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An abstract two-level Schwarz method for systems with high contrast coefficients Clemens Pechstein Johannes Kepler University Linz (A) joint work with Nicole Spillane , Fr ed eric Nataf (Univ. Paris VI) Victorita Dolean (Univ. Nice

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SLIDE 1

An abstract two-level Schwarz method for systems with high contrast coefficients

Clemens Pechstein

Johannes Kepler University Linz (A) joint work with Nicole Spillane, Fr´ ed´ eric Nataf (Univ. Paris VI) Victorita Dolean (Univ. Nice Sophia-Antipolis) Patrice Hauret (Michelin, Clermont-Ferrand) Robert Scheichl (Univ. Bath)

RICAM Special Semester Multiscale Simulation & Analysis in Energy & Environment October 8, 2011

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Motivation

Large discretized system of PDEs strongly heterogeneous coefficients (high contrast, nonlinear, multiscale) E.g. Darcy pressure equation, P1-finite elements: Au = f cond(A) ∼ κmax κmin h−2 Goal: iterative solvers robust in size and heterogeneities

Applications: flow in heterogeneous / stochastic / layered media structural mechanics electromagnetics etc.

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abstract two-level Schwarz 2 / 24

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SLIDE 3

Relation to other methods

Graham, Lechner & Scheichl Scheichl & Vainikko Overlapping Schwarz MS coarse space Pechstein & Scheichl FETI (2 papers) boundary layers WPI for some patterns Overlapping Schwarz gen.EVP & Xiang / & Scheichl Dolean, Nataf, Spillane

THIS TALK

Galvis & Efendiev b) gen.EVP K a) std. coarse space, WPI S = M

λ

= λ M

κ κ Γ κ κ ξ κ

λ

=

κ

gen.EVP K K Overlapping Schwarz Pechstein & Scheichl Weighted Poincare Ineq. Efendiev, Galvis, Lazarov & Willems Overlapping Schwarz, abstract SPD problems

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abstract two-level Schwarz 3 / 24

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SLIDE 4

Problem setting – I

Given f ∈ (V h)∗ find u ∈ V h a(u, v) = f, v ∀v ∈ V h ⇐ ⇒ A u = f Assumption throughout: A symmetric positive definite (SPD) Examples: Darcy a(u, v) =

  • Ω κ∇u · ∇v dx

Elasticity a(u, v) =

  • Ω C ε(u) : ε(v) dx

Eddy current a(u, v) =

  • Ω ν curl u · curl v + σ u · v dx

Heterogeneities / high contrast / nonlinearities in parameters

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abstract two-level Schwarz 4 / 24

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Problem setting – II

1

V h . . . FE space of functions in Ω based on mesh T h = {τ}

2

A given as set of element stiffness matrices + connectivity (list of DOF per element) Assembling property: a(v, w) =

  • τ

aτ(v|τ, w|τ) where aτ(·, ·) symm. pos. semi-definite

3

{φk}n

k=1 (FE) basis of V h

  • n each element: unisolvence

set of non-vanishing basis functions linearly independent

fulfilled by standard FE

continuous, N´ ed´ elec, Raviart-Thomas of low/high order

4

Two more assumptions on a(·, ·) later!

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abstract two-level Schwarz 5 / 24

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SLIDE 6

Problem setting – II

1

V h . . . FE space of functions in Ω based on mesh T h = {τ}

2

A given as set of element stiffness matrices + connectivity (list of DOF per element) Assembling property: a(v, w) =

  • τ

aτ(v|τ, w|τ) where aτ(·, ·) symm. pos. semi-definite

3

{φk}n

k=1 (FE) basis of V h

  • n each element: unisolvence

set of non-vanishing basis functions linearly independent

fulfilled by standard FE

continuous, N´ ed´ elec, Raviart-Thomas of low/high order

4

Two more assumptions on a(·, ·) later!

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abstract two-level Schwarz 5 / 24

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SLIDE 7

Schwarz setting – I

Overlapping partition: Ω = N

j=1 Ωj

(Ωj union of elements) Vj := span

  • φk : supp(φk) ⊂ Ωj
  • such that every φk contained in one of those spaces, i.e.

V h =

N

  • j=1

Vj Example: adding “layers” to non-overlapping partition

(partition and adding layers based on matrix information only!)

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abstract two-level Schwarz 6 / 24

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SLIDE 8

Schwarz setting – II

Local subspaces: Vj ⊂ V h j = 1, . . . , N Coarse space (defined later): V0 ⊂ V h Additive Schwarz preconditioner: M−1

AS,2 = N

  • j=0

R⊤

j A−1 j

Rj where Aj = R⊤

j ARj

and R⊤

j ↔ R⊤ j

: Vj → V h natural embedding

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abstract two-level Schwarz 7 / 24

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SLIDE 9

Partition of unity

Definitions: dof(Ωj) :=

  • k : supp(φk) ∩ Ωj = ∅
  • idof(Ωj) :=
  • k : supp(φk) ⊂ Ωj
  • Vj = span{φk}k∈idof(Ωj)

imult(k) := #

  • j : k ∈ idof(Ωj)
  • Partition of unity:

(used for design of coarse space and for stable splitting)

Ξjv =

  • k∈idof(Ωj)

1 imult(k)vk φk for v =

n

  • k=1

vkφk Properties:

N

  • j=1

Ξjv = v Ξjv ∈ Vj

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abstract two-level Schwarz 8 / 24

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SLIDE 10

Overlapping zone / Choice of coarse space

Overlapping zone: Ω◦

j

= {x ∈ Ωj : ∃i = j : x ∈ Ωi} Observation: Ξj|Ωj\Ω◦

j

= id Coarse space should be local: V0 =

N

  • j=1

V0, j where V0, j ⊂ Vj E.g. V0, j = span{Ξjpj,k}

mj k=1

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abstract two-level Schwarz 9 / 24

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SLIDE 11

Overlapping zone / Choice of coarse space

Overlapping zone: Ω◦

j

= {x ∈ Ωj : ∃i = j : x ∈ Ωi} Observation: Ξj|Ωj\Ω◦

j

= id Coarse space should be local: V0 =

N

  • j=1

V0, j where V0, j ⊂ Vj E.g. V0, j = span{Ξjpj,k}

mj k=1

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abstract two-level Schwarz 9 / 24

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SLIDE 12

Choice of coarse space (continued)

ASM theory needs stable splitting: v = v0 +

N

  • j=1

vj Suppose v0 = N

j=1 ΞjΠjv|Ωj

where Πj . . . local projector | Ξj(v − Πjv)

  • vj

|2

a,Ωj = |Ξj(v − Πjv)|2 a,Ω◦

j + |Ξj(v − Πjv))|2

a,Ωj\Ω◦

j

HOW?

≤ C |v|2

a,Ωj

(a,D . . . restriction of a to D)

“Minimal” requirements: Πj be a-orthogonal Stability estimate: |Ξj(v − Πjv)|2

a,Ω◦

j

≤ c |v|2

a,Ωj

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abstract two-level Schwarz 10 / 24

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SLIDE 13

Choice of coarse space (continued)

ASM theory needs stable splitting: v = v0 +

N

  • j=1

vj Suppose v0 = N

j=1 ΞjΠjv|Ωj

where Πj . . . local projector | Ξj(v − Πjv)

  • vj

|2

a,Ωj = |Ξj(v − Πjv)|2 a,Ω◦

j + |Ξj(v − Πjv))|2

a,Ωj\Ω◦

j

HOW?

≤ C |v|2

a,Ωj

(a,D . . . restriction of a to D)

“Minimal” requirements: Πj be a-orthogonal Stability estimate: |Ξj(v − Πjv)|2

a,Ω◦

j

≤ c |v|2

a,Ωj

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abstract two-level Schwarz 10 / 24

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Abstract eigenvalue problem

Gen.EVP per subdomain: Find pj,k ∈ Vh|Ωj and λj,k ≥ 0: aΩj(pj,k, v) = λj,k aΩ◦

j (Ξjpj,k, Ξjv)

∀v ∈ Vh|Ωj Ajpj,k = λj,k XjA◦

j Xj pj,k

(Xj . . . diagonal) (properties of eigenfunctions discussed soon) aD . . . restriction of a to D

In the two-level ASM: Choose first mj eigenvectors per subdomain: V0 = span

  • Ξjpj,k

j=1,...,N

k=1,...,mj

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abstract two-level Schwarz 11 / 24

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SLIDE 15

Abstract eigenvalue problem

Gen.EVP per subdomain: Find pj,k ∈ Vh|Ωj and λj,k ≥ 0: aΩj(pj,k, v) = λj,k aΩ◦

j (Ξjpj,k, Ξjv)

∀v ∈ Vh|Ωj Ajpj,k = λj,k XjA◦

j Xj pj,k

(Xj . . . diagonal) (properties of eigenfunctions discussed soon) aD . . . restriction of a to D

In the two-level ASM: Choose first mj eigenvectors per subdomain: V0 = span

  • Ξjpj,k

j=1,...,N

k=1,...,mj

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abstract two-level Schwarz 11 / 24

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Comparison with existing works

Galvis & Efendiev (SIAM 2010):

  • Ωj

κ ∇pj,k · ∇v dx = λj,k

  • Ωj

κ pj,k v dx ∀v ∈ Vh|Ωj Efendiev, Galvis, Lazarov & Willems (submitted): aΩj(pj,k, v) = λj,k

  • i∈neighb(j)

aΩj(ξj ξi pj,k, ξj ξi v) ∀v ∈ V|Ωj

ξj . . . partition of unity, calculated adaptively (MS)

Our gen.EVP: aΩj(pj,k, v) = λj,k aΩ◦

j (Ξjpj,k, Ξjv)

∀v ∈ Vh|Ωj both matrices typically singular = ⇒ λj,k ∈ [0, ∞]

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abstract two-level Schwarz 12 / 24

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Convergence theory

Definition: Each element appears in at most k0 of the subdomains Ωj Assumption 1: aΩj(·, ·) SPD on span{φk|Ωj}k∈dof(Ωj)\idof(Ωj) Assumption 2: aΩ◦

j (·, ·) SPD on span{φk|Ωj}k∈idof(Ωj)\idof(Ωj\Ω◦ j )

Theorem If for all j: 0 < λj,mj+1 < ∞: κ(M−1

AS,2A) ≤ (1 + k0)

  • 2 + k0 (2k0 + 1)

N

max

j=1

  • 1 +

1 λj,mj+1

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abstract two-level Schwarz 13 / 24

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SLIDE 18

Properties of eigenfunctions

Gen.EVP per subdomain: Find pj,k ∈ Vh|Ωj and λj,k ∈ [0, ∞]: aΩj(pj,k, v) = λj,k aΩ◦

j (Ξjpj,k, Ξjv)

  • =:bj(pj,k, v)

∀v ∈ Vh|Ωj Eigenfunctions with λ < ∞ are

1

aΩj(·, ·)-harmonic in Ωj \ Ω◦

j

2

aΩj(·, ·)-harmonic w.r.t. ker(Ξj|Ωj)

  • Ass. 1 =

⇒ 2nd extension possible

  • Ass. 2 =

⇒ bj(·, ·) SPD on space

  • f eigenfunctions with λ < ∞

Free DOFs:

here for continuous Q1-elements

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abstract two-level Schwarz 14 / 24

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SLIDE 19

Sufficient conditions for Assumptions 1 and 2

Assumptions 1 and 2 hold if all aτ(·, ·) definite curl (νcurl u) + u

  • r if

certain mixed “boundary” value problems solvable: Last conditions easy to check for concrete PDEs: kernel of Darcy controlled by one Dirichlet DOF kernel of 3D Elasticity controlled by three non-collinear DOFs

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abstract two-level Schwarz 15 / 24

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SLIDE 20

Sketch of proof – I

λmax ≤ 1 + k0 λmin: stable splitting v = z0 + N

j=1 zj

Lemma If for every v ∈ Vh and ∀j = 1, . . . , N: |zj|a ≤ C1 |v|a then λ−1

min ≤ 2 + C1 k0 (2 k0 + 1)

Local coarse projector: Πjv =

mj

  • k=1

bj(v, pj,k)pj,k where bj(v, w) = aΩ◦

j (Ξjv, Ξjw)

Our stable splitting: z0 =

N

  • j=1

Ξj(Πjv|Ωj) zj = Ξj(v − Πjv|Ωj)

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abstract two-level Schwarz 16 / 24

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SLIDE 21

Sketch of proof – I

λmax ≤ 1 + k0 λmin: stable splitting v = z0 + N

j=1 zj

Lemma If for every v ∈ Vh and ∀j = 1, . . . , N: |zj|a ≤ C1 |v|a then λ−1

min ≤ 2 + C1 k0 (2 k0 + 1)

Local coarse projector: Πjv =

mj

  • k=1

bj(v, pj,k)pj,k where bj(v, w) = aΩ◦

j (Ξjv, Ξjw)

Our stable splitting: z0 =

N

  • j=1

Ξj(Πjv|Ωj) zj = Ξj(v − Πjv|Ωj)

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abstract two-level Schwarz 16 / 24

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SLIDE 22

Sketch of proof – I

λmax ≤ 1 + k0 λmin: stable splitting v = z0 + N

j=1 zj

Lemma If for every v ∈ Vh and ∀j = 1, . . . , N: |zj|a ≤ C1 |v|a then λ−1

min ≤ 2 + C1 k0 (2 k0 + 1)

Local coarse projector: Πjv =

mj

  • k=1

bj(v, pj,k)pj,k where bj(v, w) = aΩ◦

j (Ξjv, Ξjw)

Our stable splitting: z0 =

N

  • j=1

Ξj(Πjv|Ωj) zj = Ξj(v − Πjv|Ωj)

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abstract two-level Schwarz 16 / 24

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SLIDE 23

Sketch of proof – II

Central property of local coarse projector: |Ξj(v − Πjv)|2

a,Ω◦

j

≤ 1 λj,mj+1 |v|2

a,Ωj

proof by standard spectral theory (on space of eigenfunctions with λ < ∞) + properties of aΩj(·, ·)-harmonic extension Central property and aΩj-orthogonality of Πj = ⇒ |zj|2

a = |Ξj(v − Πjv)|2 a,Ω◦

j + |Ξj(v − Πjv)|2

a,Ωj\Ω◦

j

≤ 1 λj,mj+1 |v|2

a,Ωj + |v − Πjv|2 a,Ωj

  • ≤ |v|a,Ωj

Lemma (last slide) = ⇒ splitting is stable

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abstract two-level Schwarz 17 / 24

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SLIDE 24

Sketch of proof – II

Central property of local coarse projector: |Ξj(v − Πjv)|2

a,Ω◦

j

≤ 1 λj,mj+1 |v|2

a,Ωj

proof by standard spectral theory (on space of eigenfunctions with λ < ∞) + properties of aΩj(·, ·)-harmonic extension Central property and aΩj-orthogonality of Πj = ⇒ |zj|2

a = |Ξj(v − Πjv)|2 a,Ω◦

j + |Ξj(v − Πjv)|2

a,Ωj\Ω◦

j

≤ 1 λj,mj+1 |v|2

a,Ωj + |v − Πjv|2 a,Ωj

  • ≤ |v|a,Ωj

Lemma (last slide) = ⇒ splitting is stable

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abstract two-level Schwarz 17 / 24

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SLIDE 25

Theory (repetition)

Two technical assumptions. Theorem If for all j: 0 < λj,mj+1 < ∞: κ(M−1

AS,2A) ≤ (1 + k0)

  • 2 + k0 (2k0 + 1)

N

max

j=1

  • 1 +

1 λj,mj+1

  • Possible criterion for picking mj:

(used in our Numerics) λj,mj+1 < δj Hj

Hj . . . subdomain diameter, δj . . . overlap

  • C. Pechstein et al

abstract two-level Schwarz 18 / 24

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SLIDE 26

Numerics – Darcy – I

Domain & Partitions Coefficient Iterations (CG) vs. jumps Code: Matlab & FreeFem++ κ2 AS-1 AS-2-low dim(VH) NEW dim(VH) 1 22 16 (8) 16 (8) 102 31 24 (8) 17 (15) 104 37 30 (8) 21 (15) 106 36 29 (8) 18 (15)

AS-1: 1-level ASM AS-2-low: mj = 1 NEW: λj,mj +1 < δj/Hj

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abstract two-level Schwarz 19 / 24

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SLIDE 27

Numerics – Darcy – II

Iterations (CG) vs. number of subdomains regular partition subd. dofs AS-1 AS-2-low dim(VH) NEW dim(VH) 4 4840 14 15 (4) 10 (6) 8 9680 26 22 (8) 11 (14) 16 19360 51 36 (16) 13 (30) 32 38720 > 100 61 (32) 13 (62) METIS partition subd. dofs AS-1 AS-2-low dim(VH) NEW dim(VH) 4 4840 21 18 (4) 15 (7) 8 9680 36 29 (8) 18 (15) 16 19360 65 45 (16) 22 (31) 32 38720 >100 79 (32) 34 (63)

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abstract two-level Schwarz 20 / 24

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SLIDE 28

Numerics – Darcy – III

Iterations (CG) vs. overlap (added) layers AS-1 AS-2-low (VH) NEW (VH) 1 26 22 (8) 11 (14) 2 22 18 (8) 9 (14) 3 16 15 (8) 9 (14)

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SLIDE 29

Numerics – 2D Elasticity

IsoValue

  • 1.05053e+10

5.28263e+09 1.58079e+10 2.63332e+10 3.68584e+10 4.73837e+10 5.79089e+10 6.84342e+10 7.89595e+10 8.94847e+10 1.0001e+11 1.10535e+11 1.21061e+11 1.31586e+11 1.42111e+11 1.52636e+11 1.63162e+11 1.73687e+11 1.84212e+11 2.10525e+11 E

E1 = 2 · 1011 ν1 = 0.3 E2 = 2 · 107 ν2 = 0.45 METIS partitions with 2 layers added subd. dofs AS-1 AS-2-low (VH) NEW (VH) 4 13122 93 134 (12) 42 (42) 16 13122 164 165 (48) 45 (159) 25 13122 211 229 (75) 47 (238) 64 13122 279 167 (192) 45 (519)

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abstract two-level Schwarz 22 / 24

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SLIDE 30

Numerics – 3D Elasticity

Iterations (CG) vs. number of subdomains E1 = 2 · 1011 ν1 = 0.3 E2 = 2 · 107 ν2 = 0.45

Relative error vs. iterations 16 regular subdomains

subd. dofs AS-1 AS-2-low (VH) NEW (VH) 4 1452 79 54 (24) 16 (46) 8 29040 177 87 (48) 16 (102) 16 58080 378 145 (96) 16 (214)

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abstract two-level Schwarz 23 / 24

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SLIDE 31

Conclusion & Outlook

Remarks: Implementation requires only element stiffness matrices + connectivity Proof works for any partition of unity

(changes the eigenproblem and coarse space)

Outlook: More testing & comparison to other methods Solution of the Eigenproblems (LAPACK → LOBPCG) Coarse space dimension reduction? Coarse problem satisfies assembling property → multilevel method — link to σAMGe ? More applications

Short Preprint (Notes au CRAS): http://hal.archives-ouvertes.fr/hal-00630892/fr/ Detailed Preprint: http://www.numa.uni-linz.ac.at/Publications/List/2011/2011-07.pdf

THANK YOU FOR YOUR ATTENTION!

  • C. Pechstein et al

abstract two-level Schwarz 24 / 24

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SLIDE 32

Conclusion & Outlook

Remarks: Implementation requires only element stiffness matrices + connectivity Proof works for any partition of unity

(changes the eigenproblem and coarse space)

Outlook: More testing & comparison to other methods Solution of the Eigenproblems (LAPACK → LOBPCG) Coarse space dimension reduction? Coarse problem satisfies assembling property → multilevel method — link to σAMGe ? More applications

Short Preprint (Notes au CRAS): http://hal.archives-ouvertes.fr/hal-00630892/fr/ Detailed Preprint: http://www.numa.uni-linz.ac.at/Publications/List/2011/2011-07.pdf

THANK YOU FOR YOUR ATTENTION!

  • C. Pechstein et al

abstract two-level Schwarz 24 / 24

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SLIDE 33

Conclusion & Outlook

Remarks: Implementation requires only element stiffness matrices + connectivity Proof works for any partition of unity

(changes the eigenproblem and coarse space)

Outlook: More testing & comparison to other methods Solution of the Eigenproblems (LAPACK → LOBPCG) Coarse space dimension reduction? Coarse problem satisfies assembling property → multilevel method — link to σAMGe ? More applications

Short Preprint (Notes au CRAS): http://hal.archives-ouvertes.fr/hal-00630892/fr/ Detailed Preprint: http://www.numa.uni-linz.ac.at/Publications/List/2011/2011-07.pdf

THANK YOU FOR YOUR ATTENTION!

  • C. Pechstein et al

abstract two-level Schwarz 24 / 24