Two-level Additive Schwarz Preconditioners for a ...

Cha-Cha Days 2010 September 24-26, 2010

Two-level Additive Schwarz Preconditioners for a Weakly Over-Penalized Symmetric Interior Penalty Method

Eun-Hee Park Mathematics & Center for Computation and Technology

Joint work with A.T. Barker, S.C. Brenner, and L.-Y. Sung

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Outline

Domain Decomposition Methods Weakly Over-Penalized Symmetric Interior Penalty (WOPSIP) Method Two-level Additive Schwarz Preconditioners Numerical Results

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Outline


Cha-Cha Days 2010

Eun-Hee Park – 2

Outline


Cha-Cha Days 2010

Eun-Hee Park – 2

Outline


Cha-Cha Days 2010

Eun-Hee Park – 2

Outline


Cha-Cha Days 2010

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Domain Decomposition Methods Why DDM ? What’s DDM ? WOPSIP Method Two-level Additive Schwarz Method Numerical Results

Domain Decomposition Methods

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Why DDM ?

Domain Decomposition Methods Why DDM ? What’s DDM ?

Toy Problem

WOPSIP Method Two-level Additive Schwarz Method Numerical Results

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−Δu

=

f

in Ω

u

=

0

on ∂Ω

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Why DDM ?


Toy Problem

WOPSIP Method Two-level Additive Schwarz Method 1

Numerical Results

0.8 0.6 1 0.4

0.8 0.6

0.2

0.4 0

0.2 0

Solved in the blink of an eye !

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−Δu

=

f

in Ω

u

=

0

on ∂Ω

Eun-Hee Park – 5

Why DDM ?


Real Application

WOPSIP Method

◆ Large scale problems ◆ How to solve in limited

Two-level Additive Schwarz Method Numerical Results

time and memory ?

Domain Decomposition ◆ Divide into many of

smaller problems ◆ Solve subdomain prob-

lems in parallel

- Solution for limitation of memory and computing time Cha-Cha Days 2010

Eun-Hee Park – 6

What’s DDM ?

Classical Schwarz Alternating Method (Schwarz, 1869) ◆ Iterative Method

Domain Decomposition Methods Why DDM ? What’s DDM ? WOPSIP Method

- Classical solutions smooth regions

on

Two-level Additive Schwarz Method

Ω

non-

Numerical Results

◆ Alternating Procedure Given an initial u0 , solve successively: −Δu2k+1 = f u2k+1 = u2k −Δu2k+2 = f u2k+2 = u2k+1 ◆ Numerical Method ?

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in Ω1

Ω1

on ∂Ω1 , in Ω2 on ∂Ω2 . Ω2

Eun-Hee Park – 7

What’s DDM ?


Discretized Problem k+1

u

Ah u = f k −1 k f − Ah u =u + P

Ω

WOPSIP Method Two-level Additive Schwarz Method

P −1 Ah u = P −1 f

Numerical Results

DD-based Preconditioner

[Dryja & Widlund (1987)]

- Additive Schwarz Preconditioner

Ω1

Ω2

k+1

u

k

=u +

A−1 1

+

−1 A2

k

f − Ah u

- Two-level Additive Schwarz Preconditioner Cha-Cha Days 2010

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Domain Decomposition Methods WOPSIP Method Two-level Additive Schwarz Method Numerical Results

WOPSIP Method

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Discontinuous Galerkin (DG) methods

Why DG methods ◆ Need for Nonconforming Meshes

- Adaptive methods e.g. L-shaped domain

Poisson Model Problem

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−Δu

=

f

in Ω

u

=

0

on ∂Ω

Eun-Hee Park – 10

Discontinuous Galerkin (DG) methods

DG methods For v, w ∈ Vh , a± h (v, w) = ±

T ∈Th

e∈Eh

T

∇v · ∇w dx −

e∈Eh

e

{{∇v}} · [ w]] ds

η {∇w}} · [ v]] ds + [ v]] · [ w]] ds, e{ |e| e e∈E h

where

Vh = v ∈ L2 (Ω) : v|T ∈ P1 (T ) ∀T ∈ Th .

◆ a− h : SIPG (Symmetric Interior Penalty Galerkin) method [Douglas & Dupont (1976), Wheeler (1978), Arnold (1982)]

◆ a+ h : NIPG (Nonsymmetric Interior Penalty Galerkin) method [Rivi` ere, Wheeler & Girault (1999)]

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Comparison: SIPG & NIPG


SIPG ◆ Correct error estimates in both energy norm and L2 norm ◆ Stability: η > η0 for a sufficiently large η0

WOPSIP Method Two-level Additive Schwarz Method Numerical Results

NIPG ◆ Correct error estimate only in energy norm ◆ Stability: η > η0 for an arbitrary η0

WOPSIP ◆

Weakly Over-Penalized Symmetric Interior Penalty

◆

Brenner, Owens & Sung (2008)

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WOPSIP method

WOPSIP Formulation To find uh ∈ Vh such that, for any vh ∈ Vh , 1 0 0 ∇uh · ∇vh dx + η Π [ u ] · Π h e e [ vh ] ds = (f, vh )L2 (Ω) , 3 |e| T e T ∈T e∈E h

h

where Π0e is the orthogonal projection from [L2 (e)]2 onto [P0 (e)]2 .

Advantages ◆ SPD ◆ No need for tuning of a penalty parameter (η = 1) ◆ Correct error estimates in both energy norm and L2 norm ◆ Meshes with hanging nodes ◆ Simple programming ◆ Intrinsic parallelism

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Domain Decomposition Methods WOPSIP Method Two-level Additive Schwarz Method Two-level Preconditioner Intergrid Transfer Operator Convergence Analysis


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Numerical Results

Eun-Hee Park – 14

Two-level Additive Schwarz Preconditioners


Linear System

WOPSIP Method

Ah uh = f

Two-level Additive Schwarz Method Two-level Preconditioner Intergrid Transfer Operator Convergence Analysis

Iterative Solver

Numerical Results

◆ Preconditioned Conjugate Gradient Method

B −1/2 Ah B −1/2 v = B −1/2 f,

◆ uh − uk Ah

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v = B 1/2 uh

k √ −1 κ(B Ah )−1 ≤2 √ uh − u0 Ah κ(B −1 Ah )+1

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Two-level Additive Schwarz Preconditioners (cont.)

Two-level Additive Schwarz Preconditioner

Domain Decomposition Methods WOPSIP Method

B

=

Bc + Bf

=

h h t IH AH −1 (IH )

+

N

Ii Ah,i −1 Iit

i=1

Two-level Additive Schwarz Method Two-level Preconditioner Intergrid Transfer Operator Convergence Analysis Numerical Results

◆ AH : Coarse Solver (associated w/ VH ) ◆ Ah,i : Subdomain Solvers (associated w/ Vi ) ◆ Ii : Natural injection operator h ◆ IH : Intergrid transfer operator

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Two-level Preconditioners (cont.)

Subdomain Solvers


Ah,i : Vi → Vi defined by ∇u · ∇v dx + Ah,i u, v =

WOPSIP Method

T ∈Thi

T

i e∈Eh

1 |e|3

e

Π0e [ u]] · Π0e [ v]] ds


Vi = {v ∈ Vh : v = 0 on Ω \ Ωi }

TH

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Th

˜ i → Ωi Ω

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Coarse Solver ◆ VH,1 :

AH v, w =

◆ VH,2 :

AH v, w =

◆ VH,3 :

AH v, w =

◆ VH,4 :

AH v, w = T ∈TH

conforming P1

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T

T ∈TH T ∈TH

WOPSIP Method

T T

1 e∈EH |e|

∇v · ∇w dx ∇v · ∇w dx 0 0 e Πe [ v]] · Πe [ w]] ds


∇v · ∇w dx +

1 e∈EH |e|

non-conforming P1 discontinuous P0

e

Π0e [ v]] · Π0e [ w]] ds

discontinuous P1

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Preconditioned System BAh =

h h t IH AH −1 (IH ) +

N

Ii Ah,i −1 Iit

i=1

WOPSIP Method

Ah


Intergrid Transfer Operator h IH : VH → Vh such that h h v, IH v ≤ CAH v, v Ah IH

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∀v ∈ VH .

Eun-Hee Park – 19

Intergrid Transfer Operator

Difference in Scales For a piecewise constant function vH ∈ VH,3 ⊂ VH,4 ⊂ Vh , Ah vH , vH

=

e∈Eh

AH vH , vH

=

E∈EH

1 |e|3 1 |E|

0 2 |Π [ v ] | ds H e e

0 2 |Π [ v ] | ds H E E

Domain Decomposition Methods WOPSIP Method Two-level Additive Schwarz Method Two-level Preconditioner Intergrid Transfer Operator Convergence Analysis Numerical Results

Edge by Edge ◆ Fine Solver

1 |e|3

◆ Coarse Solver

1 |E| Cha-Cha Days 2010

e

|Π0e [ vH ] |2 ds ∼ O

E

1 |e|2

|Π0E [ vH ] |2 ds ∼ O(1) Eun-Hee Park – 20

Need for Enriching Operator BAh =

h h t IH AH −1 (IH ) +

t Ii A−1 h,i Ii

i=1

CAh =

N

I0 AH −1 (I0 )t +

N


Ah

t Ii A−1 h,i Ii

Ah

i=1

CAh ignoring difference in scales h 2−6 2−7 2−8 2−9

N =4 CG its. 184 303 491 803

κ(CAh ) 1.14·105 1.36·106 1.72·107 2.43·108

N = 32 CG its. 184 288 494 936

WOPSIP Method Two-level Additive Schwarz Method Two-level Preconditioner Intergrid Transfer Operator Convergence Analysis Numerical Results

κ(CAh ) 1.11·105 1.38·106 1.92·107 2.93·108

- I0 : Interpolation without enrichment - Discontinuous P 1 coarse space - Coarse mesh size H = 2−5 Cha-Cha Days 2010

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Intergrid Transfer Operator (cont.)


Enriching Operator EH : VH,4 → VH,4 ∩ H01 (Ω)

WOPSIP Method Two-level Additive Schwarz Method Two-level Preconditioner Intergrid Transfer Operator Convergence Analysis

◆ Averaging Operator

At a vertex p, we define 1 (EH v)(p) = v|T (p), |Tp | T ∈T

Numerical Results

p

where |Tp | is the number of triangles sharing p as a vertex. ◆ Ran(EH ) ⊂ C 0 (Ω)

For each e ∈ Eh , |Π0e [ EH vH ] |2 ds = 0. e

cf. Cha-Cha Days 2010

1 |e|3

0 e |Πe

[ vH

] |2

ds = O

1 |e|2

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Intergrid Transfer Operator (cont.)

Ih H : Coarse Space → Fine Space h IH = Πh ◦ EH

◆ EH : Enriching operator ◆ Πh : Crouzeix-Raviart interpolation operator - Πh : H 1 (Ω) → Vh (Πh v)|T = ΠT (v|T ) - ΠT : H 1 (T ) → P1 (T ) 1 (ΠT v)(me ) = |e|

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∀T ∈ Th

e

v ds

∀e ∈ ET

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Convergence Analysis

Requirement on Ih H : VH → Vh


h h Ah IH v, IH v

≤

CAH v, v

h v − IH v L2 (Ω)

≤

CHAH v, v

∀v ∈ VH ∀v ∈ VH

Cf. Abstract Schwarz theory [Dryja & Widlund (1987)]

WOPSIP Method Two-level Additive Schwarz Method Two-level Preconditioner Intergrid Transfer Operator Convergence Analysis Numerical Results

Theorem The condition number of the preconditioned system BAh is estimated as

κ(BAh ) ≤ C 1 +

H δ

,

where C is a constant independent of H, h, δ and N . - H: coarse mesh size - δ: overlap width - N : number of subdomains [Barker, Brenner, Park & Sung, to appear]

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Convergence Analysis

H κ(BAh ) ≤ C 1 + δ

Domain Decomposition Methods WOPSIP Method Two-level Additive Schwarz Method Two-level Preconditioner Intergrid Transfer Operator Convergence Analysis

- Linear growth - Optimal preconditioner if H/δ ∼ 1

Numerical Results

- Scalable algorithm - B: preconditioner with enriching process for WOPSIP method

Remark Two-level Schwarz preconditioners for super penalty DGM 1 κ(PT LAS Ah ) ≤ C 2 h

H 1+ h

[Antonietti & Ayuso (2009)]

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Numerical Results

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Numerical Results

Poisson Problem

Computing Environment ◆ LONI Supercomputers

Find a FE approximation uh of the exact solution u such that −Δu

=

f

in Ω

u

=

0

on ∂Ω


where Ω = (0, 1) × (0, 1) ⊂ R2 and u(x, y) = xy(1 − x)(1 − y). (Louisiana Optical Network Initiative)

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One-level Preconditioner


BAh = (Bf + Bc ) Ah

WOPSIP Method

W/O Coarse Solver


Bf Ah =

N

Numerical Results

Ii Ah,i −1 Iit

Ah

i=1

N 2 4 8 16 32

CG its. 42 65 73 90 109

κ(Bf Ah ) 223.39 739.55 876.95 1.14·103 1.47·103

Table 1: N( of subdomains), fine mesh h = 2−8 , overlap width δ=h Cha-Cha Days 2010

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Comparison: CG iteration counts

N 1 2 4 8 16 32 64 128 256 512

one-level 1 42 65 73 90 109 135 149 193 220

VH,1 2 11 11 11 11 11 11 11 11 10

VH,2 3 14 15 14 14 14 14 15 14 14

VH,3 4 13 14 14 14 14 15 15 15 15

VH,4 8 13 13 14 14 14 13 12 14 14


Table 2: H/δ = 8, fine mesh h = 2−8 , coarse mesh H = 2−5 , overlap: δ = h

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VH,4 : Dependency on H and δ


Coarse mesh size H H 2−4 2−5 2−6 2−7

N =4 CG its. 19 13 10 8

wall clock time 3.9 3.6 4.8 12.7

N = 32 CG its. 20 14 10 7

WOPSIP Method

wall clock time .96 .91 1.0 1.8

Two-level Additive Schwarz Method Numerical Results

Table 3: fine mesh h = 2−8 , overlap width δ = h Overlap width δ δ 0 h 2h 3h

N =4 CG its. 22 12 9 10

wall clock time 4.7 3.5 3.2 3.4

N = 32 CG its. 17 12 8 9

wall clock time 1.3 1.0 .72 .70

Table 4: fine mesh h = 2−8 , coarse mesh H = 2−5 Cha-Cha Days 2010

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Concluding Remarks

◆

Conclusion - Extension of two-level additive Schwarz preconditioner to the WOPSIP method - Barker, Brenner, Park & Sung, to appear

◆

Current / Future Projects Development of nonoverlapping DD based preconditioners for DG methods including the WOPSIP method: BDDC method, FETI-DP method

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Concluding Remarks

◆


◆


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Concluding Remarks

◆


◆


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Concluding Remarks

◆


◆


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Concluding Remarks

◆


◆


Ω1

Ω2 Γ

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References

[1] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), pp. 742–760. [2] S. C. Brenner, T. Gudi, L. Owens, and L.-Y. Sung, An intrinsically parallel finite element method, J. Sci. Comput., 42 (2010), pp. 118–121. [3] S. C. Brenner, L. Owens, and L.-Y. Sung, A weakly over-penalized symmetric interior penalty method, Electron. Trans. Numer. Anal., 30 (2008), pp. 107–127. [4] M. Dryja and O. B. Widlund, An additive variant of the Schwarz alternating method in the case of many subregions, Technical Report 339, Department of Computer Science, Courant Institute, (1987). [5] J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, in Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976. `re, M. F. Wheeler, and V. Girault, Improved energy estimates for interior [6] B. Rivie penalty, constrained and discontinuous Galerkin methods for elliptic problems I, Comput. Geosci., 3 (1999), pp. 337–360. [7] M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., 15 (1978), pp. 152–161.

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Intrinsic Parallelism of WOPSIP method

a(v, w) =

T ∈Th

T

1 ∇v · ∇w dx + [ v(me )]] · [ w(me )]] |e|2

- Element-wise ordering

e∈Eh

- Edge-wise ordering e

v k+1

v k+2

wk+1

T

wk

vk

Ah w = P t DP w + Jw - D: Block-diagonal matrix where each block is 3 × 3 - J: Block-diagonal matrix where each block is either 1 × 1 or 2 × 2

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