K 1 : block Cholesky coarse solve + subdomain solves B D contains scalings Clemens Pechstein Substructuring for multiscale problems 15 / 38

19 Definition of jump operators jump operators: B : W U B D : W U W... torn FE space U... Lagrange multiplier space x h... interface DOF (node) on Ω i Ω k : (B w) ik (x h ) = w i (x h ) w k (x h ) Ω i x h Ω j Ω k Clemens Pechstein Substructuring for multiscale problems 16 / 38

20 Definition of jump operators jump operators: B : W U B D : W U W... torn FE space U... Lagrange multiplier space x h... interface DOF (node) on Ω i Ω k : (B D w) ik (x h ) = 1 [ ] ρ j (x h ρ k (x h ) w i (x h ) ρ i (x h ) w k (x h ) ) j N x h scalings ρ j (x h ) per subdomain j per interface DOF x h possible choices discussed later on partition of unity property averaging property: I BD B = E D : W Ŵ Clemens Pechstein Substructuring for multiscale problems 16 / 38

21 Analysis framework For FETI, FETI-DP (also BDD, BDDC): λ min 1 λ max P D {}}{ B sup D B w 2 S w W sub w 2 S W sub W suitable Common technique: S... Schur complement norm on W splitting into subdomain face, edge, vertex contributions using cut-off functions transfer operators (Sobolev extension + Scott-Zhang) Poincaré inequality [Mandel & Tezaur] [Klawonn & Widlund] [Klawonn, Widlund, Dryja] [Dohrmann, Mandel, Tezaur] [Mandel & Sousedik] [Klawonn, Rheinbach, Widlund] Clemens Pechstein Substructuring for multiscale problems 17 / 38

22 Theory vs. Implementation FETI, FETI DP, BDD, BDDC Theory Implementation cut off estimates transfer operators scalings? Weighted Poincare Inequalities (WPI) primal DOFs (FETI DP, BDDC) robust condition number bound robust method Clemens Pechstein Substructuring for multiscale problems 18 / 38

23 Outline 1 Introduction 2 Weighted Poincaré Inequalities (WPI) 3 FETI Basics 4 Multiscale Toolkit 5 TFETI 6 FETI-DP Clemens Pechstein Substructuring for multiscale problems 19 / 38

24 Boundary layers & patch decompositions subdomain Ω i Assumptions througout: patch decomposition globally conforming mesh T η patch decomposition T η (Ω i,η ) quasi-uniform variation on single patch c noise small Clemens Pechstein Substructuring for multiscale problems 20 / 38

25 Boundary layers & patch decompositions boundary layer Ω i,η η 4 Assumptions througout: patch decomposition globally conforming mesh T η patch decomposition T η (Ω i,η ) quasi-uniform variation on single patch c noise small Clemens Pechstein Substructuring for multiscale problems 20 / 38

26 Boundary layers & patch decompositions boundary layer Ω i,η patch decomposition: Ω i,η = k Y (k) i h η H η 2 (k) Y i Assumptions througout: patch decomposition globally conforming mesh T η patch decomposition T η (Ω i,η ) quasi-uniform variation on single patch c noise small Clemens Pechstein Substructuring for multiscale problems 20 / 38

27 Need good scalings ρ j (x h ) Theory Implementation (a) 1 1 (multiplicity scaling) (b) max(k diag (c) (d) (e) α max Ω j max τ Ω j :x h τ max Y (k) j :x h Y (k) j α τ α max Y (k) j j ) K diag j (x h ) (stiffness scaling) processed stiffness scaling? new promising technique, C. Dohrmann s talk, Mo, M13 Choice (a): does not work for certain jumps across interfaces Choice (b): κ may depend on max i α max Ω i α min Ω i Choice (c): κ may depend on oscillations of α or deteriorate for ragged interfaces! Clemens Pechstein Substructuring for multiscale problems 21 / 38

28 Intermediate result Lemma (P. & Scheichl 11, P. 12*) For (theoretical) choice (d), ρ j (x h ) = w W : P D w 2 S C c noise Proof: N j=1 finer splitting into patch globs transfer operators (carefully!) conventional cut-off estimates (d) difficult to mimic in practice (edge detection, TV minimization) Y (k) j max :x h Y (k) j α max Y (k) j [ (1 + log( η h ))2 w j 2 H 1 (Ω j,η ),α + (1 + log( η h )) η 2 w j 2 L 2 (Ω j,η ),α H h η ] faces edges vertices Clemens Pechstein Substructuring for multiscale problems 22 / 38

29 WPI in boundary layers Assumption: nice subdomains Ω j paths not extremely long Then: C P,α (Ω j ) 2 C P,α (Ω j,η ) 2 ( H η ) d 1 ( H η ) d 2 Clemens Pechstein Substructuring for multiscale problems 23 / 38

30 Outline 1 Introduction 2 Weighted Poincaré Inequalities (WPI) 3 FETI Basics 4 Multiscale Toolkit 5 TFETI 6 FETI-DP Clemens Pechstein Substructuring for multiscale problems 24 / 38

31 Condition number bounds I TFETI, Q = B D S B D [P. & Scheichl 08, 11, P. 12] Theorem α constant in boundary layers Ω i,η κ C c noise ( H η ) 2 (1 ( η )) 2 + log h improves to H/η if α inside larger than in Ω i,η Theorem α quasi-monotone in boundary layers Ω i,η κ C c noise ( H η ) d (1 ( η )) 2 + log h Similar: Q = Q diag ; type-m quasi-monotone Clemens Pechstein Substructuring for multiscale problems 25 / 38

32 Condition number bounds I TFETI, Q = B D S B D [P. & Scheichl 08, 11, P. 12] Theorem α constant in boundary layers Ω i,η κ C c noise ( H η ) 2 (1 ( η )) 2 + log h improves to H/η if α inside larger than in Ω i,η Theorem α quasi-monotone in boundary layers Ω i,η κ C c noise ( H η ) d (1 ( η )) 2 + log h Similar: Q = Q diag ; type-m quasi-monotone Clemens Pechstein Substructuring for multiscale problems 25 / 38

33 Condition number bounds II artificial coefficient α art : η α art = α α art α in Ω j,η elsewhere Enough to have inf c R w c 2 L 2 (Ω j,η ),α C j H 2 w 2 H 1 (Ω j ),α art Theorem α art quasi-monotone on Ω j : κ C c noise ( H η ) d+1 (1 ( η )) 2 + log h [P. & Scheichl 08, 11] [Dryja & Sarkis 11] [Gippert, Klawonn, Rheinbach 12*] Clemens Pechstein Substructuring for multiscale problems 26 / 38

34 Necessity of quasi-monotonicity? Q: is quasi-monotonicity is necessary for the robust of TFETI? Conjecture TFETI is robust iff for each subdomain, there exists a quasi-monotone artificial coefficient. is wrong TFETI... p Clemens Pechstein Substructuring for multiscale problems 27 / 38

35 Necessity of quasi-monotonicity? Q: is quasi-monotonicity is necessary for the robust of TFETI? Conjecture TFETI is robust iff for each subdomain, there exists a quasi-monotone artificial coefficient. That s wrong! TFETI has more robustness properties! Clemens Pechstein Substructuring for multiscale problems 27 / 38

36 Inclusion features I Docking inclusions can be elminated! condition FETI, many edge-island example, H/h=32 1e+08 1e+07 1e e+06 1e+07 ALPHA Theoretical reason: We actually have to estimate Ω i,η min(α i, α neighbors )... 2 dx [work in progress] Clemens Pechstein Substructuring for multiscale problems 28 / 38

37 Inclusion features II Even docking channels can be elminated! condition FETI, docking channels example, H/h=32 1e+08 1e+07 1e e+06 1e+07 ALPHA Theoretical reason: We actually have to estimate Ω i,η min(α i, α neighbors )... 2 dx [work in progress] Clemens Pechstein Substructuring for multiscale problems 29 / 38

38 Inclusion features III Face inclusions can be elminated! condition FETI, many edge-island-plus-channel example, H/h=32 1e+08 1e+07 1e e+06 1e+07 ALPHA Theoretical reasons: [work in progress] Special cut-off function [Graham, Lechner, Scheichl 07] Special transfer operator, L 2 α, H 1 α-stable for binary media Clemens Pechstein Substructuring for multiscale problems 30 / 38

1e+08 1e+07 1e+06 100000 10000 1000 100 10 1 1 10 100 1000 10000 100000 1e+06 1e+07 ALPHA condition

39 What s bad... E.g. two vertex inclusions or long channels condition FETI, many edge-island-plus-2vertex example, H/h=32 1e+08 1e+07 1e e+06 1e+07 ALPHA condition FETI, long channels example, H/h=32 1e+08 1e+07 1e e+06 1e+07 ALPHA Clemens Pechstein Substructuring for multiscale problems 31 / 38

40 Outline 1 Introduction 2 Weighted Poincaré Inequalities (WPI) 3 FETI Basics 4 Multiscale Toolkit 5 TFETI 6 FETI-DP Clemens Pechstein Substructuring for multiscale problems 32 / 38

41 Choice of primal DOFs Choice of primal DOFs should ensure that K is invertible make WPI applicable (on neighborhoods of edges/faces) With tools from above this will allow for robustness analysis Adapted tool: WPI with weighted averages [P., Sarkis, Scheichl, DD20 proc.] u u X, α L 2 (D),α C P,α (D, X, α) diam(d) u H 1 (D),α Essential requirements: α quasi-monotone averaging manifold X must see largest coefficient α and α have to match Geometry dependence can again be made explicit Clemens Pechstein Substructuring for multiscale problems 33 / 38

42 Choice of primal DOFs Choice of primal DOFs should ensure that K is invertible make WPI applicable (on neighborhoods of edges/faces) With tools from above this will allow for robustness analysis Adapted tool: WPI with weighted averages [P., Sarkis, Scheichl, DD20 proc.] u u X, α L 2 (D),α C P,α (D, X, α) diam(d) u H 1 (D),α Essential requirements: α quasi-monotone averaging manifold X must see largest coefficient α and α have to match Geometry dependence can again be made explicit Clemens Pechstein Substructuring for multiscale problems 33 / 38

43 Weighted edge averages? Simple examples: Ωi E E Ωk Ω i Ω k Primal DOF associated to edge E: u E, α E α u ds := E α ds or u E, α,alg := x h E α(x h ) u(x h ) x h E α(x h ) [Klawonn & Rheinbach, 2006] Clemens Pechstein Substructuring for multiscale problems 34 / 38

44 FETI-DP coarse space What to do here? Ωi E Ω k Ω E i Ωk Problems: Primal DOF should have same weight for both sides Usage of multiple primal DOFs dead end? Recall: for edge E, we actually have to bound Ω i,η U E min(α i, α k )... 2 dx If minimum coefficient has quasi-monotone extensions then α(x h ) = min(k diag i (x h ), K diag k (x h )) (algebraic average) does the job, at least robust w.r.t. contrast! Clemens Pechstein Substructuring for multiscale problems 35 / 38

45 FETI-DP coarse space What to do here? Ωi E Ω k Ω E i Ωk Problems: Primal DOF should have same weight for both sides Usage of multiple primal DOFs dead end? Recall: for edge E, we actually have to bound Ω i,η U E min(α i, α k )... 2 dx If minimum coefficient has quasi-monotone extensions then α(x h ) = min(k diag i (x h ), K diag k (x h )) (algebraic average) does the job, at least robust w.r.t. contrast! Clemens Pechstein Substructuring for multiscale problems 35 / 38

Numerical test condition 1e+08 1e+07 1e+06 100000 10000 1000 100 10 FETI-DP, edge crosspoint example, H/h=32 1 1 10 100 1000 10000

46 Numerical test condition 1e+08 1e+07 1e FETI-DP, edge crosspoint example, H/h= e+06 1e+07 ALPHA vertices unweighted max weight min weight Clemens Pechstein Substructuring for multiscale problems 36 / 38

47 Summary In this talk: Weighted Poincaré Inequalities Robustness theory for TFETI Robustness theory for FETI-DP (yet to be completed) Theory and testing guides to more insight / new methods! Open problems: Choice of scalings Choice of primal constraints Incorporation of eigensolves monograph: Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems Springer LNCSE series, to appear Clemens Pechstein Substructuring for multiscale problems 37 / 38

48 Summary In this talk: Weighted Poincaré Inequalities Robustness theory for TFETI Robustness theory for FETI-DP (yet to be completed) Theory and testing guides to more insight / new methods! Open problems: Choice of scalings Choice of primal constraints Incorporation of eigensolves monograph: Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems Springer LNCSE series, to appear Clemens Pechstein Substructuring for multiscale problems 37 / 38

49 Interconnecting... THANKS FOR YOUR ATTENTION Clemens Pechstein Substructuring for multiscale problems 38 / 38

50 Interconnecting... THANKS FOR YOUR ATTENTION Clemens Pechstein Substructuring for multiscale problems 38 / 38

On Iterative Substructuring Methods for Multiscale Problems

On Iterative Substructuring Methods for Multiscale Problems Clemens Pechstein Introduction Model Problem Let Ω R 2 or R 3 be a Lipschitz polytope with boundary Ω = Γ D Γ N, where Γ D Γ N = /0. We are interested