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Generalised FEs: Domain Decomposition, Optimal Local Approximation - - PowerPoint PPT Presentation

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Generalised FEs: Domain Decomposition, Optimal Local Approximation & Model Order Reduction Robert Scheichl Institute of Applied Mathematics & Interdisciplinary Centre for Scientific Computing Universit at Heidelberg Collaborators:

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SLIDE 1

Generalised FEs: Domain Decomposition, Optimal Local Approximation & Model Order Reduction

Robert Scheichl

Institute of Applied Mathematics & Interdisciplinary Centre for Scientific Computing Universit¨ at Heidelberg

Collaborators: Victorita Dolean (Strathclyde), Frederic Nataf (Sorbonne), Clemens Pechstein (Dassault Syst` emes), Daniel Peterseim (Augsburg), Nicole Spillane (´ Ecole Polytechnique), Panayot Vassilevski (LLNL), Ludmil Zikatanov (Penn State)

Parallel Solution Methods for Systems Arising from PDEs CIRM – Luminy, Marseille, September 16th, 2019

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 1 / 38

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SLIDE 2

Overview

Problem Formulation & Motivation Robust Subspace Correction vs. Multiscale Discretisation Key message: use weighted norms for contrast independence Three Theoretical Tools (Three) Practical ‘Knobs’ Beyond scalar elliptic problem: anisotropic linear elasticity High performance implementation of GenEO Some Numerical Results Outlook – GenEO as a surrogate in Multilevel MCMC

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 2 / 38

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SLIDE 3

Overview

Problem Formulation & Motivation Robust Subspace Correction vs. Multiscale Discretisation Key message: use weighted norms for contrast independence Three Theoretical Tools (Three) Practical ‘Knobs’ Beyond scalar elliptic problem: anisotropic linear elasticity High performance implementation of GenEO Some Numerical Results Outlook – GenEO as a surrogate in Multilevel MCMC

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 2 / 38

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SLIDE 4

Overview

Problem Formulation & Motivation Robust Subspace Correction vs. Multiscale Discretisation Key message: use weighted norms for contrast independence Three Theoretical Tools (Three) Practical ‘Knobs’ Beyond scalar elliptic problem: anisotropic linear elasticity High performance implementation of GenEO Some Numerical Results Outlook – GenEO as a surrogate in Multilevel MCMC

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 2 / 38

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SLIDE 5

Overview

Problem Formulation & Motivation Robust Subspace Correction vs. Multiscale Discretisation Key message: use weighted norms for contrast independence Three Theoretical Tools (Three) Practical ‘Knobs’ Beyond scalar elliptic problem: anisotropic linear elasticity High performance implementation of GenEO Some Numerical Results Outlook – GenEO as a surrogate in Multilevel MCMC

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 2 / 38

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Overview

Problem Formulation & Motivation Robust Subspace Correction vs. Multiscale Discretisation Key message: use weighted norms for contrast independence Three Theoretical Tools (Three) Practical ‘Knobs’ Beyond scalar elliptic problem: anisotropic linear elasticity High performance implementation of GenEO Some Numerical Results Outlook – GenEO as a surrogate in Multilevel MCMC

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 2 / 38

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SLIDE 7

Model Problem

Elliptic PDE in bounded domain Ω ⊂ Rd, d = 2, 3 −∇ · (α∇u) = f

+ suitable BCs on ∂Ω

Issues adressed even more pronounced in other eqns., e.g. linear elasticity.

Strongly varying coefficient α(x) ≥ 1

(otherwise rescale eqn.)

(scalar α, or quasi-isotropic tensor α with λmax(α) ∼ λmin(α))

FE discretisation (p.w. lin. V h): a(uh, vh) = (f , vh) ∀vh ∈ Vh Two possible aims:

h-optimal, α-robust parallel solver (fine mesh T h, α resolved) H-optimal, α-robust approximation in coarse space V H (α not resolved: “Upscaling” – no scale separation!)

Key Question (for both): Robust coarse space

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 3 / 38

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SLIDE 8

Model Problem

Elliptic PDE in bounded domain Ω ⊂ Rd, d = 2, 3 −∇ · (α∇u) = f

+ suitable BCs on ∂Ω

Issues adressed even more pronounced in other eqns., e.g. linear elasticity.

Strongly varying coefficient α(x) ≥ 1

(otherwise rescale eqn.)

(scalar α, or quasi-isotropic tensor α with λmax(α) ∼ λmin(α))

FE discretisation (p.w. lin. V h): a(uh, vh) = (f , vh) ∀vh ∈ Vh Two possible aims:

h-optimal, α-robust parallel solver (fine mesh T h, α resolved) H-optimal, α-robust approximation in coarse space V H (α not resolved: “Upscaling” – no scale separation!)

Key Question (for both): Robust coarse space

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 3 / 38

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SLIDE 9

Model Problem

Elliptic PDE in bounded domain Ω ⊂ Rd, d = 2, 3 −∇ · (α∇u) = f

+ suitable BCs on ∂Ω

Issues adressed even more pronounced in other eqns., e.g. linear elasticity.

Strongly varying coefficient α(x) ≥ 1

(otherwise rescale eqn.)

(scalar α, or quasi-isotropic tensor α with λmax(α) ∼ λmin(α))

FE discretisation (p.w. lin. V h): a(uh, vh) = (f , vh) ∀vh ∈ Vh Two possible aims:

h-optimal, α-robust parallel solver (fine mesh T h, α resolved) H-optimal, α-robust approximation in coarse space V H (α not resolved: “Upscaling” – no scale separation!)

Key Question (for both): Robust coarse space

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 3 / 38

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SLIDE 10

Applications: Simulation in Heterogeneous Media

Elasticity, e.g. in bone or carbon fibre composites Subsurface flow, e.g. in an oil reservoir

(SPE10 benchmark)

... many more ...

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 4 / 38

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SLIDE 11

Initial Remarks

Complicated variation of α(x) on many scales (h ≪ diam(Ω)) Hard to resolve by “geometric” coarse mesh! Goal A: Efficient & scalable multilevel parallel solver

robust w.r.t. mesh size h (⇔ w.r.t. problem size n: O(n) cost) robust w.r.t. coefficients α(x) !

Goal B: Simulate on coarse mesh where α is not resolved !

Discretisation in “special” coarse space V H → Upscaling Approximation depends on (subgrid) variation & contrast in α !

Robust multiscale space is expensive for general coefficients Unless we have periodicity, scale separation, multiple RHSs, parameter dependence, not clear why Goal B over Goal A Coefficient-robust theory for Goal B much less well developed !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 5 / 38

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Initial Remarks

Complicated variation of α(x) on many scales (h ≪ diam(Ω)) Hard to resolve by “geometric” coarse mesh! Goal A: Efficient & scalable multilevel parallel solver

robust w.r.t. mesh size h (⇔ w.r.t. problem size n: O(n) cost) robust w.r.t. coefficients α(x) !

Goal B: Simulate on coarse mesh where α is not resolved !

Discretisation in “special” coarse space V H → Upscaling Approximation depends on (subgrid) variation & contrast in α !

Robust multiscale space is expensive for general coefficients Unless we have periodicity, scale separation, multiple RHSs, parameter dependence, not clear why Goal B over Goal A Coefficient-robust theory for Goal B much less well developed !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 5 / 38

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SLIDE 13

Initial Remarks

Complicated variation of α(x) on many scales (h ≪ diam(Ω)) Hard to resolve by “geometric” coarse mesh! Goal A: Efficient & scalable multilevel parallel solver

robust w.r.t. mesh size h (⇔ w.r.t. problem size n: O(n) cost) robust w.r.t. coefficients α(x) !

Goal B: Simulate on coarse mesh where α is not resolved !

Discretisation in “special” coarse space V H → Upscaling Approximation depends on (subgrid) variation & contrast in α !

Robust multiscale space is expensive for general coefficients Unless we have periodicity, scale separation, multiple RHSs, parameter dependence, not clear why Goal B over Goal A Coefficient-robust theory for Goal B much less well developed !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 5 / 38

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SLIDE 14

Initial Remarks

Complicated variation of α(x) on many scales (h ≪ diam(Ω)) Hard to resolve by “geometric” coarse mesh! Goal A: Efficient & scalable multilevel parallel solver

robust w.r.t. mesh size h (⇔ w.r.t. problem size n: O(n) cost) robust w.r.t. coefficients α(x) !

Goal B: Simulate on coarse mesh where α is not resolved !

Discretisation in “special” coarse space V H → Upscaling Approximation depends on (subgrid) variation & contrast in α !

Robust multiscale space is expensive for general coefficients Unless we have periodicity, scale separation, multiple RHSs, parameter dependence, not clear why Goal B over Goal A Coefficient-robust theory for Goal B much less well developed !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 5 / 38

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SLIDE 15

Initial Remarks

Complicated variation of α(x) on many scales (h ≪ diam(Ω)) Hard to resolve by “geometric” coarse mesh! Goal A: Efficient & scalable multilevel parallel solver

robust w.r.t. mesh size h (⇔ w.r.t. problem size n: O(n) cost) robust w.r.t. coefficients α(x) !

Goal B: Simulate on coarse mesh where α is not resolved !

Discretisation in “special” coarse space V H → Upscaling Approximation depends on (subgrid) variation & contrast in α !

Robust multiscale space is expensive for general coefficients Unless we have periodicity, scale separation, multiple RHSs, parameter dependence, not clear why Goal B over Goal A Coefficient-robust theory for Goal B much less well developed !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 5 / 38

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Domain Decomposition / Multigrid Theory for Varying Coefficients

Coarse grids resolve coefficient

Bramble, Pasciak & Schatz, 88 & 89; Mandel, 93; Dryja, Smith & Widlund, 94; Wang & Xie, 94; Chan & Mathew, 94; Dryja, Sarkis & Widlund, 96; Sarkis, 97; Klawonn & Widlund, 01, Mandel & Dohrmann, 03; Toselli & Widlund, 05; Xu & Zhu, 08; etc

Coarse grids do not resolve coefficient

Graham & Hagger, 99; Graham, Lechner & RS, 07; Pechstein & RS, 08; Van lent, RS & Graham 09; Galvis & Efendiev 10; Dolean, Nataf, RS & Spillane, 11; RS, Vassilevski & Zikatanov, 11; Efendiev, Galvis, Lazarov & Willems, 12; Spillane, Dolean, Hauret, Nataf et al, 14; Heinlein, Klawonn & Rheinbach, 16; Gander & Loneland, 17; etc

Ideas from Algebraic Multigrid literature

Alcouffe, Brandt, Dendy et al, 81; Ruge, St¨ uben, 87; Vassilevski, 92; Vanek, Mandel & Brezina, 96; Chartier, Falgout, Henson et al, 03; Falgout, Vassilevski & Zikatanov 05; Vassilevski, 08; etc

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 6 / 38

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SLIDE 17

Domain Decomposition / Multigrid Theory for Varying Coefficients

Coarse grids resolve coefficient

Bramble, Pasciak & Schatz, 88 & 89; Mandel, 93; Dryja, Smith & Widlund, 94; Wang & Xie, 94; Chan & Mathew, 94; Dryja, Sarkis & Widlund, 96; Sarkis, 97; Klawonn & Widlund, 01, Mandel & Dohrmann, 03; Toselli & Widlund, 05; Xu & Zhu, 08; etc

Coarse grids do not resolve coefficient

Graham & Hagger, 99; Graham, Lechner & RS, 07; Pechstein & RS, 08; Van lent, RS & Graham 09; Galvis & Efendiev 10; Dolean, Nataf, RS & Spillane, 11; RS, Vassilevski & Zikatanov, 11; Efendiev, Galvis, Lazarov & Willems, 12; Spillane, Dolean, Hauret, Nataf et al, 14; Heinlein, Klawonn & Rheinbach, 16; Gander & Loneland, 17; etc

Ideas from Algebraic Multigrid literature

Alcouffe, Brandt, Dendy et al, 81; Ruge, St¨ uben, 87; Vassilevski, 92; Vanek, Mandel & Brezina, 96; Chartier, Falgout, Henson et al, 03; Falgout, Vassilevski & Zikatanov 05; Vassilevski, 08; etc

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 6 / 38

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SLIDE 18

Domain Decomposition / Multigrid Theory for Varying Coefficients

Coarse grids resolve coefficient

Bramble, Pasciak & Schatz, 88 & 89; Mandel, 93; Dryja, Smith & Widlund, 94; Wang & Xie, 94; Chan & Mathew, 94; Dryja, Sarkis & Widlund, 96; Sarkis, 97; Klawonn & Widlund, 01, Mandel & Dohrmann, 03; Toselli & Widlund, 05; Xu & Zhu, 08; etc

Coarse grids do not resolve coefficient

Graham & Hagger, 99; Graham, Lechner & RS, 07; Pechstein & RS, 08; Van lent, RS & Graham 09; Galvis & Efendiev 10; Dolean, Nataf, RS & Spillane, 11; RS, Vassilevski & Zikatanov, 11; Efendiev, Galvis, Lazarov & Willems, 12; Spillane, Dolean, Hauret, Nataf et al, 14; Heinlein, Klawonn & Rheinbach, 16; Gander & Loneland, 17; etc

Ideas from Algebraic Multigrid literature

Alcouffe, Brandt, Dendy et al, 81; Ruge, St¨ uben, 87; Vassilevski, 92; Vanek, Mandel & Brezina, 96; Chartier, Falgout, Henson et al, 03; Falgout, Vassilevski & Zikatanov 05; Vassilevski, 08; etc

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 6 / 38

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Types of Multiscale Methods & Theory (incomplete list)

Adaptive FEs

..., [Babuska, Rheinboldt, 1978]

Generalised FEs

[Babuska, Osborn, 1983]

Numerical Upscaling

..., [Durlofsky, 1991]

Multiscale Finite Elements

[Hou, Wu, 1997], ...

Variational Multiscale Method

[Hughes et al, 1998]

Multigrid Based Upscaling

[Moulton, Dendy, Hyman, 1998]

Multiscale Finite Volume Methods

[Jenny, Lee, Tchelepi, 2003]

Heterogeneous Multiscale Method

[E, Engquist, 2003]

Multiscale Mortar Spaces

[Arbogast, Wheeler et al, 2007] (& other DD based methods)

Adaptive Multiscale FVMs/FEs [Durlovsky, Efendiev, Ginting, 2007] Energy minimising bases

[Dubois, Mishev, Zikatanov, 2009]

Locally spectral (Generalised MsFEM) [Efendiev, Galvis, Wu, 2010] ... etc ...

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 7 / 38

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Simplifying Assumptions & Theory (incomplete list of refs)

1 Periodicity ⇒ Homogenisation theory

..., [Hou, Wu, 1997], ...

2 Scale Separation

..., [Abdulle, 2005], ...

3 Inclusions and simple interfaces

[Chu, Graham, Hou, 2010] (high contrast, no periodicity, no scale separation)

4 Bound in special flux norm

[Berlyand, Owhadi, 2010] (high contrast, no periodicity, no scale separation)

5 Low contrast

[Larson, Malqvist, ’07], [Owhadi, Zhang, ’11], [Grasedyck et al, ’11], [Babuska, Lipton, ’11], [Malqvist, Peterseim, ’14] (no periodicity, no scale separation)

6 Exploit links to DD theory [RS, Vassilevski, Zikatanov, 2011]

(weighted Poincar´ e, stable quasi-interpolant, weighted Bramble-Hilbert)

Combine 5 and 6 to cover more general high contrast coeffs.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 8 / 38

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SLIDE 21

Simplifying Assumptions & Theory (incomplete list of refs)

1 Periodicity ⇒ Homogenisation theory

..., [Hou, Wu, 1997], ...

2 Scale Separation

..., [Abdulle, 2005], ...

3 Inclusions and simple interfaces

[Chu, Graham, Hou, 2010] (high contrast, no periodicity, no scale separation)

4 Bound in special flux norm

[Berlyand, Owhadi, 2010] (high contrast, no periodicity, no scale separation)

5 Low contrast

[Larson, Malqvist, ’07], [Owhadi, Zhang, ’11], [Grasedyck et al, ’11], [Babuska, Lipton, ’11], [Malqvist, Peterseim, ’14] (no periodicity, no scale separation)

6 Exploit links to DD theory [RS, Vassilevski, Zikatanov, 2011]

(weighted Poincar´ e, stable quasi-interpolant, weighted Bramble-Hilbert)

Combine 5 and 6 to cover more general high contrast coeffs.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 8 / 38

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SLIDE 22

Simplifying Assumptions & Theory (incomplete list of refs)

1 Periodicity ⇒ Homogenisation theory

..., [Hou, Wu, 1997], ...

2 Scale Separation

..., [Abdulle, 2005], ...

3 Inclusions and simple interfaces

[Chu, Graham, Hou, 2010] (high contrast, no periodicity, no scale separation)

4 Bound in special flux norm

[Berlyand, Owhadi, 2010] (high contrast, no periodicity, no scale separation)

5 Low contrast

[Larson, Malqvist, ’07], [Owhadi, Zhang, ’11], [Grasedyck et al, ’11], [Babuska, Lipton, ’11], [Malqvist, Peterseim, ’14] (no periodicity, no scale separation)

6 Exploit links to DD theory [RS, Vassilevski, Zikatanov, 2011]

(weighted Poincar´ e, stable quasi-interpolant, weighted Bramble-Hilbert)

Combine 5 and 6 to cover more general high contrast coeffs.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 8 / 38

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SLIDE 23

Simplifying Assumptions & Theory (incomplete list of refs)

1 Periodicity ⇒ Homogenisation theory

..., [Hou, Wu, 1997], ...

2 Scale Separation

..., [Abdulle, 2005], ...

3 Inclusions and simple interfaces

[Chu, Graham, Hou, 2010] (high contrast, no periodicity, no scale separation)

4 Bound in special flux norm

[Berlyand, Owhadi, 2010] (high contrast, no periodicity, no scale separation)

5 Low contrast

[Larson, Malqvist, ’07], [Owhadi, Zhang, ’11], [Grasedyck et al, ’11], [Babuska, Lipton, ’11], [Malqvist, Peterseim, ’14] (no periodicity, no scale separation)

6 Exploiting links to DD [RS, Vassilevski, Zikatanov, 2011]

(weighted Poincar´ e, stable quasi-interpolant, weighted Bramble-Hilbert)

Combine 5 and 6 to cover more general high contrast coeffs.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 8 / 38

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SLIDE 24

Classical theory and more recent ideas Classical theory

in H1 and H1/2-norm based on standard Poincar´ e inequalities and robustness of weighted L2-projections

[Bramble, Xu, Math Comp 91], . . . (for resolving coarse grids!)

More recent ideas

directly in the energy norm based on weighted Poincar´ e type inequalities

[Galvis, Efendiev, 2010], [Pechstein, RS, 2011 & 2012]

and an abstract Bramble-Hilbert Lemma ← − This Talk!

(for energy minimising coarse spaces) [RS, Vassilevski, Zikatanov, MMS 2011]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 9 / 38

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SLIDE 25

Classical theory and more recent ideas Classical theory

in H1 and H1/2-norm based on standard Poincar´ e inequalities and robustness of weighted L2-projections

[Bramble, Xu, Math Comp 91], . . . (for resolving coarse grids!)

More recent ideas

directly in the energy norm based on weighted Poincar´ e type inequalities

[Galvis, Efendiev, 2010], [Pechstein, RS, 2011 & 2012]

and an abstract Bramble-Hilbert Lemma ← − This Talk!

(for energy minimising coarse spaces) [RS, Vassilevski, Zikatanov, MMS 2011]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 9 / 38

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SLIDE 26

Subspace Correction Methods (e.g. two-level Schwarz or multigrid )

Problem (in variational form): Find uh ∈ Vh s.t. a(uh, vh) ≡

  • Ω

α∇uh · ∇vh = (f , vh) for all vh ∈ Vh. Precondition by solving (exactly or approximately) in subspaces V0, V1, . . . VL ⊂ Vh

in parallel (additive) or successively (multiplicative)

Two-level overlapping Schwarz Ω2 Ω Ω3

1

χ1

2

χ3 χ

Vℓ = {vh ∈ Vh : supp(vh) ⊂ Ωℓ} with

  • verlapping partitioning {Ωℓ}L

ℓ=1 of Ω

and V0 = span{Φj ∈ Vh : j = 1, . . . , N} (abstract) M−1

add A = L

  • ℓ=0

RT

ℓ A−1 ℓ Rℓ A

  • = Pℓ

Aℓ = restriction of A to subspace Ωℓ

(assume overlap δ H)

Geometric Multigrid & BPX

similar with Vℓ = p.w. lin. FE space on nested triangulations {Thℓ}L

ℓ=0

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 10 / 38

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SLIDE 27

Subspace Correction Methods (e.g. two-level Schwarz or multigrid )

Problem (in variational form): Find uh ∈ Vh s.t. a(uh, vh) ≡

  • Ω

α∇uh · ∇vh = (f , vh) for all vh ∈ Vh. Precondition by solving (exactly or approximately) in subspaces V0, V1, . . . VL ⊂ Vh

in parallel (additive) or successively (multiplicative)

Two-level overlapping Schwarz Ω2 Ω Ω3

1

χ1

2

χ3 χ

Vℓ = {vh ∈ Vh : supp(vh) ⊂ Ωℓ} with

  • verlapping partitioning {Ωℓ}L

ℓ=1 of Ω

and V0 = span{Φj ∈ Vh : j = 1, . . . , N} (abstract) M−1

add A = L

  • ℓ=0

RT

ℓ A−1 ℓ Rℓ A

  • = Pℓ

Aℓ = restriction of A to subspace Ωℓ

(assume overlap δ H)

Geometric Multigrid & BPX

similar with Vℓ = p.w. lin. FE space on nested triangulations {Thℓ}L

ℓ=0

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 10 / 38

slide-28
SLIDE 28

Subspace Correction Methods (e.g. two-level Schwarz or multigrid )

Problem (in variational form): Find uh ∈ Vh s.t. a(uh, vh) ≡

  • Ω

α∇uh · ∇vh = (f , vh) for all vh ∈ Vh. Precondition by solving (exactly or approximately) in subspaces V0, V1, . . . VL ⊂ Vh

in parallel (additive) or successively (multiplicative)

Two-level overlapping Schwarz Ω2 Ω Ω3

1

χ1

2

χ3 χ

Vℓ = {vh ∈ Vh : supp(vh) ⊂ Ωℓ} with

  • verlapping partitioning {Ωℓ}L

ℓ=1 of Ω

and V0 = span{Φj ∈ Vh : j = 1, . . . , N} (abstract) M−1

add A = L

  • ℓ=0

RT

ℓ A−1 ℓ Rℓ A

  • = Pℓ

Aℓ = restriction of A to subspace Ωℓ

(assume overlap δ H)

Geometric Multigrid & BPX

similar with Vℓ = p.w. lin. FE space on nested triangulations {Thℓ}L

ℓ=0

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 10 / 38

slide-29
SLIDE 29

Subspace Correction Methods (e.g. two-level Schwarz or multigrid )

Problem (in variational form): Find uh ∈ Vh s.t. a(uh, vh) ≡

  • Ω

α∇uh · ∇vh = (f , vh) for all vh ∈ Vh. Precondition by solving (exactly or approximately) in subspaces V0, V1, . . . VL ⊂ Vh

in parallel (additive) or successively (multiplicative)

Two-level overlapping Schwarz Ω2 Ω Ω3

1

χ1

2

χ3 χ

Vℓ = {vh ∈ Vh : supp(vh) ⊂ Ωℓ} with

  • verlapping partitioning {Ωℓ}L

ℓ=1 of Ω

and V0 = span{Φj ∈ Vh : j = 1, . . . , N} (abstract) M−1

add A = L

  • ℓ=0

RT

ℓ A−1 ℓ Rℓ A

  • = Pℓ

Aℓ = restriction of A to subspace Ωℓ

(assume overlap δ H)

Geometric Multigrid & BPX

similar with Vℓ = p.w. lin. FE space on nested triangulations {Thℓ}L

ℓ=0

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 10 / 38

slide-30
SLIDE 30

Two-level Overlapping Schwarz – Abstract Theory

Let v2

0,α =

  • Ω

αv 2 dx

(weighted L2-norm)

Analogously to classical theory (in H1-seminorm and L2-norm) we have:

Theorem (Two-level Schwarz)

[RS, Vassilevski, Zikatanov, MMS 2011]

If there exists an operator Π : Vh → V0 such that, for all v ∈ Vh, Πv2

a ≤ C1 v2 a

and v − Πv2

0,α ≤ C2 v2 a

(1)

(stability) (weak approximation)

then κ(M−1

addA) C1 + C2. The hidden constant is independent of α, L, h.

Similar result for geometric multigrid (different norm · ∗ induced by smoother)

Main question: How to choose Π and how to prove (1)?

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 11 / 38

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SLIDE 31

Two-level Overlapping Schwarz – Abstract Theory

Let v2

0,α =

  • Ω

αv 2 dx

(weighted L2-norm)

Analogously to classical theory (in H1-seminorm and L2-norm) we have:

Theorem (Two-level Schwarz)

[RS, Vassilevski, Zikatanov, MMS 2011]

If there exists an operator Π : Vh → V0 such that, for all v ∈ Vh, Πv2

a ≤ C1 v2 a

and v − Πv2

0,α ≤ C2 v2 a

(1)

(stability) (weak approximation)

then κ(M−1

addA) C1 + C2. The hidden constant is independent of α, L, h.

Similar result for geometric multigrid (different norm · ∗ induced by smoother)

Main question: How to choose Π and how to prove (1)?

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 11 / 38

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SLIDE 32

“Nodal” Coarse Spaces, e.g. Piecewise Linears

V0 = VH cts. p.w. linears on a shape-regular grid TH

No assumption that coefficient is resolved on TH !

Theorem

[RS, Vassilevski, Zikatanov, SINUM 2012]

For all K ∈ TH, let C P

K > 0 be the best constant s.t. for all v ∈ Vh

infξ∈R v − ξ2

0,α,ωK ≤ C P K H2 v2 a

(WPI)

(with a slight variation near Dirichlet boundaries). Then

κ(M−1

addA) maxK∈TH C P K

with Πv =

j vα j Φj

and vα

j

=

  • supp(Φj) αv dx
  • supp(Φj) α dx .

(WPI) linked directly to local quasi-monotonicity [Pechstein, RS, 2012]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 12 / 38

slide-33
SLIDE 33

“Nodal” Coarse Spaces, e.g. Piecewise Linears

V0 = VH cts. p.w. linears on a shape-regular grid TH

No assumption that coefficient is resolved on TH !

Theorem

[RS, Vassilevski, Zikatanov, SINUM 2012]

For all K ∈ TH, let C P

K > 0 be the best constant s.t. for all v ∈ Vh

infξ∈R v − ξ2

0,α,ωK ≤ C P K H2 v2 a

(WPI)

(with a slight variation near Dirichlet boundaries). Then

κ(M−1

addA) maxK∈TH C P K

with Πv =

j vα j Φj

and vα

j

=

  • supp(Φj) αv dx
  • supp(Φj) α dx .

(WPI) linked directly to local quasi-monotonicity [Pechstein, RS, 2012]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 12 / 38

slide-34
SLIDE 34

“Nodal” Coarse Spaces, e.g. Piecewise Linears

V0 = VH cts. p.w. linears on a shape-regular grid TH

No assumption that coefficient is resolved on TH !

Theorem

[RS, Vassilevski, Zikatanov, SINUM 2012]

For all K ∈ TH, let C P

K > 0 be the best constant s.t. for all v ∈ Vh

infξ∈R v − ξ2

0,α,ωK ≤ C P K H2 v2 a

(WPI)

(with a slight variation near Dirichlet boundaries). Then

κ(M−1

addA) maxK∈TH C P K

with Πv =

j vα j Φj

and vα

j

=

  • supp(Φj) αv dx
  • supp(Φj) α dx .

(WPI) linked directly to local quasi-monotonicity [Pechstein, RS, 2012]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 12 / 38

slide-35
SLIDE 35

When is Poincar´ e constant independent of contrast in α?

Careful theory in [Pechstein, RS, 2012] linking robustness to quasi-monotonicity. Bounds for the effective Poincar´ e constant C P

T :

Darker colour means higher permeability.

* X

min

η

min

η X*

min

η

O(1) O(1 + log( H

η ))

O( H

η )

O( η

h ) * X 4 9 6 1 2 3 7 9 8 6 5 4 * X 1 8 5 7 3 2

(a) (b)

η η η H

C P

T α2 α1

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 13 / 38

slide-36
SLIDE 36

When is Poincar´ e constant independent of contrast in α?

Careful theory in [Pechstein, RS, 2012] linking robustness to quasi-monotonicity. Bounds for the effective Poincar´ e constant C P

T :

Darker colour means higher permeability.

* X

min

η

min

η X*

min

η

O(1) O(1 + log( H

η ))

O( H

η )

O( η

h ) * X 4 9 6 1 2 3 7 9 8 6 5 4 * X 1 8 5 7 3 2

(a) (b)

η η η H

C P

T α2 α1

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 13 / 38

slide-37
SLIDE 37

When is Poincar´ e constant independent of contrast in α?

Careful theory in [Pechstein, RS, 2012] linking robustness to quasi-monotonicity. Bounds for the effective Poincar´ e constant C P

T :

Darker colour means higher permeability.

* X

min

η

min

η X*

min

η

O(1) O(1 + log( H

η ))

O( H

η )

O( η

h ) * X 4 9 6 1 2 3 7 9 8 6 5 4 * X 1 8 5 7 3 2

(a) (b)

η η η H

C P

T α2 α1

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 13 / 38

slide-38
SLIDE 38

Numerical Example (Geometric Multigrid)

Ω = (0, 1)2, uniform grids {Tℓ}L

ℓ=0 with L = 4 and hL = 1/384.

Two “islands” not alligned with T0 and T1 where α(x) = α

(α(x) = 1 elsewhere) C P

K bounded for all ωK

C P

K not bdd. on some ωK

  • α

λ−1

1

#MG Its (tol = 10−8) λ−1

1

#MG Its (tol = 10−8) 101 1.69 10 1.72 10 102 2.75 14 3.87 19 103 3.32 12 14.5 23 104 3.42 10 115.5 70 105 3.42 10 1125 76 In right table islands closer to each other!

Guiding principle for choice of “nodal” coarse spaces TH sufficiently fine (locally) s.t. α(x) quasi-monotone on all ωK When it is difficult to ensure quasi-monotonicity on all ωK − → Coefficient-dependent Coarse Spaces !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 14 / 38

slide-39
SLIDE 39

Numerical Example (Geometric Multigrid)

Ω = (0, 1)2, uniform grids {Tℓ}L

ℓ=0 with L = 4 and hL = 1/384.

Two “islands” not alligned with T0 and T1 where α(x) = α

(α(x) = 1 elsewhere) C P

K bounded for all ωK

C P

K not bdd. on some ωK

  • α

λ−1

1

#MG Its (tol = 10−8) λ−1

1

#MG Its (tol = 10−8) 101 1.69 10 1.72 10 102 2.75 14 3.87 19 103 3.32 12 14.5 23 104 3.42 10 115.5 70 105 3.42 10 1125 76 In right table islands closer to each other!

Guiding principle for choice of “nodal” coarse spaces TH sufficiently fine (locally) s.t. α(x) quasi-monotone on all ωK When it is difficult to ensure quasi-monotonicity on all ωK − → Coefficient-dependent Coarse Spaces !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 14 / 38

slide-40
SLIDE 40

Numerical Example (Geometric Multigrid)

Ω = (0, 1)2, uniform grids {Tℓ}L

ℓ=0 with L = 4 and hL = 1/384.

Two “islands” not alligned with T0 and T1 where α(x) = α

(α(x) = 1 elsewhere) C P

K bounded for all ωK

C P

K not bdd. on some ωK

  • α

λ−1

1

#MG Its (tol = 10−8) λ−1

1

#MG Its (tol = 10−8) 101 1.69 10 1.72 10 102 2.75 14 3.87 19 103 3.32 12 14.5 23 104 3.42 10 115.5 70 105 3.42 10 1125 76 In right table islands closer to each other!

Guiding principle for choice of “nodal” coarse spaces TH sufficiently fine (locally) s.t. α(x) quasi-monotone on all ωK When it is difficult to ensure quasi-monotonicity on all ωK − → Coefficient-dependent Coarse Spaces !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 14 / 38

slide-41
SLIDE 41

Numerical Example (Geometric Multigrid)

Ω = (0, 1)2, uniform grids {Tℓ}L

ℓ=0 with L = 4 and hL = 1/384.

Two “islands” not alligned with T0 and T1 where α(x) = α

(α(x) = 1 elsewhere) C P

K bounded for all ωK

C P

K not bdd. on some ωK

  • α

λ−1

1

#MG Its (tol = 10−8) λ−1

1

#MG Its (tol = 10−8) 101 1.69 10 1.72 10 102 2.75 14 3.87 19 103 3.32 12 14.5 23 104 3.42 10 115.5 70 105 3.42 10 1125 76 In right table islands closer to each other!

Guiding principle for choice of “nodal” coarse spaces TH sufficiently fine (locally) s.t. α(x) quasi-monotone on all ωK When it is difficult to ensure quasi-monotonicity on all ωK − → Coefficient-dependent Coarse Spaces !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 14 / 38

slide-42
SLIDE 42

Energy minimising coarse spaces = Generalised FEs

Suppose {Ωℓ}L

ℓ=1 is an overlapping partition of Ω and {χℓ}L ℓ=1 an

associate partition of unity w. χℓ∞ 1 & ∇χℓ∞ δ−1

ℓ

H−1

ℓ

(This could be a set of FE basis functions and their supports.)

Local Energy Minimization subject to Functional Constraints For each subdomain Ωℓ, assume that we have a collection of linear functionals {fℓ,j}mℓ

j=1 ⊂ Vh(Ωℓ)′ and let

Ψℓ,j = arg min

v∈Vh(Ωℓ) v2 a,Ωℓ

subject to fℓ,k(Ψℓ,j) = δjk . (2) Now, with Ih the standard nodal interpolant onto Vh, let VH = span{Φℓ,j} with Φℓ,j = Ih (χℓΨℓ,j) , ℓ = 1, L, j = 1, mℓ .

(“glueing” together the locally energy minimising bases via a partition of unity)

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 15 / 38

slide-43
SLIDE 43

Energy minimising coarse spaces = Generalised FEs

Suppose {Ωℓ}L

ℓ=1 is an overlapping partition of Ω and {χℓ}L ℓ=1 an

associate partition of unity w. χℓ∞ 1 & ∇χℓ∞ δ−1

ℓ

H−1

ℓ

(This could be a set of FE basis functions and their supports.)

Local Energy Minimization subject to Functional Constraints For each subdomain Ωℓ, assume that we have a collection of linear functionals {fℓ,j}mℓ

j=1 ⊂ Vh(Ωℓ)′ and let

Ψℓ,j = arg min

v∈Vh(Ωℓ) v2 a,Ωℓ

subject to fℓ,k(Ψℓ,j) = δjk . (2) Now, with Ih the standard nodal interpolant onto Vh, let VH = span{Φℓ,j} with Φℓ,j = Ih (χℓΨℓ,j) , ℓ = 1, L, j = 1, mℓ .

(“glueing” together the locally energy minimising bases via a partition of unity)

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 15 / 38

slide-44
SLIDE 44

Energy minimising coarse spaces = Generalised FEs

Suppose {Ωℓ}L

ℓ=1 is an overlapping partition of Ω and {χℓ}L ℓ=1 an

associate partition of unity w. χℓ∞ 1 & ∇χℓ∞ δ−1

ℓ

H−1

ℓ

(This could be a set of FE basis functions and their supports.)

Local Energy Minimization subject to Functional Constraints For each subdomain Ωℓ, assume that we have a collection of linear functionals {fℓ,j}mℓ

j=1 ⊂ Vh(Ωℓ)′ and let

Ψℓ,j = arg min

v∈Vh(Ωℓ) v2 a,Ωℓ

subject to fℓ,k(Ψℓ,j) = δjk . (2) Now, with Ih the standard nodal interpolant onto Vh, let VH = span{Φℓ,j} with Φℓ,j = Ih (χℓΨℓ,j) , ℓ = 1, L, j = 1, mℓ .

(“glueing” together the locally energy minimising bases via a partition of unity)

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 15 / 38

slide-45
SLIDE 45

Energy minimising coarse spaces = Generalised FEs

Importance of energy minimization noted in AMG literature:

Explicitly: [Mandel, Brezina & Vanek, 99]; [Wan, Chan & Smith, 99]; [Xu & Zikatanov, 04]; [Brannick, Brezina et al, 05] (implicitly in all AMG methods)

Theorem [RS, Vassilevski, Zikatanov, MMS 2011] If ∀v ∈ Vh(Ωℓ) the local quasi-interpolant Πℓv =

j fℓ,j(v)Ψℓ,j

satisfies Πℓva,Ωℓ

  • va,Ωℓ
  • Ωℓ α|v − Πℓv|2 dx
  • H2

ℓ u2 a,Ωℓ

then κ(M−1

addA) 1 with Πv = L ℓ=1

mℓ

j=1 fℓ,j(v)Φℓ,j .

The assumptions of this theorem follow from a ’novel’ abstract approximation result related to the Bramble-Hilbert Lemma.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 16 / 38

slide-46
SLIDE 46

Energy minimising coarse spaces = Generalised FEs

Importance of energy minimization noted in AMG literature:

Explicitly: [Mandel, Brezina & Vanek, 99]; [Wan, Chan & Smith, 99]; [Xu & Zikatanov, 04]; [Brannick, Brezina et al, 05] (implicitly in all AMG methods)

Theorem [RS, Vassilevski, Zikatanov, MMS 2011] If ∀v ∈ Vh(Ωℓ) the local quasi-interpolant Πℓv =

j fℓ,j(v)Ψℓ,j

satisfies Πℓva,Ωℓ

  • va,Ωℓ
  • Ωℓ α|v − Πℓv|2 dx
  • H2

ℓ u2 a,Ωℓ

then κ(M−1

addA) 1 with Πv = L ℓ=1

mℓ

j=1 fℓ,j(v)Φℓ,j .

The assumptions of this theorem follow from a ’novel’ abstract approximation result related to the Bramble-Hilbert Lemma.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 16 / 38

slide-47
SLIDE 47

Energy minimising coarse spaces = Generalised FEs

Importance of energy minimization noted in AMG literature:

Explicitly: [Mandel, Brezina & Vanek, 99]; [Wan, Chan & Smith, 99]; [Xu & Zikatanov, 04]; [Brannick, Brezina et al, 05] (implicitly in all AMG methods)

Theorem [RS, Vassilevski, Zikatanov, MMS 2011] If ∀v ∈ Vh(Ωℓ) the local quasi-interpolant Πℓv =

j fℓ,j(v)Ψℓ,j

satisfies Πℓva,Ωℓ

  • va,Ωℓ
  • Ωℓ α|v − Πℓv|2 dx
  • H2

ℓ u2 a,Ωℓ

then κ(M−1

addA) 1 with Πv = L ℓ=1

mℓ

j=1 fℓ,j(v)Φℓ,j .

The assumptions of this theorem follow from a ’novel’ abstract approximation result related to the Bramble-Hilbert Lemma.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 16 / 38

slide-48
SLIDE 48

An abstract Bramble–Hilbert Lemma – Tool 1

Suppose V ⊂ H and H Hilbert with norm · , a is an abstract symmetric continuous bilinear form on V ×V and {fk}m

k=1 ⊂V ′

and define (as in the specific case above), for all v ∈ V , ψk = arg min

v∈V |v|2 a,

subject to fj(ψk) = δjk j, k = 1, . . . , m . Make the following assumptions:

  • A1. a is positive semi-definite and defines a semi-norm | · |a on V

and

  • v2 + |v|2

a defines a norm on V .

  • A2. For all q ∈ Rm there exists a vq ∈ V with

fk(vq) = qk, and vq cqql2(Rm).

  • A3. v2 ≤ ca|v|2

a + cf

m

k=1 |fk(v)|2 ,

for all v ∈ V .

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 17 / 38

slide-49
SLIDE 49

An abstract Bramble–Hilbert Lemma – Tool 1

Suppose V ⊂ H and H Hilbert with norm · , a is an abstract symmetric continuous bilinear form on V ×V and {fk}m

k=1 ⊂V ′

and define (as in the specific case above), for all v ∈ V , ψk = arg min

v∈V |v|2 a,

subject to fj(ψk) = δjk j, k = 1, . . . , m . Make the following assumptions:

  • A1. a is positive semi-definite and defines a semi-norm | · |a on V

and

  • v2 + |v|2

a defines a norm on V .

  • A2. For all q ∈ Rm there exists a vq ∈ V with

fk(vq) = qk, and vq cqql2(Rm).

  • A3. v2 ≤ ca|v|2

a + cf

m

k=1 |fk(v)|2 ,

for all v ∈ V .

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 17 / 38

slide-50
SLIDE 50

An abstract Bramble–Hilbert Lemma – Tool 1

Theorem (RS, Vassilevski, Zikatanov, MMS 2011) Let Assumptions A1-3 hold. Then πu =

k fk(u)ψk satisfies

|πu|a ≤ |u|a and u − πu ≤ √ca|u|a for all u ∈ V .

(Note that this is independent of the constants cq and cf in A2 and A3.)

Proof. First one notes that given u ∈ V , πu minimizes energy subject to fk(v) = fk(u). Thus |πu|a ≤ |u|a by construction. Secondly, from A3, the fact that fk(v − Πv) = 0 ∀k and the stability estimate, we get v − Πv2 ≤ ca|v − Πv|2

a + cf m

  • l=1

|f (v − Πv)|2 = ca|v − Πv|2

a ≤ 2ca(|v|2 a + |Πv|2 a) ≤ 4ca|v|2 a .

(can lose factor 2 by more careful bound)

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 18 / 38

slide-51
SLIDE 51

An abstract Bramble–Hilbert Lemma – Tool 1

Theorem (RS, Vassilevski, Zikatanov, MMS 2011) Let Assumptions A1-3 hold. Then πu =

k fk(u)ψk satisfies

|πu|a ≤ |u|a and u − πu ≤ √ca|u|a for all u ∈ V .

(Note that this is independent of the constants cq and cf in A2 and A3.)

Proof. First one notes that given u ∈ V , πu minimizes energy subject to fk(v) = fk(u). Thus |πu|a ≤ |u|a by construction. Secondly, from A3, the fact that fk(v − Πv) = 0 ∀k and the stability estimate, we get v − Πv2 ≤ ca|v − Πv|2

a + cf m

  • l=1

|f (v − Πv)|2 = ca|v − Πv|2

a ≤ 2ca(|v|2 a + |Πv|2 a) ≤ 4ca|v|2 a .

(can lose factor 2 by more careful bound)

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 18 / 38

slide-52
SLIDE 52

An abstract Bramble–Hilbert Lemma – Tool 1

Theorem (RS, Vassilevski, Zikatanov, MMS 2011) Let Assumptions A1-3 hold. Then πu =

k fk(u)ψk satisfies

|πu|a ≤ |u|a and u − πu ≤ √ca|u|a for all u ∈ V .

(Note that this is independent of the constants cq and cf in A2 and A3.)

Proof. First one notes that given u ∈ V , πu minimizes energy subject to fk(v) = fk(u). Thus |πu|a ≤ |u|a by construction. Secondly, from A3, the fact that fk(v − Πv) = 0 ∀k and the stability estimate, we get v − Πv2 ≤ ca|v − Πv|2

a + cf m

  • l=1

|f (v − Πv)|2 = ca|v − Πv|2

a ≤ 2ca(|v|2 a + |Πv|2 a) ≤ 4ca|v|2 a .

(can lose factor 2 by more careful bound)

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 18 / 38

slide-53
SLIDE 53

In our specific model problem considered above Assumption A1 is naturally satisfied on any subdomain Ωℓ with H = L2(Ωℓ) and v =

  • Ωℓ αv2 dx (weighted L2-norm !)

Assumption A2 simply means the functionals {fk} should be linearly independent. Coarse space robustness reduced to verifying Assumption A3

For one functional reduces to (WPI) and quasi-monotonicity. For more then one functional opens possibility of coefficient robustness even for non-quasi-monotone coefficients.

More importantly: can be applied also to other problems, e.g. elasticity, Stokes, ...

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 19 / 38

slide-54
SLIDE 54

In our specific model problem considered above Assumption A1 is naturally satisfied on any subdomain Ωℓ with H = L2(Ωℓ) and v =

  • Ωℓ αv2 dx (weighted L2-norm !)

Assumption A2 simply means the functionals {fk} should be linearly independent. Coarse space robustness reduced to verifying Assumption A3

For one functional reduces to (WPI) and quasi-monotonicity. For more then one functional opens possibility of coefficient robustness even for non-quasi-monotone coefficients.

More importantly: can be applied also to other problems, e.g. elasticity, Stokes, ...

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 19 / 38

slide-55
SLIDE 55

Choice of functionals – ‘Knob’ 1

1

[Galvis, Efendiev ’10]: fℓ,j(v) =

  • Ωℓ αΨℓ,jv dx where Ψℓ,j is the

jth eigenfunction of matrix pencil of local stiffness & mass matrix

2

[RS, Vassilevski, Zikatanov ’11]: fℓ,j(v) =

  • Ωℓ,j αv dx/
  • Ωℓ,j α dx

where {Ωℓ,j}mℓ

j=1 is partitioning of Ωℓ s.t. (WPI) holds on each Ωℓ,j

(Construction of Ψℓ,j requires solution of mℓ local saddle point systems.)

3

[Dolean, Nataf, RS, Spillane ’12]: fℓ,j(v) =

  • ∂Ωℓ αΨℓ,jv ds where

Ψℓ,j is jth eigenfunction of Dirichlet-to-Neumann operator on ∂Ωℓ

4

[Spillane et al ’14]: fℓ,j(v) =

  • Ωℓ α∇
  • χℓΨℓ,j
  • · ∇
  • χℓv
  • dx where

Ψℓ,j is jth eigenfct. of matrix pencil stiffness matrix & a(χℓ · , χℓ ·) ← − GenEO – see below! But also in multiscale literature:

1

[Babuska, Lipton ’11]: fℓ,j(v) =

  • ωℓ α∇Ψℓ,j · ∇v dx with ωℓ ⊂ Ωℓ

2

[Peterseim, RS ’16] (LOD): fℓ,j(v) =

  • Ωℓ αχjv dx/
  • Ωℓ αχj dx

3

[Owhadi ’17] (Gamblets): Hierarchy of functionals similar to Case 2

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 20 / 38

slide-56
SLIDE 56

Choice of functionals – ‘Knob’ 1

1

[Galvis, Efendiev ’10]: fℓ,j(v) =

  • Ωℓ αΨℓ,jv dx where Ψℓ,j is the

jth eigenfunction of matrix pencil of local stiffness & mass matrix

2

[RS, Vassilevski, Zikatanov ’11]: fℓ,j(v) =

  • Ωℓ,j αv dx/
  • Ωℓ,j α dx

where {Ωℓ,j}mℓ

j=1 is partitioning of Ωℓ s.t. (WPI) holds on each Ωℓ,j

(Construction of Ψℓ,j requires solution of mℓ local saddle point systems.)

3

[Dolean, Nataf, RS, Spillane ’12]: fℓ,j(v) =

  • ∂Ωℓ αΨℓ,jv ds where

Ψℓ,j is jth eigenfunction of Dirichlet-to-Neumann operator on ∂Ωℓ

4

[Spillane et al ’14]: fℓ,j(v) =

  • Ωℓ α∇
  • χℓΨℓ,j
  • · ∇
  • χℓv
  • dx where

Ψℓ,j is jth eigenfct. of matrix pencil stiffness matrix & a(χℓ · , χℓ ·) ← − GenEO – see below! But also in multiscale literature:

1

[Babuska, Lipton ’11]: fℓ,j(v) =

  • ωℓ α∇Ψℓ,j · ∇v dx with ωℓ ⊂ Ωℓ

2

[Peterseim, RS ’16] (LOD): fℓ,j(v) =

  • Ωℓ αχjv dx/
  • Ωℓ αχj dx

3

[Owhadi ’17] (Gamblets): Hierarchy of functionals similar to Case 2

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 20 / 38

slide-57
SLIDE 57

Choice of functionals – ‘Knob’ 1

1

[Galvis, Efendiev ’10]: fℓ,j(v) =

  • Ωℓ αΨℓ,jv dx where Ψℓ,j is the

jth eigenfunction of matrix pencil of local stiffness & mass matrix

2

[RS, Vassilevski, Zikatanov ’11]: fℓ,j(v) =

  • Ωℓ,j αv dx/
  • Ωℓ,j α dx

where {Ωℓ,j}mℓ

j=1 is partitioning of Ωℓ s.t. (WPI) holds on each Ωℓ,j

(Construction of Ψℓ,j requires solution of mℓ local saddle point systems.)

3

[Dolean, Nataf, RS, Spillane ’12]: fℓ,j(v) =

  • ∂Ωℓ αΨℓ,jv ds where

Ψℓ,j is jth eigenfunction of Dirichlet-to-Neumann operator on ∂Ωℓ

4

[Spillane et al ’14]: fℓ,j(v) =

  • Ωℓ α∇
  • χℓΨℓ,j
  • · ∇
  • χℓv
  • dx where

Ψℓ,j is jth eigenfct. of matrix pencil stiffness matrix & a(χℓ · , χℓ ·) ← − GenEO – see below! But also in multiscale literature:

1

[Babuska, Lipton ’11]: fℓ,j(v) =

  • ωℓ α∇Ψℓ,j · ∇v dx with ωℓ ⊂ Ωℓ

2

[Peterseim, RS ’16] (LOD): fℓ,j(v) =

  • Ωℓ αχjv dx/
  • Ωℓ αχj dx

3

[Owhadi ’17] (Gamblets): Hierarchy of functionals similar to Case 2

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 20 / 38

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SLIDE 58

Choice of functionals – ‘Knob’ 1

1

[Galvis, Efendiev ’10]: fℓ,j(v) =

  • Ωℓ αΨℓ,jv dx where Ψℓ,j is the

jth eigenfunction of matrix pencil of local stiffness & mass matrix

2

[RS, Vassilevski, Zikatanov ’11]: fℓ,j(v) =

  • Ωℓ,j αv dx/
  • Ωℓ,j α dx

where {Ωℓ,j}mℓ

j=1 is partitioning of Ωℓ s.t. (WPI) holds on each Ωℓ,j

(Construction of Ψℓ,j requires solution of mℓ local saddle point systems.)

3

[Dolean, Nataf, RS, Spillane ’12]: fℓ,j(v) =

  • ∂Ωℓ αΨℓ,jv ds where

Ψℓ,j is jth eigenfunction of Dirichlet-to-Neumann operator on ∂Ωℓ

4

[Spillane et al ’14]: fℓ,j(v) =

  • Ωℓ α∇
  • χℓΨℓ,j
  • · ∇
  • χℓv
  • dx where

Ψℓ,j is jth eigenfct. of matrix pencil stiffness matrix & a(χℓ · , χℓ ·) ← − GenEO – see below! But also in multiscale literature:

1

[Babuska, Lipton ’11]: fℓ,j(v) =

  • ωℓ α∇Ψℓ,j · ∇v dx with ωℓ ⊂ Ωℓ

2

[Peterseim, RS ’16] (LOD): fℓ,j(v) =

  • Ωℓ αχjv dx/
  • Ωℓ αχj dx

3

[Owhadi ’17] (Gamblets): Hierarchy of functionals similar to Case 2

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 20 / 38

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SLIDE 59

Choice of functionals – ‘Knob’ 1

1

[Galvis, Efendiev ’10]: fℓ,j(v) =

  • Ωℓ αΨℓ,jv dx where Ψℓ,j is the

jth eigenfunction of matrix pencil of local stiffness & mass matrix

2

[RS, Vassilevski, Zikatanov ’11]: fℓ,j(v) =

  • Ωℓ,j αv dx/
  • Ωℓ,j α dx

where {Ωℓ,j}mℓ

j=1 is partitioning of Ωℓ s.t. (WPI) holds on each Ωℓ,j

(Construction of Ψℓ,j requires solution of mℓ local saddle point systems.)

3

[Dolean, Nataf, RS, Spillane ’12]: fℓ,j(v) =

  • ∂Ωℓ αΨℓ,jv ds where

Ψℓ,j is jth eigenfunction of Dirichlet-to-Neumann operator on ∂Ωℓ

4

[Spillane et al ’14]: fℓ,j(v) =

  • Ωℓ α∇
  • χℓΨℓ,j
  • · ∇
  • χℓv
  • dx where

Ψℓ,j is jth eigenfct. of matrix pencil stiffness matrix & a(χℓ · , χℓ ·) ← − GenEO – see below! But also in multiscale literature:

1

[Babuska, Lipton ’11]: fℓ,j(v) =

  • ωℓ α∇Ψℓ,j · ∇v dx with ωℓ ⊂ Ωℓ

2

[Peterseim, RS ’16] (LOD): fℓ,j(v) =

  • Ωℓ αχjv dx/
  • Ωℓ αχj dx

3

[Owhadi ’17] (Gamblets): Hierarchy of functionals similar to Case 2

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 20 / 38

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SLIDE 60

Verification of Assumption A3 for Cases 1 and 2

Recall on subdomain Ωℓ chose H = L2(Ωℓ) and v =

  • Ωℓ αv 2 dx

Case 2: Applying (WPI) on each subsubdomain Ωℓ,j: v2 ≤

mℓ

  • j=1

C P

ℓ,jH2 ℓ

  • Ωℓ,j

α|∇v|2 dx + αmaxHd

ℓ

=: cf |fℓ,j(v)|2 ≤ cav2

a + cf mℓ

  • j=1

|fℓ,j(v)|2 with ca := (max

j

C P

ℓ,j)H2 ℓ

Case 1: Recall fℓ,j(v) =

  • Ωℓ αΨℓ,jv dx with Ψℓ,j the eigenfunction

related to pair of energy and weighted L2-inner product. Thus: v2 ≤ cav2

a + cf mℓ

  • j=1

|fℓ,j(v)|2 with ca := λ−1

ℓ,mℓ+1

Also random energy minimisation methods possible [Buhr, Smetana ’18]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 21 / 38

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SLIDE 61

Verification of Assumption A3 for Cases 1 and 2

Recall on subdomain Ωℓ chose H = L2(Ωℓ) and v =

  • Ωℓ αv 2 dx

Case 2: Applying (WPI) on each subsubdomain Ωℓ,j: v2 ≤

mℓ

  • j=1

C P

ℓ,jH2 ℓ

  • Ωℓ,j

α|∇v|2 dx + αmaxHd

ℓ

=: cf |fℓ,j(v)|2 ≤ cav2

a + cf mℓ

  • j=1

|fℓ,j(v)|2 with ca := (max

j

C P

ℓ,j)H2 ℓ

Case 1: Recall fℓ,j(v) =

  • Ωℓ αΨℓ,jv dx with Ψℓ,j the eigenfunction

related to pair of energy and weighted L2-inner product. Thus: v2 ≤ cav2

a + cf mℓ

  • j=1

|fℓ,j(v)|2 with ca := λ−1

ℓ,mℓ+1

Also random energy minimisation methods possible [Buhr, Smetana ’18]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 21 / 38

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SLIDE 62

Verification of Assumption A3 for Cases 1 and 2

Recall on subdomain Ωℓ chose H = L2(Ωℓ) and v =

  • Ωℓ αv 2 dx

Case 2: Applying (WPI) on each subsubdomain Ωℓ,j: v2 ≤

mℓ

  • j=1

C P

ℓ,jH2 ℓ

  • Ωℓ,j

α|∇v|2 dx + αmaxHd

ℓ

=: cf |fℓ,j(v)|2 ≤ cav2

a + cf mℓ

  • j=1

|fℓ,j(v)|2 with ca := (max

j

C P

ℓ,j)H2 ℓ

Case 1: Recall fℓ,j(v) =

  • Ωℓ αΨℓ,jv dx with Ψℓ,j the eigenfunction

related to pair of energy and weighted L2-inner product. Thus: v2 ≤ cav2

a + cf mℓ

  • j=1

|fℓ,j(v)|2 with ca := λ−1

ℓ,mℓ+1

Also random energy minimisation methods possible [Buhr, Smetana ’18]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 21 / 38

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SLIDE 63

Verification of Assumption A3 for Cases 1 and 2

Recall on subdomain Ωℓ chose H = L2(Ωℓ) and v =

  • Ωℓ αv 2 dx

Case 2: Applying (WPI) on each subsubdomain Ωℓ,j: v2 ≤

mℓ

  • j=1

C P

ℓ,jH2 ℓ

  • Ωℓ,j

α|∇v|2 dx + αmaxHd

ℓ

=: cf |fℓ,j(v)|2 ≤ cav2

a + cf mℓ

  • j=1

|fℓ,j(v)|2 with ca := (max

j

C P

ℓ,j)H2 ℓ

Case 1: Recall fℓ,j(v) =

  • Ωℓ αΨℓ,jv dx with Ψℓ,j the eigenfunction

related to pair of energy and weighted L2-inner product. Thus: v2 ≤ cav2

a + cf mℓ

  • j=1

|fℓ,j(v)|2 with ca := λ−1

ℓ,mℓ+1

Also random energy minimisation methods possible [Buhr, Smetana ’18]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 21 / 38

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SLIDE 64

Contrast-robust approximation theory for LOD & GFEM

How can we use this abstract Bramble-Hilbert Lemma to obtain a contrast-robust approximation theory for LOD or GFEM?

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 22 / 38

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SLIDE 65

Localizable Orthogonal Decomposition (LOD)

FE space VH := span{Φj} associated with (coarse) FE mesh TH Quasi-interpolation operator Π : Vh → VH with Πv :=

  • j

(v, Φj)L2(Ω) (1, Φj)L2(Ω) Φj

(Π invertible on VH!)

Decomposition Vh = VH ⊕ V f

h

with V f

h := kernel Π = {v ∈ Vh | Πv = 0}

For each v ∈ Vh define the fine scale projection Pfv ∈ V f

h by

a(Pfv, w) = a(v, w) for all w ∈ V f

h

(global!)

a–Orthogonal Decomposition [Malqvist, Peterseim, ’11] Vh = V ms

H ⊕ V f h

and a(V ms

H , V f h) = 0

with V ms

H

:= (1 − Pf)VH

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 23 / 38

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SLIDE 66

Localizable Orthogonal Decomposition (LOD)

FE space VH := span{Φj} associated with (coarse) FE mesh TH Quasi-interpolation operator Π : Vh → VH with Πv :=

  • j

(v, Φj)L2(Ω) (1, Φj)L2(Ω) Φj

(Π invertible on VH!)

Decomposition Vh = VH ⊕ V f

h

with V f

h := kernel Π = {v ∈ Vh | Πv = 0}

For each v ∈ Vh define the fine scale projection Pfv ∈ V f

h by

a(Pfv, w) = a(v, w) for all w ∈ V f

h

(global!)

a–Orthogonal Decomposition [Malqvist, Peterseim, ’11] Vh = V ms

H ⊕ V f h

and a(V ms

H , V f h) = 0

with V ms

H

:= (1 − Pf)VH

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 23 / 38

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SLIDE 67

Modified (multiscale) nodal basis

{Φj | j = 1, . . . , N} ⊂ VH denotes classical nodal basis ϕf

j := PfΦj ∈ V f h denotes the fine scale correction of Φj

Ideal multiscale FE space V ms

H

= span

  • Φj − ϕf

j | j = 1, . . . , N

  • Example

Φj − ϕf

j ∈V ms

H

= Φj

  • ∈VH
  • ϕf

j

  • ∈V f

h

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 24 / 38

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SLIDE 68

Exponential decay and localisation

Define nodal patches ωj,k of k-th order around vertex xj of TH

x

ωj,1

x

ωj,2

x

ωj,3

x

ωj,4 Can show that |ϕf

j |H1(Ω\ωj,k) γk|ϕf j |H1(Ω)

(with γ < 1). Define ϕf

j,k ∈ V f h(ωj,k) := {v ∈ V f h | supp v ⊂ ωj,k} (the

localised correction) s.t. a(ϕf

j,k, w) = a(Φj, w)

for all w ∈ V f

h(ωj,k)

Localized multiscale FE spaces V ms

H,k := span{ΦH j − ϕf j,k | j = 1, . . . , N}

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 25 / 38

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SLIDE 69

Multiscale Coarse Problem & Approximation Result

Multiscale approximation Seek ums

H,k ∈ V ms H,k such that

a(ums

H,k, v) = (f , v)

for all v ∈ V ms

H,k

dim V ms

H,k = dim VH = N & basis functions have local support

Overlap of the supports is proportional to the parameter k Theorem (Malqvist & Peterseim, 2011) |u − ums

H,k|H1(Ω) kdγkf H−1(Ω) + Hf L2(Ω) + |u − uh|H1(Ω)

Thus, provided k logγ( 1

H ) and h is suff’ly small we have optimal

O(H) convergence without any assumptions on scales or regularity.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 26 / 38

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SLIDE 70

Multiscale Coarse Problem & Approximation Result

Multiscale approximation Seek ums

H,k ∈ V ms H,k such that

a(ums

H,k, v) = (f , v)

for all v ∈ V ms

H,k

dim V ms

H,k = dim VH = N & basis functions have local support

Overlap of the supports is proportional to the parameter k Theorem (Malqvist & Peterseim, 2011) |u − ums

H,k|H1(Ω) kdγkf H−1(Ω) + Hf L2(Ω) + |u − uh|H1(Ω)

Thus, provided k logγ( 1

H ) and h is suff’ly small we have optimal

O(H) convergence without any assumptions on scales or regularity.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 26 / 38

slide-71
SLIDE 71

But unfortunately γ → 1 as the contrast αmax αmin → ∞ and the hidden constant depends also on αmax αmin ⇓ Thus for high contrast (in theory) no localization !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 27 / 38

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SLIDE 72

But unfortunately γ → 1 as the contrast αmax αmin → ∞ and the hidden constant depends also on αmax αmin ⇓ Thus for high contrast (in theory) no localization !

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 27 / 38

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SLIDE 73

A contrast-robust theory (work again in weighted norms)

Theorem (Peterseim & RS, 2016) If ∃ linear, cont. quasi-interpolation operator Π : Vh → VH s.t. (QI1) (Π|VH)−1ΠvH = vH, for all vH ∈ VH (QI2) H−2

T v − Πv2 0,α,T + v − Πv2 a,T ≤ C2v2 a,ωT ,

for all v ∈ Vh and T ∈ TH (QI3) for all vH ∈ VH there exists a v ∈ Vh, s.t. Πv = vH, supp v ⊂ supp vH and va ≤ C3vHa. then (with some universal constant m 1) u −ums

H,ka

  • αmax

αmin

m e−k H f H−1(Ω) + H α−1/2

min

f L2(Ω) +u −uha Thus, provided k ln( αmax

αmin 1 H ) and h suff’ly small we have optimal

O(H) convergence without assumptions on regularity or contrast.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 28 / 38

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SLIDE 74

A suitable quasi-interpolation operator

For simplicity assume α p.w. constant w.r.t. some grid Tη, with h < η < H, but not by TH

(TH ⊂ Tη ⊂ TH nested)

Choose Πv :=

N

  • j=1

(αv, Φj)L2(Ω) (α, Φj)L2(Ω) Φj

(again weighted!)

Theorem [Peterseim, RS ’16] For all T ∈ TH, let C P

T > 0 be the best constant s.t.

infξ∈R v − ξ2

0,α,ωT ≤ C P T H2 Tv2 a,ωT

∀v ∈ Vh . (WPI) Then H−2

T v − Πv2 0,α,T + v − Πv2 a,T C2 v2 a

where C2 H

η maxT∈TH C P T , i.e. Assumption (QI2).

Moreover, (QI1) and (QI3) hold with C3

  • H

η

2 .

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 29 / 38

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SLIDE 75

A suitable quasi-interpolation operator

For simplicity assume α p.w. constant w.r.t. some grid Tη, with h < η < H, but not by TH

(TH ⊂ Tη ⊂ TH nested)

Choose Πv :=

N

  • j=1

(αv, Φj)L2(Ω) (α, Φj)L2(Ω) Φj

(again weighted!)

Theorem [Peterseim, RS ’16] For all T ∈ TH, let C P

T > 0 be the best constant s.t.

infξ∈R v − ξ2

0,α,ωT ≤ C P T H2 Tv2 a,ωT

∀v ∈ Vh . (WPI) Then H−2

T v − Πv2 0,α,T + v − Πv2 a,T C2 v2 a

where C2 H

η maxT∈TH C P T , i.e. Assumption (QI2).

Moreover, (QI1) and (QI3) hold with C3

  • H

η

2 .

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 29 / 38

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SLIDE 76

A suitable quasi-interpolation operator

For simplicity assume α p.w. constant w.r.t. some grid Tη, with h < η < H, but not by TH

(TH ⊂ Tη ⊂ TH nested)

Choose Πv :=

N

  • j=1

(αv, Φj)L2(Ω) (α, Φj)L2(Ω) Φj

(again weighted!)

Theorem [Peterseim, RS ’16] For all T ∈ TH, let C P

T > 0 be the best constant s.t.

infξ∈R v − ξ2

0,α,ωT ≤ C P T H2 Tv2 a,ωT

∀v ∈ Vh . (WPI) Then H−2

T v − Πv2 0,α,T + v − Πv2 a,T C2 v2 a

where C2 H

η maxT∈TH C P T , i.e. Assumption (QI2).

Moreover, (QI1) and (QI3) hold with C3

  • H

η

2 .

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 29 / 38

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SLIDE 77

Discussion & Tool 2

In summary, we do get contrast independent convergence rates, but so far only under fairly stringent assumptions on the type of coefficient variation

(i.e. locally quasi-monotone & p.w. constant w.r.t. Tη for moderate H/η)

Key tool: Weighted Caccioppoli-type Inequality Let ω ⊂ Ω s.t. dist(∂ω, ∂Ω) > δ > 0. Then ua,ω ≤ 2δ−1 u0,α,Ω for all a-harmonic u on Ω.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 30 / 38

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SLIDE 78

Ideas for non-quasi-monotone coefficients – Work in Progress!

Adapt grid or enrich local space or change functionals ! LOD: Refine base grid TH locally where C P

T depends on contrast

(similar to multiresolution idea in gamblets)

GFEM: Use eigenproblem with Ωℓ ⊂ Ω∗

ℓ and combine (abstract)

Bramble-Hilbert (Tool 1) with (weighted) Caccioppoli (Tool 2) [Babuska, Lipton ’11], [Smetana, Patera ’16], [Buhr, Smetana ’18]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 31 / 38

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SLIDE 79

Ideas for non-quasi-monotone coefficients – Work in Progress!

Adapt grid or enrich local space or change functionals ! LOD: Refine base grid TH locally where C P

T depends on contrast

(similar to multiresolution idea in gamblets)

GFEM: Use eigenproblem with Ωℓ ⊂ Ω∗

ℓ and combine (abstract)

Bramble-Hilbert (Tool 1) with (weighted) Caccioppoli (Tool 2) [Babuska, Lipton ’11], [Smetana, Patera ’16], [Buhr, Smetana ’18]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 31 / 38

slide-80
SLIDE 80

Ideas for non-quasi-monotone coefficients – Work in Progress!

Adapt grid or enrich local space or change functionals ! LOD: Refine base grid TH locally where C P

T depends on contrast

(similar to multiresolution idea in gamblets)

GFEM: Use eigenproblem with Ωℓ ⊂ Ω∗

ℓ and combine (abstract)

Bramble-Hilbert (Tool 1) with (weighted) Caccioppoli (Tool 2) [Babuska, Lipton ’11], [Smetana, Patera ’16], [Buhr, Smetana ’18]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 31 / 38

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SLIDE 81

Beyond scalar elliptic problems

Linear elasticity equations: a(u, v) :=

  • Ω

C(x)ε(u) : ε(v) dx =

  • Ω

f · v dx +

  • Γ

(σ · n) · v dx ∀v ∈ V small length scales (<mm), high contrast and strongly anisotropic CERTEST (EPSRC Project)

Bristol, Bath, Exeter, Heidelberg,...

STEAM (Turing/Royce Project)

Exeter, Heidelberg, Imperial,...

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 32 / 38

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SLIDE 82

Beyond scalar elliptic problems

Linear elasticity equations: a(u, v) :=

  • Ω

C(x)ε(u) : ε(v) dx =

  • Ω

f · v dx +

  • Γ

(σ · n) · v dx ∀v ∈ V small length scales (<mm), high contrast and strongly anisotropic CERTEST (EPSRC Project)

Bristol, Bath, Exeter, Heidelberg,...

STEAM (Turing/Royce Project)

Exeter, Heidelberg, Imperial,...

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 32 / 38

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SLIDE 83

Change Eigenproblem: GenEO

[Spillane et al ’14] – Tool 3

Key lemma in subspace correction theory to bound κ(M−1

addA):

Lions’ Lemma – Stable splitting ∃C0 > 0 : ∀v ∈ Vh : ∃vℓ ∈ Vℓ : v =

L

  • ℓ=0

vℓ and

L

  • ℓ=0

vℓ2

a ≤ C 2 0 v2 a

Key observation in [Spillane, Dolean, Hauret, Nataf, Pechstein, RS ’14]: Lemma (Local sufficient condition) – Tool 3 Suppose that ∃C1 > 0 : ∀ℓ = 1, . . . , L : vℓ2

a,Ωℓ ≤ C 2 1 v2 a,Ωℓ. Then

the splitting above is stable with C 2

0 = 2 + k0C 2 1 + 2k2 0C 2 1

(where k0 is the maximal #subdomains any degree of freedom belongs to)

Choose vℓ := χℓ(v − v0). Motivates following (variational) eigenproblem: aΩℓ

  • ψℓ,j, v
  • = λjaΩℓ
  • χℓψℓ,j, χℓv
  • ∀v ∈ Vh(Ωℓ)

(full overlap case)

and w. V0 := span

  • Ih(χℓψℓ,j) : ℓ ≤ L, j ≤ mℓ
  • get vℓ2

a,Ωℓ ≤ λ−1 ℓ,mℓ+1v2 a,Ωℓ

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 33 / 38

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SLIDE 84

Change Eigenproblem: GenEO

[Spillane et al ’14] – Tool 3

Key lemma in subspace correction theory to bound κ(M−1

addA):

Lions’ Lemma – Stable splitting ∃C0 > 0 : ∀v ∈ Vh : ∃vℓ ∈ Vℓ : v =

L

  • ℓ=0

vℓ and

L

  • ℓ=0

vℓ2

a ≤ C 2 0 v2 a

Key observation in [Spillane, Dolean, Hauret, Nataf, Pechstein, RS ’14]: Lemma (Local sufficient condition) – Tool 3 Suppose that ∃C1 > 0 : ∀ℓ = 1, . . . , L : vℓ2

a,Ωℓ ≤ C 2 1 v2 a,Ωℓ. Then

the splitting above is stable with C 2

0 = 2 + k0C 2 1 + 2k2 0C 2 1

(where k0 is the maximal #subdomains any degree of freedom belongs to)

Choose vℓ := χℓ(v − v0). Motivates following (variational) eigenproblem: aΩℓ

  • ψℓ,j, v
  • = λjaΩℓ
  • χℓψℓ,j, χℓv
  • ∀v ∈ Vh(Ωℓ)

(full overlap case)

and w. V0 := span

  • Ih(χℓψℓ,j) : ℓ ≤ L, j ≤ mℓ
  • get vℓ2

a,Ωℓ ≤ λ−1 ℓ,mℓ+1v2 a,Ωℓ

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 33 / 38

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SLIDE 85

Change Eigenproblem: GenEO

[Spillane et al ’14] – Tool 3

Key lemma in subspace correction theory to bound κ(M−1

addA):

Lions’ Lemma – Stable splitting ∃C0 > 0 : ∀v ∈ Vh : ∃vℓ ∈ Vℓ : v =

L

  • ℓ=0

vℓ and

L

  • ℓ=0

vℓ2

a ≤ C 2 0 v2 a

Key observation in [Spillane, Dolean, Hauret, Nataf, Pechstein, RS ’14]: Lemma (Local sufficient condition) – Tool 3 Suppose that ∃C1 > 0 : ∀ℓ = 1, . . . , L : vℓ2

a,Ωℓ ≤ C 2 1 v2 a,Ωℓ. Then

the splitting above is stable with C 2

0 = 2 + k0C 2 1 + 2k2 0C 2 1

(where k0 is the maximal #subdomains any degree of freedom belongs to)

Choose vℓ := χℓ(v − v0). Motivates following (variational) eigenproblem: aΩℓ

  • ψℓ,j, v
  • = λjaΩℓ
  • χℓψℓ,j, χℓv
  • ∀v ∈ Vh(Ωℓ)

(full overlap case)

and w. V0 := span

  • Ih(χℓψℓ,j) : ℓ ≤ L, j ≤ mℓ
  • get vℓ2

a,Ωℓ ≤ λ−1 ℓ,mℓ+1v2 a,Ωℓ

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 33 / 38

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SLIDE 86

Change Eigenproblem: GenEO

[Spillane et al ’14] – Tool 3

Key lemma in subspace correction theory to bound κ(M−1

addA):

Lions’ Lemma – Stable splitting ∃C0 > 0 : ∀v ∈ Vh : ∃vℓ ∈ Vℓ : v =

L

  • ℓ=0

vℓ and

L

  • ℓ=0

vℓ2

a ≤ C 2 0 v2 a

Key observation in [Spillane, Dolean, Hauret, Nataf, Pechstein, RS ’14]: Lemma (Local sufficient condition) – Tool 3 Suppose that ∃C1 > 0 : ∀ℓ = 1, . . . , L : vℓ2

a,Ωℓ ≤ C 2 1 v2 a,Ωℓ. Then

the splitting above is stable with C 2

0 = 2 + k0C 2 1 + 2k2 0C 2 1

(where k0 is the maximal #subdomains any degree of freedom belongs to)

Choose vℓ := χℓ(v − v0). Motivates following (variational) eigenproblem: aΩℓ

  • ψℓ,j, v
  • = λjaΩℓ
  • χℓψℓ,j, χℓv
  • ∀v ∈ Vh(Ωℓ)

(full overlap case)

and w. V0 := span

  • Ih(χℓψℓ,j) : ℓ ≤ L, j ≤ mℓ
  • get vℓ2

a,Ωℓ ≤ λ−1 ℓ,mℓ+1v2 a,Ωℓ

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 33 / 38

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SLIDE 87

Toy Composite Example for Demonstration - Cantilever

Flat composite plate [0, 100mm] × [0, 20mm] Cantilever under uniform pressure (top surface) 12 Layers - 11 weak interfaces [∓45◦/0◦/90◦/ ± 45◦/ ∓ 45◦/90◦/0◦/ ± 45◦] . 20-node serendipity elements

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 34 / 38

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SLIDE 88

GenEO Modes & Numerical Results (Benchmarking)

[Butler, Dodwell, Reinarz, Sandhu, RS, Seelinger ’19]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 35 / 38

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SLIDE 89

Industrially motivated problem

(with over 2 × 108 DOFs)

Wingbox section with defect under internal fuel pressure

(ply-scale stress resolution!!)

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 36 / 38

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SLIDE 90

Parallel Efficiency of HPC Implementation up to 15,360 cores

HPC implementation of GenEO within Dune Parallel performance on UK National HPC Cluster

[Butler, Dodwell, Reinarz, Sandhu, RS, Seelinger ’19]

This scale of computations brings composites problems that would otherwise be unthinkable into the feasible range.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 37 / 38

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SLIDE 91

Current Work & Final Remarks

Extend to nonlinear elasticity & composite failure More complicated geometries & bigger overlap GenEO as a GFEM: first results in [Dodwell, Sandhu, RS ’17]

(different functional as [Babuska, Lipton], [Buhr, Smetana]: ‘Knob’ 1)

Bayesian inference: surrogate in multilevel MCMC ‘Knob’ 2: Choice of partition of unity (seems to have big effect) ‘Knob’ 3: ARPACK eigensolver vs. randomised eigensolver

Some initial experiments below!

Theoretical Aim: Prove contrast-independent approximation results for (versions of) LOD and GFEM !

THANK YOU!

[If anybody is up for rock climbing on Wed afternoon, please let me know!]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

slide-92
SLIDE 92

Current Work & Final Remarks

Extend to nonlinear elasticity & composite failure More complicated geometries & bigger overlap GenEO as a GFEM: first results in [Dodwell, Sandhu, RS ’17]

(different functional as [Babuska, Lipton], [Buhr, Smetana]: ‘Knob’ 1)

Bayesian inference: surrogate in multilevel MCMC ‘Knob’ 2: Choice of partition of unity (seems to have big effect) ‘Knob’ 3: ARPACK eigensolver vs. randomised eigensolver

Some initial experiments below!

Theoretical Aim: Prove contrast-independent approximation results for (versions of) LOD and GFEM !

THANK YOU!

[If anybody is up for rock climbing on Wed afternoon, please let me know!]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

slide-93
SLIDE 93

Current Work & Final Remarks

Extend to nonlinear elasticity & composite failure More complicated geometries & bigger overlap GenEO as a GFEM: first results in [Dodwell, Sandhu, RS ’17]

(different functional as [Babuska, Lipton], [Buhr, Smetana]: ‘Knob’ 1)

Bayesian inference: surrogate in multilevel MCMC ‘Knob’ 2: Choice of partition of unity (seems to have big effect) ‘Knob’ 3: ARPACK eigensolver vs. randomised eigensolver

Some initial experiments below!

Theoretical Aim: Prove contrast-independent approximation results for (versions of) LOD and GFEM !

THANK YOU!

[If anybody is up for rock climbing on Wed afternoon, please let me know!]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

slide-94
SLIDE 94

Current Work & Final Remarks

Extend to nonlinear elasticity & composite failure More complicated geometries & bigger overlap GenEO as a GFEM: first results in [Dodwell, Sandhu, RS ’17]

(different functional as [Babuska, Lipton], [Buhr, Smetana]: ‘Knob’ 1)

Bayesian inference: surrogate in multilevel MCMC ‘Knob’ 2: Choice of partition of unity (seems to have big effect) ‘Knob’ 3: ARPACK eigensolver vs. randomised eigensolver

Some initial experiments below!

Theoretical Aim: Prove contrast-independent approximation results for (versions of) LOD and GFEM !

THANK YOU!

[If anybody is up for rock climbing on Wed afternoon, please let me know!]

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

slide-95
SLIDE 95

References

1

RS, PS Vassilevski & LT Zikatanov, Weak Approximation Properties of Elliptic Projections with Functional Constraints, Multiscale Model Sim (SIAM) 9, 2011.

2

N Spillane, V Dolean, P Hauret, F Nataf, C Pechstein & RS, Abstract Robust Coarse Spaces for Systems of PDEs via Generalized Eigenproblems in the Overlaps, Numer Math 126, 2014.

3

D Peterseim & RS, Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast, Comput Meth Appl Math 16, 2016.

4

TJ Dodwell, A Sandhu & RS, Customized Coarse Models for Highly Heterogeneous Materials, in ”Bifurcation and Degradation of Geomaterials with Engineering Applications” (Papamichos et al Eds.), Springer Series in Geomechanics and Geoengineering, 2017.

5

R Butler, TJ Dodwell, A Reinarz, A Sandhu, RS & L Seelinger, High- performance dune modules for solving large-scale, strongly anisotropic elliptic problems with applications to aerospace composites arXiv:1901.05188, 2019.

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

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SLIDE 96

Eigenfunctions for Different Partitions of Unity (scalar elliptic)

Coefficient function Harmonic POU - 1st Mode First 6 eigenmodes in each domain:

[Arne Strehlow] P.O.U. λ1 λ2 λ3 λ4 λ5 λ6 p.w. const. Ω1 0.00272 0.00536 0.19743 0.22077 0.28599 0.34094 Ω2 0.00315 0.00749 0.20680 0.22085 0.30189 0.34094 Sarkis Ω1 0.01963 0.05366 0.09788 1.05319 1.05517 1.05974 Ω2 0.01599 0.04153 0.09416 0.99473 1.05318 1.05327 Harmonic Ω1 0.03357 0.21091 0.78878 1.05086 1.05326 1.05974 Ω2 0.03444 0.28577 0.83536 1.00547 1.00852 1.00915

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

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SLIDE 97

Eigenfunctions for Different Partitions of Unity (scalar elliptic)

Coefficient function Harmonic POU - 1st Mode First 6 eigenmodes in each domain:

[Arne Strehlow] P.O.U. λ1 λ2 λ3 λ4 λ5 λ6 p.w. const. Ω1 0.00272 0.00536 0.19743 0.22077 0.28599 0.34094 Ω2 0.00315 0.00749 0.20680 0.22085 0.30189 0.34094 Sarkis Ω1 0.01963 0.05366 0.09788 1.05319 1.05517 1.05974 Ω2 0.01599 0.04153 0.09416 0.99473 1.05318 1.05327 Harmonic Ω1 0.03357 0.21091 0.78878 1.05086 1.05326 1.05974 Ω2 0.03444 0.28577 0.83536 1.00547 1.00852 1.00915

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

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SLIDE 98

Eigenfunctions for Different Partitions of Unity (scalar elliptic)

’Disconnected’ cross

[Arne Strehlow]

(4 subdomains) Harmonic POU Piecewise constant POU

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

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SLIDE 99

GenEO as GFEM for scalar elliptic problem

[Tim Dodwell]

First 5 eigenfunctions on Ω6 (16 subdomains; a(x) is log-normal sample): Fine model dim Vh = 4 × 104 Coarse Model dim VH = 320

(m = 20, O = 5)

err = uh − RT

HUH2

uh2

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

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SLIDE 100

Coarse Approximation Error (channels & islands)

Figure: Parameter Distribution

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

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SLIDE 101

Coarse Approximation Error (channels & islands)

[Linus Seelinger]

Figure: Coarse Error – 1 EV/subdomain

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

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SLIDE 102

Coarse Approximation Error (channels & islands)

[Linus Seelinger]

Figure: Coarse Error – 2 EV/subdomain

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

slide-103
SLIDE 103

Coarse Approximation Error (channels & islands)

[Linus Seelinger]

Figure: Coarse Error – 3 EV/subdomain

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

slide-104
SLIDE 104

Coarse Approximation Error (channels & islands)

[Linus Seelinger]

Figure: Coarse Error – 4 EV/subdomain

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

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SLIDE 105

Coarse Approximation Error (channels & islands)

[Linus Seelinger]

Figure: Coarse Error – 5 EV/subdomain

Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38