Coupled FETI/BETI Solvers for Nonlinear Potential Problems in - - PowerPoint PPT Presentation

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Coupled FETI/BETI Solvers for Nonlinear Potential Problems in - - PowerPoint PPT Presentation

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Outline Coupled FETI/BETI Solvers for Nonlinear Potential Problems in Unbounded Domains Clemens Pechstein 1 Ulrich Langer 1 , 2 1 Special Research Programm SFB F013 on Numerical and Symbolic Scientific Computing 2 Institute of Computational

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SLIDE 1

Outline

Coupled FETI/BETI Solvers for Nonlinear Potential Problems in Unbounded Domains

Clemens Pechstein1 Ulrich Langer1,2

1Special Research Programm SFB F013 on

Numerical and Symbolic Scientific Computing

2Institute of Computational Mathematics

Johannes Kepler University Linz

DD 17, Strobl/Wolfgangsee, 2006

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 2

Outline

Outline

1

FETI/BETI Introduction Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

2

Generalizations Unbounded Domains Nonlinear Problems

3

Conclusion

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 3

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Outline

1

FETI/BETI Introduction Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

2

Generalizations Unbounded Domains Nonlinear Problems

3

Conclusion

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 4

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Motivation – Electromagnetic Field Computations

Nonlinear Magnetostatics in 2D: − ∇ ·

  • ν(|∇u|)∇u
  • = f

+ boundary or radiation conditions + transmission conditions

Characteristics: Nonlinear in ferromagnetic materials Linear behavior in surrounding air / air gaps Typically large jumps over material interfaces

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 5

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Motivation – Electromagnetic Field Computations

Nonlinear Magnetostatics in 2D: − ∇ ·

  • ν(|∇u|)∇u
  • = f

+ boundary or radiation conditions + transmission conditions

Characteristics: Nonlinear in ferromagnetic materials Linear behavior in surrounding air / air gaps Typically large jumps over material interfaces

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 6

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Finite Element Tearing and Interconnecting – Overview

Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 7

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Finite Element Tearing and Interconnecting – Overview

Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 8

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Finite Element Tearing and Interconnecting – Overview

Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 9

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Finite Element Tearing and Interconnecting – Overview

Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration Tearing

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 10

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Finite Element Tearing and Interconnecting – Overview

Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration Interconnecting

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 11

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Finite Element Tearing and Interconnecting – Overview

Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 12

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Finite Element Tearing and Interconnecting – Overview

Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 13

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Finite Element Tearing and Interconnecting – Overview

FETI – Features PCG iteration and preconditioning via local Dirichlet and Neumann solvers Allows massive parallelization Condition number O((1 + log(H/h))2) Robust w.r.t. coefficient jumps

Mandel/Tezaur, 1996 Klawonn/Widlund, 2001

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Boundary Element Tearing and Interconnecting – Overview

FETI-technique can be carried over to the BEM → BETI

Langer/Steinbach, 2003

Coupling BETI and FETI:

Benetit from advantages of both techniques air gaps / surrounding air → BEM nonlinear materials → FEM Langer/Steinbach, 2004

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 15

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Boundary Element Tearing and Interconnecting – Overview

FETI-technique can be carried over to the BEM → BETI

Langer/Steinbach, 2003

Coupling BETI and FETI:

Benetit from advantages of both techniques air gaps / surrounding air → BEM nonlinear materials → FEM Langer/Steinbach, 2004

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 16

Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Outline

1

FETI/BETI Introduction Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

2

Generalizations Unbounded Domains Nonlinear Problems

3

Conclusion

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Quasi-regular non-overlapping Domain Decomposition Ω ⊂ Rd bounded Γ = ∂Ω Ω =

i∈I Ωi

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Quasi-regular non-overlapping Domain Decomposition Ω ⊂ Rd bounded Γ = ∂Ω Ω =

i∈I Ωi

Γi = ∂Ωi ni : outward unit normal vector Γjk = Γj ∩ Γk ΓS =

i∈I Γi

Hi = diam Ωi ≃ H

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Quasi-regular non-overlapping Domain Decomposition Ω ⊂ Rd bounded Γ = ∂Ω Ω =

i∈I Ωi

Γi = ∂Ωi ni : outward unit normal vector Γjk = Γj ∩ Γk ΓS =

i∈I Γi

Hi = diam Ωi ≃ H

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Quasi-regular non-overlapping Domain Decomposition Ω ⊂ Rd bounded Γ = ∂Ω Ω =

i∈I Ωi

Γi = ∂Ωi ni : outward unit normal vector Γjk = Γj ∩ Γk ΓS =

i∈I Γi

Hi = diam Ωi ≃ H

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Quasi-regular non-overlapping Domain Decomposition Ω ⊂ Rd bounded Γ = ∂Ω Ω =

i∈I Ωi

Γi = ∂Ωi ni : outward unit normal vector Γjk = Γj ∩ Γk ΓS =

i∈I Γi

Hi = diam Ωi ≃ H

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Model Problem: −∇ · [αi∇u] = f in Ωi ∀i ∈ I u = 0

  • n Γ

αi ∂u ∂ni + αj ∂u ∂nj = 0

  • n Γi j

∀i = j Solution fulfills Continuity: [u]Γi j = 0 Transmission: ti + tj = 0

  • n Γi j

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Model Problem: −∇ · [αi∇u] = f in Ωi ∀i ∈ I u = 0

  • n Γ

αi ∂u ∂ni

ti

+ αj ∂u ∂nj

tj

= 0

  • n Γi j

∀i = j Solution fulfills Continuity: [u]Γi j = 0 Transmission: ti + tj = 0

  • n Γi j

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Fixed sub-domain Ωi: Solution of −∇ · [αi∇ui] = fi in Ωi ui = gi

  • n Γi

defines co-normal derivative ti = αi

∂ui ∂ni

Dirichlet-to-Neumann map: gi → ti = Si gi − Ni fi Steklov-Poincar´ e operator: Si : H1/2(Γi) → H−1/2(Γi) linear, bounded, injective Newton potential: Ni : H−1(Ωi) → H−1/2(Γi)

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Fixed sub-domain Ωi: Solution of −∇ · [αi∇ui] = fi in Ωi ui = gi

  • n Γi

defines co-normal derivative ti = αi

∂ui ∂ni

Dirichlet-to-Neumann map: gi → ti = Si gi − Ni fi Steklov-Poincar´ e operator: Si : H1/2(Γi) → H−1/2(Γi) linear, bounded, injective Newton potential: Ni : H−1(Ωi) → H−1/2(Γi)

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

A Continuous Formulation

Model Problem −∇ · [αi∇u] = f in Ωi ∀i ∈ I u = 0

  • n Γ

αi ∂u ∂ni

ti

+ αj ∂u ∂nj

tj

= 0

  • n Γi j

∀i = j “equivalent” to Variational Formulation: Find u ∈ H1/2

0,Γ (ΓS) :

  • i∈I

Si u|Γi, v|Γi =

  • i∈I

Ni f|Ωi, v|Γi ∀v ∈ H1/2

0,Γ (ΓS)

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Dirichlet-to-Neumann map – FEM

Fixed sub-domain Ωi, FEM-triangulation Ti,h Given: Dirichlet data gi ∈ Vh(Γi). Find: Galerkin representation ti ∈ V ∗

h (Γi) of Neumann trace.

via Schur complement: ti = SFEM

i,h

gi − NFEM

i,h

f i with SFEM

i,h

= K ΓΓ

i,h −

  • K IΓ

i,h

⊤ K II

i,h

−1K IΓ

i,h

NFEM

i,h

f i = f Γ

i −

  • K IΓ

i,h

⊤ K II

i,h

−1f I

i

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Dirichlet-to-Neumann map – BEM

Fixed sub-domain Ωi, BEM-triangulation Ti,h of Γi Homogeneous b.v.p. is characterized by the Cald´ eron projector gi ti

  • =

1

2I − Ki

Vi Di

1 2I + K ′ i

gi ti

  • Symmetric and stable approximation:

SBEM

i,h

= Di,h + 1 2M⊤

i,h + K ⊤ i,h

  • V −1

i,h

1 2Mi,h + Ki,h

  • General Dirichlet-to-Neumann map:

ti = SBEM

i,h

gi − NBEM

i,h

fi

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Minimization Problem

Variational Problem ↔ Minimization Problem: min

u∈V0,Γ,h(ΓS)

  • i∈I

1 2

  • SFEM/BEM

i,h

Aiu, Aiu

  • −
  • i∈I
  • NFEM/BEM

i,h

f i, Aiu

  • where Ai are connectivity matrices

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Constrained Minimization Problem

Variational Problem ↔ Constrained Minimization Problem: min

ui∈V0,Γ,h(Γi)

  • i∈I

1 2

  • SFEM/BEM

i,h

ui, ui

  • −
  • i∈I
  • NFEM/BEM

i,h

f i, ui

  • subject to the continuity constraints
  • i∈I

Biui = 0 with (Bi)mn ∈ {0, −1, +1}

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Lagrange Multipliers

B B B B B B B B B B B B B B B @ SFEM

1,h

B⊤

1

... . . . SFEM

q,h

B⊤

q

SBEM

q+1,h

B⊤

q+1

... . . . SBEM

p,h

B⊤

p

B1 · · · Bq Bq+1 · · · Bp 1 C C C C C C C C C C C C C C C A B B B B B B B B B B B B B @ u1 . . . uq uq+1 . . . up λ 1 C C C C C C C C C C C C C A = B B B B B B B B B B B B B B @ f FEM

1

. . . f FEM

q

f BEM

q+1

. . . f BEM

p

1 C C C C C C C C C C C C C C A

Elimination of primal variables ui =

  • SFEM/BEM

i,h

† f FEM/BEM

i

− B⊤

i λ

  • + Riγi
  • Clemens Pechstein, Ulrich Langer

FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Lagrange Multipliers

B B B B B B B B B B B B B B B @ SFEM

1,h

B⊤

1

... . . . SFEM

q,h

B⊤

q

SBEM

q+1,h

B⊤

q+1

... . . . SBEM

p,h

B⊤

p

B1 · · · Bq Bq+1 · · · Bp 1 C C C C C C C C C C C C C C C A B B B B B B B B B B B B B @ u1 . . . uq uq+1 . . . up λ 1 C C C C C C C C C C C C C A = B B B B B B B B B B B B B B @ f FEM

1

. . . f FEM

q

f BEM

q+1

. . . f BEM

p

1 C C C C C C C C C C C C C C A

Elimination of primal variables ui =

  • SFEM/BEM

i,h

† f FEM/BEM

i

− B⊤

i λ

  • + Riγi
  • Clemens Pechstein, Ulrich Langer

FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Dual Problem

Reduced system:

  • F

−G G⊤ λ γ

  • =

d e

  • with

F =

  • i∈I

Bi

  • SFEM/BEM

i,h

†B⊤

i ,

G = (BiRi)i:Γi∩Γ=∅ P : suitable projection onto (ker G)⊥ → spd Problem Find ˜ λ ∈ (ker G)⊥ : P⊤F P ˜ λ = ˜ d → PCG sub-space iteration on (ker G)⊥

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Dual Problem

Reduced system:

  • F

−G G⊤ λ γ

  • =

d e

  • with

F =

  • i∈I

Bi

  • SFEM/BEM

i,h

†B⊤

i ,

G = (BiRi)i:Γi∩Γ=∅ P : suitable projection onto (ker G)⊥ → spd Problem Find ˜ λ ∈ (ker G)⊥ : P⊤F P ˜ λ = ˜ d → PCG sub-space iteration on (ker G)⊥

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

Outline

1

FETI/BETI Introduction Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

2

Generalizations Unbounded Domains Nonlinear Problems

3

Conclusion

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

FETI/BETI Preconditioners

F =

  • i∈I

Bi

  • SFEM/BEM

i,h

†B⊤

i

Construct Preconditioner M−1 = D−1

B i∈I

Bi D−1

α,i SFEM/BEM i,h

  • r DBEM

i,h

D−1

α,i B⊤ i

  • D−1

B

with suitable scaling matrices Dα,i and [DB]i = BiD−1

α,i B⊤ i

Theorem If projection P and scaling Dα,i suitably chosen then κ

  • P M−1P⊤P⊤F P
  • ≤ C
  • 1 + log

H

h

2, C independent of h, H and coefficient jumps.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

FETI/BETI Preconditioners

F =

  • i∈I

Bi

  • SFEM/BEM

i,h

†B⊤

i

Construct Preconditioner M−1 = D−1

B i∈I

Bi D−1

α,i SFEM/BEM i,h

  • r DBEM

i,h

D−1

α,i B⊤ i

  • D−1

B

with suitable scaling matrices Dα,i and [DB]i = BiD−1

α,i B⊤ i

Theorem If projection P and scaling Dα,i suitably chosen then κ

  • P M−1P⊤P⊤F P
  • ≤ C
  • 1 + log

H

h

2, C independent of h, H and coefficient jumps.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

FETI/BETI Preconditioners

Proof. FETI: Klawonn/Widlund 2001 Preconditioner based on scaling Dα,i(x) =

αi P

j∈Nx αj .

Proof based on partition of unity into face/edge/vertex terms and inequalities in H1/2(Γi)-norms. BETI and Coupling: Langer/Steinbach 2003/2004 based on spectral equivalences SFEM

i,h

≃ SBEM

i,h

≃ DBEM

i,h

and

  • ther properties of SBEM

i,h

.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

FETI/BETI Preconditioners

Proof. FETI: Klawonn/Widlund 2001 Preconditioner based on scaling Dα,i(x) =

αi P

j∈Nx αj .

Proof based on partition of unity into face/edge/vertex terms and inequalities in H1/2(Γi)-norms. BETI and Coupling: Langer/Steinbach 2003/2004 based on spectral equivalences SFEM

i,h

≃ SBEM

i,h

≃ DBEM

i,h

and

  • ther properties of SBEM

i,h

.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

FETI/BETI Preconditioners

F =

  • i∈I

Bi

  • SFEM/BEM

i,h

†B⊤

i

Preconditioner M−1 = D−1

B i∈I

BiD−1

α,i SFEM/BEM i,h

  • r DBEM

i,h

D−1

α,i B⊤ i

  • D−1

B

Crucial steps in PCG can be done in parallel FEM: local Dirichlet and Neumann solvers BEM: application of SBEM

i,h

and its pseudo-inverse

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

FETI/BETI Preconditioners

F =

  • i∈I

Bi

  • SFEM/BEM

i,h

†B⊤

i

Preconditioner M−1 = D−1

B i∈I

BiD−1

α,i SFEM/BEM i,h

  • r DBEM

i,h

D−1

α,i B⊤ i

  • D−1

B

Crucial steps in PCG can be done in parallel FEM: local Dirichlet and Neumann solvers BEM: application of SBEM

i,h

and its pseudo-inverse

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

FETI/BETI – Numerical Results

2D magnetic valve

  • glob. dof

PCG it. FETI

FETI/BETI

Lagr. H/h FETI

FETI/BETI

806 484 408 3 12 11 3539 1612 777 6 14 13 14357 5662 1515 12 17 16 57833 20983 2991 24 19 18 232145 80194 5943 48 21 20

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Outline

1

FETI/BETI Introduction Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

2

Generalizations Unbounded Domains Nonlinear Problems

3

Conclusion

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Unbounded Domains

Exterior Domain Ωc = Rd \ Ω −∇ · [αext∇u] = 0 in Ωc ui = g

  • n Γ

|u(x) − u0| = O(|x|−1) for |x| → ∞ Dirichlet-to-Neumann map: g → t := αext ∂u

∂nc

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Unbounded Domains

Interior Domain Ω: g t

  • =

1

2I − K

V D

1 2I + K ′

g t

  • +

N0 f N1 f

  • Dirichlet-to-Neumann map:

t = Sint g − Nint f ker Sint = span{1} Symmetric and stable approximation: SBEM

int,h = Dh +

1 2M⊤

h + K ⊤ h

  • V −1

h

1 2Mh + Kh

  • Clemens Pechstein, Ulrich Langer

FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Unbounded Domains

Exterior Domain Ωc: g t

  • =

1

2I + K

V D

1 2I − K ′

g t

  • +

u0

  • Dirichlet-to-Neumann map:

t = Sext g − V −1u0 ker Sext = {0} Symmetric and stable approximation: SBEM

ext,h = Dh +

1 2M⊤

h − K ⊤ h

  • V −1

h

1 2Mh − Kh

  • Clemens Pechstein, Ulrich Langer

FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Standard FETI/BETI theory

Core part of the condition number estimate: |PDu|2

S ≤ C

  • 1 + log

H h 2 |u|2

S,

where |u|2

S :=

  • i∈I

|ui|2

SFEM/BEM

i,h

and PD denotes the Dα,i-weighted projection.

  • E. g. for a face F = Γi ∩ Γj:

(PDu)i(x) = ± αj αi + αj

  • ui(x) − uj(x)
  • Proof performed by a partition of unity on Γi:

1 =

  • F∈FΓi

θF +

  • E∈EΓi

θE +

  • V∈VΓi

θV

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Generalization to unbounded domains

FETI/BETI theory might work quite analogous when the exterior domain is treated as a subdomain. But following problems arise: Sext has a trivial kernel different scalings: Hi = diam Ωi and HΓ = diam Ω number of subdomains Ωi touching the outer boundary Γ may be arbitrarily large processor balancing

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Unbounded Domains – Numerical Results

HF = maxF∈FDD diam F maximal diameter of a face HΩ = diam Ω 2D, unbounded FETI/BETI (including the exterior domain) HF/h = 4 fixed # subdom. HΩ/HF PCG it. 9 3 8 36 6 9 144 12 9 576 24 11 2304 48 12

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Condition Estimate for Unbounded Domains

HF = maxF∈FDD diam F maximal diameter of a face HΩ = diam Ω Conjecture For the unbounded FETI/BETI setting the following condition number estimate holds: κ

  • P M−1P⊤P⊤F P
  • ≤ C
  • 1 + log

HF

h

2 1 + log HΩ

HF

α, C independent of h, HF, HΩ and coefficient jumps. Proof in progess ...

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Outline

1

FETI/BETI Introduction Motivation and Overview Coupled FETI/BETI Formulation Preconditioners

2

Generalizations Unbounded Domains Nonlinear Problems

3

Conclusion

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 52

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Nonlinear Problems

Nonlinear Potential Problem: − ∇ ·

  • νi(|∇u|)∇u
  • = f

in Ωi

+ boundary / radiation conditions + transmission conditions

where νi ∈ C1(R+

0 → R+)

with s → νi(s)s strongly monotonic increasing and Lipschitz continuous

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 53

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Nonlinear Problems

Nonlinear Potential Problem: − ∇ ·

  • νi(|∇u|)∇u
  • = f

in Ωi

+ boundary / radiation conditions + transmission conditions

where νi ∈ C1(R+

0 → R+)

with s → νi(s)s strongly monotonic increasing and Lipschitz continuous

  • e. g. νi determined from approximated

B-H-curve (Electromagnetics) J¨ uttler/Pechstein, 2005

reluctivity νi

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 54

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Newton’s Method

→ Linearized Problem − ∇ ·

  • ζi(∇u(k))∇w(k)

= r (k) in Ωi

+ boundary / radiation conditions + transmission conditions

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 55

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Newton’s Method

→ Linearized Problem − ∇ ·

  • ζi(∇u(k))∇w(k)

= r (k) in Ωi

+ boundary / radiation conditions + transmission conditions

Problem: Spectrum of ζi(∇u(k)) Rate of Variation:

maxx∈Ωi λmax(ζi(∇u(k)(x))) minx∈Ωi λmin(ζi(∇u(k)(x)))

Rate of Anisotropy: maxx∈Ωi

λmax(ζi(∇u(k)(x))) λmin(ζi(∇u(k)(x)))

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 56

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Newton’s Method

→ Linearized Problem − ∇ ·

  • ζi(∇u(k))∇w(k)

= r (k) in Ωi

+ boundary / radiation conditions + transmission conditions reluctivity νi(|∇u(k)|)

Problem: Spectrum of ζi(∇u(k)) Rate of Variation:

maxx∈Ωi λmax(ζi(∇u(k)(x))) minx∈Ωi λmin(ζi(∇u(k)(x)))

can be large (103) Rate of Anisotropy: maxx∈Ωi

λmax(ζi(∇u(k)(x))) λmin(ζi(∇u(k)(x)))

here small

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Preconditioning

First Preconditioner M−1

1

Find bounds αi, αi: αi|ξ|2 ≤ (ζi(∇u(k)(x)) ξ, ξ) ≤ αi|ξ|2 ∀x ∈ Ωi ∀i ∈ I M−1

1

is based on Steklov Poincar´ e operators with the constant scalar coefficients αi Theorem M−1

1

fulfills κ(P M−1

1 P⊤P⊤F P)

  • maxi∈I

αi αi

  • 1 + log

H

h

2 independent of coefficient jumps.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 58

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Preconditioning

First Preconditioner M−1

1

Find bounds αi, αi: αi|ξ|2 ≤ (ζi(∇u(k)(x)) ξ, ξ) ≤ αi|ξ|2 ∀x ∈ Ωi ∀i ∈ I M−1

1

is based on Steklov Poincar´ e operators with the constant scalar coefficients αi Theorem M−1

1

fulfills κ(P M−1

1 P⊤P⊤F P)

  • maxi∈I

αi αi

  • 1 + log

H

h

2 independent of coefficient jumps.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Preconditioning

Second Preconditioner M−1

2

Introduce mean coefficient field ˆ αi(x) = max

ξ∈Rd\{0}

(ζi(∇u(k)(x)) ξ, ξ) |ξ|2 M−1

2

is based on Steklov-Poincar´ e operators with scalar but varying coefficients ˆ αi(x). Scaling matrices Dˆ

α,i involve ˆ

αi(x). Conjecture M−1

2

leads to a better condition number than M−1

1

with a moderate dependence on the spectral variation.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 60

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Preconditioning

Second Preconditioner M−1

2

Introduce mean coefficient field ˆ αi(x) = max

ξ∈Rd\{0}

(ζi(∇u(k)(x)) ξ, ξ) |ξ|2 M−1

2

is based on Steklov-Poincar´ e operators with scalar but varying coefficients ˆ αi(x). Scaling matrices Dˆ

α,i involve ˆ

αi(x). Conjecture M−1

2

leads to a better condition number than M−1

1

with a moderate dependence on the spectral variation.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Nonlinear Problems – Numerical Results

Model problem

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 62

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Nonlinear Problems – Numerical Results

fL∞ = 2.3 · 10−6 εlin = 10−8 εNewton = 10−6

PCG iterations d.o.f. Lagr. H/h anis. var.

Newt.

M−1

1

M−1

2

ref 897 178 4 6.1 13.1 2 14.5 10.0 11 3713 354 8 6.1 13.7 2 17.0 12.0 12 15105 706 16 6.1 13.8 2 19.0 13.0 13 60929 1410 32 6.1 13.9 2 19.5 15.0 14 224737 2818 64 6.1 13.9 2 20.0 15.0 14

15 subdomains, FETI

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 63

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Nonlinear Problems – Numerical Results

fL∞ = 8 · 10−6 εlin = 10−8 εNewton = 10−6

PCG iterations d.o.f. Lagr. H/h anis. var.

Newt.

M−1

1

M−1

2

ref 897 178 4 4.8 4.8 2 14.5 10.0 11 3713 354 8 4.9 8.5 2 16.5 12.0 12 15105 706 16 5.0 26.6 3 22.7 12.7 13 60929 1410 32 11.2 331.6 3 26.0 14.7 14 224737 2818 64 12.3 2657.1 3 88.3 16.7 14

15 subdomains, FETI

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 64

Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Nonlinear Problems – Numerical Results

Simplified model of a magnetic valve

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Nonlinear Problems – Numerical Results

preconditioner M−1

2

FETI εlin = 10−8 εNewton = 10−6

d.o.f. Lagr. H/h anis. var. Newt. PCG it. ref 806 408 3 12.0 187.4 6 14.0 12 2539 777 6 12.0 469.9 4 17.8 14 14357 1515 12 12.0 852.4 4 21.3 17 57833 2991 24 12.0 1495.4 3 25.3 19 232145 5943 48 12.4 2670.9 4 27.8 21

70 subdomains

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Nonlinear Problems – Numerical Results

preconditioner M−1

2

FETI/BETI εlin = 10−8 εNewton = 10−6

d.o.f. Lagr. H/h anis. var. Newt. PCG it. ref 806 408 3 12.0 187.4 6 13.7 11 2539 777 6 12.0 469.9 4 18.0 13 14357 1515 12 12.0 852.4 5 21.0 16 57833 2991 24 12.0 1495.4 4 24.0 18 232145 5943 48 12.4 2670.9 5 27.2 20

70 subdomains

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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Introduction Generalizations Conclusion Unbounded Domains Nonlinear Problems

Nonlinear Problems – Numerical Results

distribution of ν(|B|) for the solution of the nonlinear problem Newton residual (nested iteration)

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 68

Introduction Generalizations Conclusion

Conclusion

Generalization of coupled FETI/BETI framework to

1

unbounded domains

2

varying coefficients → nonlinear problems Two conjectures driven from numerical results:

1

leads to weak dependence on HΩ/HF

2

leads to weak dependence of the rate of variation in the condition estimate. Acknowledgements: Many thanks to Olaf Steinbach and G¨ unther Of for endless discussions. Financial support of the FWF within the SFB F013.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 69

Introduction Generalizations Conclusion

Conclusion

Generalization of coupled FETI/BETI framework to

1

unbounded domains

2

varying coefficients → nonlinear problems Two conjectures driven from numerical results:

1

leads to weak dependence on HΩ/HF

2

leads to weak dependence of the rate of variation in the condition estimate. Acknowledgements: Many thanks to Olaf Steinbach and G¨ unther Of for endless discussions. Financial support of the FWF within the SFB F013.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains

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SLIDE 70

Introduction Generalizations Conclusion

Conclusion

Generalization of coupled FETI/BETI framework to

1

unbounded domains

2

varying coefficients → nonlinear problems Two conjectures driven from numerical results:

1

leads to weak dependence on HΩ/HF

2

leads to weak dependence of the rate of variation in the condition estimate. Acknowledgements: Many thanks to Olaf Steinbach and G¨ unther Of for endless discussions. Financial support of the FWF within the SFB F013.

Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains