Multiscale methods for time-harmonic acoustic and elastic wave - - PowerPoint PPT Presentation

Multiscale methods for time-harmonic acoustic and elastic wave propagation

Dietmar Gallistl

(joint work with D. Brown and D. Peterseim)

Institut f¨ ur Angewandte und Numerische Mathematik Karlsruher Institut f¨ ur Technologie (KIT)

Workshop on Analysis and Numerics of Acoustic and Electromagnetic Problems RICAM, 17 –22. October 2016

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 1

Outline

Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 2

Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

Model problem: high-frequency acoustic scattering

◮ Ω ⊂ Rd bounded polytope,

diamΩ ≈ 1

◮ ∂Ω = ΓD ∪ΓR ◮ wave number κ > 0 real

parameter

◮ incident wave uin ◮ we seek u = uin +uscat ◮ positive and bounded material

parameters A(x), n(x), β(x)

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 3

Model problem: high-frequency acoustic scattering

◮ Ω ⊂ Rd bounded polytope,

diamΩ ≈ 1

◮ ∂Ω = ΓD ∪ΓR ◮ wave number κ > 0 real

parameter

◮ incident wave uin ◮ we seek u = uin +uscat ◮ positive and bounded material

parameters A(x), n(x), β(x)

−div(A∇u)−κ2 nu = f in Ω u =

• n ΓD ⊆ ∂Ω

(A∇u)·ν −iβκu = g

• n ΓR := ∂Ω\ΓD
• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Pollution effect

1d Helmholtz:

−uxx −κ2u = 0 in [−1,1], uin(x) = exp(−iκx)

Results for P1-FEM (fixed resolution H := const/κ)

κ

10 0 10 1 10 2 10 3

V-error

10 -1 10 0 10 1 10 2 P1FEM P1-best ratio

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 4

Pollution effect

1d Helmholtz:

−uxx −κ2u = 0 in [−1,1], uin(x) = exp(−iκx)

Results for P1-FEM (fixed resolution H := const/κ)

κ

10 0 10 1 10 2 10 3

V-error

10 -1 10 0 10 1 10 2 P1FEM P1-best ratio

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 4

Pollution effect

1d Helmholtz:

−uxx −κ2u = 0 in [−1,1], uin(x) = exp(−iκx)

Results for P1-FEM (fixed resolution H := const/κ)

κ

10 0 10 1 10 2 10 3

V-error

10 -1 10 0 10 1 10 2 P1FEM P1-best ratio

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 4

Pollution effect

1d Helmholtz:

−uxx −κ2u = 0 in [−1,1], uin(x) = exp(−iκx)

Results for P1-FEM (fixed resolution H := const/κ) error(FEM) error(bestapprox) κs [Babuˇ ska-Sauter 2000]

κ

10 0 10 1 10 2 10 3

V-error

10 -1 10 0 10 1 10 2 P1FEM P1-best ratio

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 4

Pollution effect

1d Helmholtz:

−uxx −κ2u = 0 in [−1,1], uin(x) = exp(−iκx)

Results for P1-FEM (fixed resolution H := const/κ) Goal: error(msPG) error(bestapprox) ≤ C

κ

10 0 10 1 10 2 10 3

V-error

10 -1 10 0 10 1 10 2 P1FEM P1-best ratio msPG

• D. Gallistl and D. Peterseim.

Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering.

• Comput. Methods Appl. Mech. Eng., 2015.
• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Other Approaches

◮ hp-FEM with p ≈ logκ

Melenk, Sauter

◮ Trefftz methods, Plane wave methods

Hiptmair, Moiola, Perugia

◮ DG FEM

Farhat, Tezaur; Feng, Wu

◮ Ultra-weak formulation

Cessenat, Despr´ es; Buffa, Monk

◮ Discontinuous Petrov-Galerkin

Demkowicz, Gopalakrishnan

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

Variational formulation

◮ Hilbert space V := H1 D(Ω;C) := {v ∈ H1(Ω;C) : v|ΓD = 0} ◮ Continuous sesquilinear form on V ×V

a(v,w) := (A∇v,∇w)L2(Ω) −κ2(nv,w)L2(Ω) −iκ(βv,w)L2(ΓR)

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 6

Variational formulation

◮ Hilbert space V := H1 D(Ω;C) := {v ∈ H1(Ω;C) : v|ΓD = 0} ◮ Continuous sesquilinear form on V ×V

a(v,w) := (A∇v,∇w)L2(Ω) −κ2(nv,w)L2(Ω) −iκ(βv,w)L2(ΓR) Weak formulation seeks u ∈ V such that a(u,v) = (f,v)L2(Ω) +(g,v)L2(ΓR) for all v ∈ V.

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 6

Variational formulation

◮ Hilbert space V := H1 D(Ω;C) := {v ∈ H1(Ω;C) : v|ΓD = 0} ◮ Continuous sesquilinear form on V ×V

a(v,w) := (A∇v,∇w)L2(Ω) −κ2(nv,w)L2(Ω) −iκ(βv,w)L2(ΓR) Weak formulation seeks u ∈ V such that a(u,v) = (f,v)L2(Ω) +(g,v)L2(ΓR) for all v ∈ V.

◮ Data f ∈ L2(Ω;C) and g ∈ L2(ΓR;C) ◮ Norm vV :=

• κ2v2

L2(Ω) +∇v2 L2(Ω)

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Well-posedness

Assumption (Polynomial well-posedness)

There exist a constant c(Ω) and a polynomial p such that c(Ω) p(κ) ≤ inf

v∈V\{0}

sup

w∈V\{0}

ℜa(v,w) vVwV (⋆)

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 7

Well-posedness

Assumption (Polynomial well-posedness)

There exist a constant c(Ω) and a polynomial p such that c(Ω) p(κ) ≤ inf

v∈V\{0}

sup

w∈V\{0}

ℜa(v,w) vVwV (⋆) Homogeneous material:

◮ assumption not satisfied in general [Betcke et al 2011] ◮ pure impedence problem + Ω convex ⇒ γ(κ,Ω) κ

[Melenk 1995]

◮ pure impedence problem + Ω Lipschitz ⇒ γ(κ,Ω) κ7/2

[Esterhazy-Melenk 2012]

◮ star-shaped scatterer ⇒ γ(κ,Ω) κ [Hetmaniuk 2007]

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 7

Well-posedness

Assumption (Polynomial well-posedness)

There exist a constant c(Ω) and a polynomial p such that c(Ω) p(κ) ≤ inf

v∈V\{0}

sup

w∈V\{0}

ℜa(v,w) vVwV (⋆) Heterogeneous material:

Theorem

There is a class of smooth coefficients A, n such that (⋆) holds.

• D. Brown, D. Gallistl, and D. Peterseim.

Multiscale Petrov-Galerkin method for high-frequency heterogeneous Helmholtz equations. Arxiv preprint, 2015.

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

Finite element spaces

Coarse scale H 1/κ

◮ GH coarse mesh ◮ VH := Q1(GH)∩V standard Q1 FE space

Fine scale h 1/κ2

◮ Gh refinement of GH ◮ Vh := Q1(Gh)∩V standard Q1 FE space

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Finite element spaces

Coarse scale H 1/κ

◮ GH coarse mesh ◮ VH := Q1(GH)∩V standard Q1 FE space

Fine scale h 1/κ2

◮ Gh refinement of GH ◮ Vh := Q1(Gh)∩V standard Q1 FE space

Quasi-interpolation

◮ IH : Vh → VH quasi-local projection ◮ Stability and L2-approximation

H−1v−IHvL2(T) +∇IHvL2(T) ≤ CIH∇vL2(N(T))

◮ Example: IH := EH ◦ΠH

ΠH ... piecewise L2 projection onto Q1(GH) EH ... nodal averaging operator

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Subscale correction

Fine scale remainder (null space of IH): Wh := {vh ∈ Vh : IH(vh) = 0} Subscale corrector C∞ : VH → Wh: Given vH ∈ VH , C∞vH solves a(w,C∞vH) = a(w,vH) for all w ∈ Wh

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 9

Subscale correction

Fine scale remainder (null space of IH): Wh := {vh ∈ Vh : IH(vh) = 0} Subscale corrector C∞ : VH → Wh: Given vH ∈ VH , C∞vH solves a(w,C∞vH) = a(w,vH) for all w ∈ Wh

0.5 1 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

→C∞

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Subscale correction

Fine scale remainder (null space of IH): Wh := {vh ∈ Vh : IH(vh) = 0} Subscale corrector C∞ : VH → Wh: Given vH ∈ VH , C∞vH solves a(w,C∞vH) = a(w,vH) for all w ∈ Wh

Lemma

Under the resolution condition Hκ 1, the corrector problems are well-posed (even coercive).

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Ideal Petrov-Galerkin method

◮ Standard trial space VH ◮ Corrected test space

VH := (1−C∞)VH

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Ideal Petrov-Galerkin method

◮ Standard trial space VH ◮ Corrected test space

VH := (1−C∞)VH

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im

Λz C∞Λz ˜ Λz = (1−C∞)Λz

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Ideal Petrov-Galerkin method

◮ Standard trial space VH ◮ Corrected test space

VH := (1−C∞)VH

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im

Λz C∞Λz ˜ Λz = (1−C∞)Λz

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 10

Ideal Petrov-Galerkin method

◮ Standard trial space VH ◮ Corrected test space

VH := (1−C∞)VH

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im

Λz C∞Λz ˜ Λz = (1−C∞)Λz

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 10

Ideal Petrov-Galerkin method

◮ Standard trial space VH ◮ Corrected test space

VH := (1−C∞)VH

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im

Λz C∞Λz ˜ Λz = (1−C∞)Λz

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 10

Ideal Petrov-Galerkin method

◮ Standard trial space VH ◮ Corrected test space

VH := (1−C∞)VH

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im

Λz C∞Λz ˜ Λz = (1−C∞)Λz

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 10

Ideal Petrov-Galerkin method

◮ Standard trial space VH ◮ Corrected test space

VH := (1−C∞)VH

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im

Λz C∞Λz ˜ Λz = (1−C∞)Λz

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 10

Ideal Petrov-Galerkin method

◮ Standard trial space VH ◮ Corrected test space

VH := (1−C∞)VH

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im

Λz C∞Λz ˜ Λz = (1−C∞)Λz Ideal multiscale Petrov-Galerkin FEM seeks ums

H ∈ VH such that

a(ums

H , ˜

vH) = (f, ˜ vH)L2(Ω) +(g, ˜ vH)L2(ΓR) for all ˜ vH ∈ VH.

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 10

Ideal Petrov-Galerkin method

◮ Standard trial space VH ◮ Corrected test space

VH := (1−C∞)VH

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im

Connections to existing methods

◮ Variational multiscale method [Hughes et al] ◮ Local orthogonal decomposition [Malqvist, Peterseim]

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Stability and quasi-optimality

Since uh −IHuh ∈ Wh, we have a(uh −IHuh

• ∈Wh

,(1−C∞)vH) = 0 for all vH ∈ VH.

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 11

Stability and quasi-optimality

Since uh −IHuh ∈ Wh, we have a(uh −IHuh, (1−C∞)vH

• =

vH∈ VH

) = 0 for all vH ∈ VH.

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 11

Stability and quasi-optimality

Since uh −IHuh ∈ Wh, we have a(uh −IHuh, (1−C∞)vH

• =

vH∈ VH

) = 0 for all vH ∈ VH. Hence: a(IHuh, vH) = a(uh, vH) = (f, vH)L2(Ω) +(g, vH)L2(ΓR)

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 11

Stability and quasi-optimality

Since uh −IHuh ∈ Wh, we have a(uh −IHuh, (1−C∞)vH

• =

vH∈ VH

) = 0 for all vH ∈ VH. Hence: a(IHuh, vH) = a(uh, vH) = (f, vH)L2(Ω) +(g, vH)L2(ΓR) = ⇒ IHuh solves Petrov-Galerkin method

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 11

Stability and quasi-optimality

Since uh −IHuh ∈ Wh, we have a(uh −IHuh, (1−C∞)vH

• =

vH∈ VH

) = 0 for all vH ∈ VH. Hence: a(IHuh, vH) = a(uh, vH) = (f, vH)L2(Ω) +(g, vH)L2(ΓR) = ⇒ IHuh solves Petrov-Galerkin method

Theorem

The resolution condition Hκ 1 implies γ(κ,Ω)−1 inf

vH∈VH\{0}

sup

˜ vH∈ VH\{0}

ℜa(vH, ˜ vH) vHV˜ vHV and uh −ums

H V = (1−IH)uhV ≈ min vH∈VH uh −vHV.

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 11

Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

Exponential decay of the correctors

Theorem

There is 0 < β < 1 such that for all vertices z and m ∈ N we have ∇(1−C∞)ΛzL2(Ω\Bm·H(z)) β m∇ΛzL2(Ω).

0.5 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1

˜ Λz = (1−C∞)Λz Λz

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Localized fine-scale correctors

0.5 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Re Im

• 16
• 14
• 12
• 10
• 8
• 6
• 4
• 2
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Multiscale FEMs for waves RICAM 2016

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Localized fine-scale correctors

◮ Parameter m ∈ N ◮ m-th order element patches ΩT := Nm(T)

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Localized fine-scale correctors

◮ Parameter m ∈ N ◮ m-th order element patches ΩT := Nm(T)

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Localized fine-scale correctors

◮ Parameter m ∈ N ◮ m-th order element patches ΩT := Nm(T)

h≪ H

◮ Localized test functions

VH := (1−Cm)VH

• D. Gallistl (KIT)

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Multiscale Petrov-Galerkin FEM

Theorem (Well-posedness)

Let h ≪ 1, Hκ 1 and m |logγ(κ,Ω)| ≈ |logκ|. Then a) Stability γ(κ,Ω)−1 inf

vH∈VH\{0}

sup

˜ vH∈ VH\{0}

ℜa(vH, ˜ vH) vHV˜ vHV b) Quasi-optimality u−ums

H V min vH∈VH u−vHV.

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Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

Time-harmonic elastic wave model

We seek a displacement field u : Ω → Rd such that σ(u) = Cε(u) −div(σ(u))−κ2u = f in Ω, σ(u)·ν −iκu = g on ΓR, u = 0 on ΓD for the strain ε := symD and the elasticity tensor CM := 2µM +λ trMId×d.

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Time-harmonic elastic wave model

We seek a displacement field u : Ω → Rd such that σ(u) = Cε(u) −div(σ(u))−κ2u = f in Ω, σ(u)·ν −iκu = g on ΓR, u = 0 on ΓD for the strain ε := symD and the elasticity tensor CM := 2µM +λ trMId×d. Variational formulation employs V := {v ∈ (H1(Ω))d : v|ΓD = 0} with norm ·V :=

• κ2·2

L2(Ω) +C1/2ε ·2 L2(Ω).

Sesquilinear form on V a(v,w) := (Cε(v),ε(w))L2(Ω) −(κ2v,w)L2(Ω) −(iκv,w)L2(ΓR∩∂Ω).

• D. Gallistl (KIT)

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Multiscale method and error analysis

Analogous to the acoustic case. Provided h ≪ 1, Hκ 1 and m |logγ(κ,Ω)| ≈ |logκ|, then the method is stable and quasi-optimal under the assumption of polynomial well-posedness c(Ω) p(κ) ≤ inf

v∈V\{0}

sup

w∈V\{0}

ℜa(v,w) vVwV . (⋆)

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

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Multiscale method and error analysis

Analogous to the acoustic case. Provided h ≪ 1, Hκ 1 and m |logγ(κ,Ω)| ≈ |logκ|, then the method is stable and quasi-optimal under the assumption of polynomial well-posedness c(Ω) p(κ) ≤ inf

v∈V\{0}

sup

w∈V\{0}

ℜa(v,w) vVwV . (⋆) Only few (conditional) results available, see Cummings & Feng (2005).

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 17

Multiscale method and error analysis

Analogous to the acoustic case. Provided h ≪ 1, Hκ 1 and m |logγ(κ,Ω)| ≈ |logκ|, then the method is stable and quasi-optimal under the assumption of polynomial well-posedness c(Ω) p(κ) ≤ inf

v∈V\{0}

sup

w∈V\{0}

ℜa(v,w) vVwV . (⋆) Only few (conditional) results available, see Cummings & Feng (2005). Brown&DG 2016: Polynomial stability holds if ΓR = ∂Ω.

• D. Gallistl (KIT)

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• p. 17

Wavenumber-explicit stability bounds

Theorem (Brown&DG 2016)

If ΓR = ∂Ω, then c(Ω) κ7/2 ≤ inf

v∈V\{0}

sup

w∈V\{0}

ℜa(v,w) vVwV .

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 18

Wavenumber-explicit stability bounds

Theorem (Brown&DG 2016)

If ΓR = ∂Ω, then c(Ω) κ7/2 ≤ inf

v∈V\{0}

sup

w∈V\{0}

ℜa(v,w) vVwV . Proof: generalizes techniques of Melenk & Sauter (2010) and Melenk & Esterhazy (2012). Newton-potential estimates κ−1Nκ(f)H2(Ω) +Nκ(f)H1(Ω) +κNκ(f)L2(Ω) ≤ CfL2(Ω), with Nκ := Gκ ∗f and the Kupradze matrix Gκ.

• D. Gallistl (KIT)

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Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

Numerical experiment for heterogeneous material

Data: Ω = (0,1)2, f = point sources −(∆+κ2 n)u = f in Ω ∇u·ν −iκu = 0

• n ∂Ω

Parameters:

◮ κ = 27 ◮ n =

• 1

1/4

• ◮ H variable

◮ m variable ◮ h = 2−9

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Numerical experiment for heterogeneous material

Data: Ω = (0,1)2, f = point sources −(∆+κ2 n)u = f in Ω ∇u·ν −iκu = 0

• n ∂Ω

Parameters:

◮ κ = 27 ◮ n =

• 1

1/4

• ◮ H variable

◮ m variable ◮ h = 2−9

• D. Gallistl (KIT)

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• p. 19

Numerical experiment for heterogeneous material

Data: Ω = (0,1)2, f = point sources −(∆+κ2 n)u = f in Ω ∇u·ν −iκu = 0

• n ∂Ω

H

10 -2 10 -1

V-norm error

10 -2 10 -1

P1FEM m = 1 m = 2 m = 3 P1-best O(H)

Parameters:

◮ κ = 27 ◮ n =

• 1

1/4

• ◮ H variable

◮ m variable ◮ h = 2−9

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 19

Numerical experiment: 2D elasticity (scattering)

Data: Ω = (0,1)2 \[0.375,0.625]2, f = point source, λ = 1 = µ displacement Parameters:

◮ κ = 27 ◮ H variable ◮ m variable ◮ h = 2−11

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 20

Numerical experiment: 2D elasticity (scattering)

Data: Ω = (0,1)2 \[0.375,0.625]2, f = point source, λ = 1 = µ

10−2.4 10−2.2 10−2 10−1.8 10−1.6 10−1 100 H FEM msPG m = 1 msPG m = 2 msPG m = 3 O(H)

V-norm errors Parameters:

◮ κ = 27 ◮ H variable ◮ m variable ◮ h = 2−11

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 20

Numerical experiment: 2D elasticity (scattering)

Data: Ω = (0,1)2 \[0.375,0.625]2, f = point source, λ = 1 = µ

10−2.4 10−2.2 10−2 10−1.8 10−1.6 10−2 10−1 100 H FEM msPG m = 1 msPG m = 2 msPG m = 3 O(H2)

L2-norm errors Parameters:

◮ κ = 27 ◮ H variable ◮ m variable ◮ h = 2−11

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 20

Numerical experiment: 3D elasticity (scattering)

Data: Ω = (0,1)3, f = 0, λ = 1 = µ, polynomial solution, ΓR = ∂Ω

10−1.4 10−1.2 10−1 10−0.5 100 100.5 H FEM msPG m = 1 msPG m = 2 O(H)

V-norm errors Parameters:

◮ κ = 25 ◮ H variable ◮ m variable ◮ h = 2−6

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 21

Numerical experiment: 3D elasticity (scattering)

Data: Ω = (0,1)3, f = 0, λ = 1 = µ, polynomial solution, ΓR = ∂Ω

10−1.4 10−1.2 10−1 10−1 100 H FEM msPG m = 1 msPG m = 2 O(H2)

L2-norm errors Parameters:

◮ κ = 25 ◮ H variable ◮ m variable ◮ h = 2−6

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 21

3D elasticity: pointwise errors

Data: Ω = (0,1)3, f = 0, λ = 1 = µ, polynomial solution, ΓR = ∂Ω

1 2 3 4 5 6 7 8 ×10 -3 1 2 3 4 5 6 7 8 ×10 -3

poinwise error: FEM (left) vs. multiscale (right)

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 22

Summary

◮ Multiscale Petrov-Galerkin FEM for the acoustic and elastic

Helmholtz problems with large wave number

◮ Under reasonable assumptions on the parameters, the method

is pollution-free

◮ In homogeneous (or more general periodic) media, the fine

scale test functions depend only on local mesh-configurations and the seemingly high cost for the computation of the test functions can be drastically reduced on structured meshes

◮ Numerical experiments in two and three space dimensions

support these observations.

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 23

References

◮ D. Brown, D. Gallistl, and D. Peterseim. Multiscale Petrov-Galerkin method for high-frequency heterogeneous Helmholtz equations. Lect. Notes Comput. Sci. Eng., 2016. ◮ D. Brown and D. Gallistl, Multiscale sub-grid correction method for time-harmonic high-frequency elastodynamics with wavenumber explicit bounds. ArXiv e-prints, 2016. ◮ D. Gallistl and D. Peterseim. Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering. CMAME, 295:1-17, 2015. ◮ D. Peterseim. Eliminating the pollution effect in Helmholtz problems by local subscale correction. Math. Comp., 2016. ◮ A. M˚ alqvist and D. Peterseim. Localization of elliptic multiscale problems.

• Math. Comp., 2014.

Thank you for your attention.

• D. Gallistl (KIT)

Multiscale FEMs for waves RICAM 2016

• p. 24