Analytical and numerical aspects of a stable DG method for ......Analytical and numerical aspects of a stable DG method for Helmholtz problems MagdalenaStrugaru Basque Center for Applied

Analytical and numerical aspects of a stable DGmethod for Helmholtz problems

Magdalena StrugaruBasque Center for Applied Mathematics, Spain

In collaboration with:

Mohamed AmaraUniversity of Pau and INRIA, France

Rabia DjellouliCalifornia State University Northridge, USA

p. 1 Magdalena Strugaru Bilbao, January 2015

Outline

Motivation and context

A new DG solution methodology forHelmholtz problems

F Solution strategy

F Performance assessment

F Error estimates

Summary and perspectives


Outline



F Solution strategy


F Error estimates



Outline



F Solution strategy


F Error estimates



Outline



F Solution strategy


F Error estimates



Outline



F Solution strategy


F Error estimates



Outline



F Solution strategy


F Error estimates



Outline



F Solution strategy


F Error estimates



Motivation and Context

Applications

Radar

Sonar

Medical imaging

Nondestructive testing

Geophysical exploration



Applications

Radar

Sonar

Medical imaging





Applications

Radar

Sonar

Medical imaging





Applications

Radar

Sonar

Medical imaging





Applications

Radar

Sonar

Medical imaging





Applications

Radar

Sonar

Medical imaging




Motivation and ContextNumerical Difficulties

ka = 1, ha

= 110



ka = 1, ha

= 110



ka = 1, ha

= 110



ka = 1, ha

= 110



ka = 1, ha

= 110



ka = 3, ha

= 110



ka = 3, ha

= 110



ka = 3, ha

= 120



ka = 3, ha

= 130



ka = 3, ha

= 130

=⇒ kh = 110



Numerical Difficulties

Resolution necessary to achieve 10% on therelative error

kh 6= constant




Resolution necessary to achieve 10% on therelative error

kh 6= constant




|u−uh|1|u|1

≤ C1kh+ C2k3h2; kh < 1

(Babuska et al (95, 00))



“Realistic” simulation (Tezaur et al (02))

ka = 10

System of about 9.6 million complexunknowns




Motivation and ContextProminent Plane Waves Based

ApproachesWeak Element MethodRose (75)Partition of Unity MethodBabuska-Melenk (97), Laghrouche-Bettes (00)Ultra-Weak Variational MethodCessenat-Desprès (98)Least-Squares Method (LSM)Monk-Wang (99)Trefftz-Type Wave-Based MethodDesmet et al (98, 02, 10), Stojek (98)Plane wave Discontinuous Galerkin MethodHiptmair et al (09)Discontinuous Galerkin Method (DGM)Farhat et al (01, 03, 04, 05). . .


Motivation and ContextDGM Formulation (Farhat et al)

Main Features

Plane waves for local approximation ofthe fieldLagrange multipliers to enforcecontinuityAnalytical evaluation of the matricesGlobal system: symmetric and sparseSize of the global system ≡# dofs forthe Lagrange multiplier



DGM Formulation (Farhat et al)2D Numerical Performance

DGM outperforms high-order FE methods:

R-4-1, R-8-2 require 5 to 7 times fewerdof than Q2Q-16-4 requires 6 times fewer dof thanQ4Q-32-8 requires 25 times fewer dof thanQ4



Issues

Inf-Sup condition: Discrete spacescompatibility

# plane waves vs. # Lagrange multipliers



Issues





Issues





Issues


Relative error, kh=1/2

R-8-2 element



Issues



R-8-2 element

R-8-3 element



Issues

Numerical instabilities

Total relative error, ka=1

R-8-2 element


Motivation and ContextUltimate objective

Design a stable DG-like method

⇓Build on top of DGM

Preserve the good features of DGMOvercome the numerical instabilities

Preserve the good features of DGM

From where to start and how to proceed?


Motivation and ContextUltimate objective

Design a stable DG-like method

⇓Build on top of DGM

Preserve the good features of DGMOvercome the numerical instabilities

Preserve the good features of DGM

From where to start and how to proceed?


A new DG solution methodology

Mathematical model


A new DG solution methodologyMathematical model












A new DG solution methodologyStep 1. Domain partition








A new DG solution methodologyStep 2. Splitting process: u(µ)=ϕ+Ψ(µ)

In each element K solve two sets of Helmholzproblems:∆ϕK + k2ϕK = 0 in K∂nϕ

K − ikϕK = 0 on ∂K ∩ Σ∂nϕ

K = −∂nuinc on ∂K ∩ Γ∂nϕ

K − i kϕK = 0 on ∂K ∩ Ωc∆ΨK(µ) + k2ΨK(µ) = 0 in K∂nΨK(µ)− ikΨK(µ) = 0 on ∂K ∩ Σ

∂nΨK(µ) = 0 on ∂K ∩ Γ∂nΨK(µ)− i kΨK(µ) = µ on ∂K ∩ Ωc

Gain: well posed local problems for α ∈ R∗+








Gain: well posed local problems for α ∈ R∗+p. 33 Magdalena Strugaru Bilbao, January 2015







Well-posed local problemsp. 34 Magdalena Strugaru Bilbao, January 2015


Step 3. Optimization process:Gain: Hermitian and positive semi-definite globalmatrix

Restore the continuity of the field and itsnormal derivative in the least-squares sense:

λ = arg minµ

∑e−interior

(‖[u(µ)]‖2

0,e + ‖[[∂nu(µ)]]‖20,e

Gain: Hermitian and positive semi-definite globalmatrixStep 4. Build the solution: u=ϕ+Ψ(λ)Gain: Hermitian and positive semi-definite globalmatrix





λ = arg minµ

∑e−interior

(‖[u(µ)]‖2

0,e + ‖[[∂nu(µ)]]‖20,e






λ = arg minµ

∑e−interior

(‖[u(µ)]‖2

0,e + ‖[[∂nu(µ)]]‖20,e



A new DG solution methodologyVariational formulation

Step 2. Compute ϕK and ΨK(µ) for each µ bysolving local variational problems in each K:

aK(v, w) = lK(w)with

aK(v, w) =

ˆ∂K

(∂nv − i kv)(∂nw + i kw)

Algebraic level: solve one linear system

F nK × nK Hermitian positive definite matrixF nK = # plane wavesF multiple right-hand sideF analytical evaluation of the entries



Step 2. Compute ϕK and ΨK(µ) for each µ bysolving local variational problems in each K:

aK(v, w) = lK(w)with

aK(v, w) =

ˆ∂K

(∂nv − i kv)(∂nw + i kw)

Algebraic level: solve one linear system

F nK × nK Hermitian positive definite matrixF nK = # plane wavesF multiple right-hand sideF analytical evaluation of the entries



Step 3. Compute the Lagrange multiplier λ bysolving one global variational problem:

A(λ, µ) = L(µ)with

A(λ, µ) =∑

e−interior

(ê

[Ψ(λ)][Ψ(µ)]

+

ê

[[∂nΨ(λ)]][[∂nΨ(µ)]]

)

Algebraic level: solve a sparse linear systemF nλ × nλ Hermitian positive semi-definitematrixF nλ = # Lagrange multipliersF analytical evaluation of the entries



Step 3. Compute the Lagrange multiplier λ bysolving one global variational problem:

A(λ, µ) = L(µ)with

A(λ, µ) =∑

e−interior

(ê

[Ψ(λ)][Ψ(µ)]

+

ê

[[∂nΨ(λ)]][[∂nΨ(µ)]]

)Algebraic level: solve a sparse linear systemF nλ × nλ Hermitian positive semi-definitematrixF nλ = # Lagrange multipliersF analytical evaluation of the entries


A new DG solution methodologyPerformance assessment

Waveguide problem

Plane wave solution: u = eikx.d∗, d∗ = (cos θ∗, sin θ∗)

Modified H1 norm:

‖·‖H1 =

√∑K∈τh

‖·‖2H1(K) +

∑e−interior

‖[·]‖2L2(e)


A new DG solution methodologyPerformance assessmentWaveguide problem


Modified H1 norm:

‖·‖H1 =

√∑K∈τh

‖·‖2H1(K) +

∑e−interior

‖[·]‖2L2(e)


A new DG solution methodologyPerformance assessmentWaveguide problem


Modified H1 norm:

‖·‖H1 =

√∑K∈τh

‖·‖2H1(K) +

∑e−interior

‖[·]‖2L2(e)



Compatibility condition

DGM


R-8-2 element

R-8-3 element



Compatibility conditionDGM


R-8-2 element

R-8-3 element



Compatibility conditionThe proposed method


R-8-2 element

R-8-3 element



Compatibility conditionThe proposed method


R-8-2 element

R-8-3 element



Sensitivity to the mesh refinement

Total relative error, ka=20

R-7-2 element



Sensitivity to the frequency

ka R-7-2 R-11-350 28% 0.05%100 51% 0.07%200 69% 0.20%errorerrorerror, kh=2

Total relative error, kh=2

R-7-2 element

R-11-3 element

Computational cost increased by 50%Gain of more than two orders of magnitude






R-7-2 element

R-11-3 element







R-7-2 element

R-11-3 element







R-7-2 element

R-11-3 element







R-7-2 element

R-11-3 element







R-7-2 element

R-11-3 element







R-7-2 element

R-11-3 element




Resolution vs. fixed accuracy

Level # elements/wavelengthof

accuracy R-11-3 R-13-410% 1.88 1.495% 2.43 1.601% 2.82 1.99

errorerrorerror, kh=2ka=400

R-11-3 element

R-13-4 element

Computational cost reduced by about40%rather than refinement




Level # degrees of freedomof # elements/wavelength

accuracy R-11-3 R-13-410% 171,360 142,8805% 286,440 164,8321% 386,640 256,032


R-11-3 element

R-13-4 element

Computational cost reduced by about40%rather than refinement




Level # degrees of freedomof # elements/wavelength

accuracy R-11-3 R-13-410% 171,360 142,8805% 286,440 164,8321% 386,640 256,032


R-11-3 element

R-13-4 element

=⇒Higher-order elements rather meshrefinement


A new DG solution methodologySensitivity to the mesh distortion

DGM, Q-8-2 The proposed method, Q-7-2

Relative error, ka=30, θ∗ = 67.5


















A new DG solution methodologyPerformance assessmentScattering problem


A new DG solution methodologySensitivity to the mesh refinementComparison with LSM

Relative error, ka=10Sensitivity to the mesh refinement

Q-7-2 element




Q-7-2 element


A new DG solution methodologySensitivity to the mesh refinementComparison with DGM


Q-7-2 element




Q-7-2 element




Q-7-2 element




# elements/wavelengthLevelof New method LSM LSM

accuracy (Q-7-2) (7pw) (8pw)10% 18.84 18.84 18.845% 25.12 31.40 31.401% 50.24 75.36 75.36


Computational cost reduced by more than50%!




# degrees of freedomLevelof New method LSM LSM

accuracy (Q-7-2) (7pw) (8pw)10% 240 252 2885% 448 700 8001% 1,920 4,032 4,608







accuracy (Q-7-2) (7pw) (8pw)10% 240 252 2885% 448 700 8001% 1,920 4,032 4,608







accuracy (Q-7-2) (7pw) (8pw)10% 240 252 2885% 448 700 8001% 1,920 4,032 4,608




A new DG solution methodologyMathematical results: error estimates

Truncated Taylor operator LN :

|v − LNv|p,K ≤ ChN+1−pK

(|v|N+1,K + hK |v|N+2,K

),

∀v ∈ HN+2, 0 ≤ p ≤ N

Interpolation operators πN :

|v −ΠNv|p,K ≤ ChN+1−pK

(N∑l=0

kN+1−l |v|l,K +

|v|N+1,K + hK |v|N+2,K

),

∀v ∈ HN+2, ∆v + k2v = 0, 0 ≤ p ≤ N


A new DG solution methodologyMathematical results: error estimates

Truncated Taylor operator LN :

|v − LNv|p,K ≤ ChN+1−pK

(|v|N+1,K + hK |v|N+2,K

),

∀v ∈ HN+2, 0 ≤ p ≤ N

Interpolation operators πN :

|v −ΠNv|p,K ≤ ChN+1−pK

(N∑l=0

kN+1−l |v|l,K +

|v|N+1,K + hK |v|N+2,K

),

∀v ∈ HN+2, ∆v + k2v = 0, 0 ≤ p ≤ N


A new DG solution methodologyMathematical results: error estimatesComputational cost reduced by more than50%!

R-7-2 element:

infvh

‖u− vh‖0,Ω ≤ C(k2h ‖Ψ(λ)‖0,Ω + kh |Ψ(λ)|1,Ω + h |Ψ(λ)|2,Ω

h

k|Ψ(λ)|3,Ω +

h

k2|Ψ(λ)|4,Ω +

h2

k2|Ψ(λ)|5,Ω

)Computational cost reduced by more than50%!Computational cost reduced by more than50%!


Summary and Perspectives

Simplicity

Local systems: small, Hermitian andpositive definite

Global system: sparse, Hermitian andpositive semi-definite

Performance: Promising numericalresults (accuracy and stability)

Unstructured and adaptive mesh



Simplicity







Simplicity







Simplicity







Simplicity






Summary and PerspectivesCurrent and future work

Pursue the numerical investigation

FHigh frequency regime andhigh-order elements

FAdaptive strategy

Extension to 3D scattering problemsExtension to 3D elasto-acousticscattering problems





FAdaptive strategy






FAdaptive strategy






FAdaptive strategy






FAdaptive strategy

Extension to 3D scattering problemsExtension to 3D elasto-acousticscattering problemsp. 69 Magdalena Strugaru Bilbao, January 2015

The End...

Thank you for your attention!


A new DG solution methodologyConnection with DGM (Farhat et al)Gain: Hermitian and positive semi-definite globalmatrix

Local formulationGain: Hermitian and positive semi-definite globalmatrix

The proposed method:ˆ∂K

(∂nv − i kv)(∂nw + i kw) ds = l(w)

DGM: ˆ∂K

(∂nv − i kvχΣ)w ds = l(w)

Gain: Hermitian and positive semi-definite globalmatrixGain: Hermitian and positive semi-definite globalmatrix


A new DG solution methodologyConnection with DGM (Farhat et al)Gain: Hermitian and positive semi-definite globalmatrixLocal formulationGain: Hermitian and positive semi-definite globalmatrix

The proposed method:ˆ∂K

(∂nv − i kv)(∂nw + i kw) ds = l(w)

DGM: ˆ∂K

(∂nv − i kvχΣ)w ds = l(w)

Gain: Hermitian and positive semi-definite globalmatrixGain: Hermitian and positive semi-definite globalmatrix


A new DG solution methodologyConnection with DGM (Farhat et al)Gain: Hermitian and positive semi-definite globalmatrixGlobal formulationGain: Hermitian and positive semi-definite globalmatrix

The proposed method:∑e⊂Ω

(βe

ê[Φ(λ)][Φ(µ)] + γe

ê[[∂nΦ(λ)]][[∂nΦ(µ)]]

)= L(µ)

DGM:∑e⊂Ω

1

|e|

ê[Φ(λ)]µ = L(µ)

Gain: Hermitian and positive semi-definite globalmatrixGain: Hermitian and positive semi-definite globalmatrixp. 72 Magdalena Strugaru Bilbao, January 2015

A new DG solution methodologyConnection with LSM (Monk et al)Gain: Hermitian and positive semi-definite globalmatrixGlobal formulationGain: Hermitian and positive semi-definite globalmatrix

The proposed method:∑e⊂Ω

(βe

ê[Φ(λ)][Φ(µ)] + γe

ê[[∂nΦ(λ)]][[∂nΦ(µ)]]

)= L(µ)

LSM:∑e⊂Ω

(k2

ê[u][v] +

ê[[∂nu]][[∂nv]]

)= L(v)

Gain: Hermitian and positive semi-definite globalmatrixGain: Hermitian and positive semi-definite globalmatrixp. 73 Magdalena Strugaru Bilbao, January 2015


Computation complexity

Uniform n× n mesh

nK shape functions at the element level

nλK dofs for the Lagrange multiplier on each

interior edge of the element K

Method Asymptotic size Stencil widthof the solution vector

New method 4nλKn2 20nλ

K

DGM 2nλKn2 7nλ

K

LSM nKn2 5nK




Uniform n× n mesh






K

DGM 2nλKn2 7nλ

K

LSM nKn2 5nK




Uniform n× n mesh






K

DGM 2nλKn2 7nλ

K

LSM nKn2 5nK




Uniform n× n mesh






K

DGM 2nλKn2 7nλ

K

LSM nKn2 5nK




Uniform n× n mesh






K

DGM 2nλKn2 7nλ

K

LSM nKn2 5nK




Reduction of the computational cost ofLevel the proposed method with respect to:of

accuracy LSM (7 pw) LSM (8 pw)10% 4.76% 16.66%5% 36.00% 44.00%1% 52.38% 58.33%



A new DG solution methodologyError estimates in L2−norm and Hp−semi-norms

R-7-2 element:

|u− πNu|p,K ≤ Ch4−pK

(k4 ‖u‖0,K + k3 |u|1,K + k2 |u|2,K

k |u|3,K + |u|4,K + hK |u|5,K)∀0 ≤ p ≤ 2

infvh


h

k|Ψ(λ)|3,Ω +

h

k2|Ψ(λ)|4,Ω +

h2

k2|Ψ(λ)|5,Ω

)Computational cost reduced by more than50%!


A new DG solution methodologyError estimates in L2−norm and Hp−semi-norms

R-7-2 element:

|u− πNu|p,K ≤ Ch4−pK

(k4 ‖u‖0,K + k3 |u|1,K + k2 |u|2,K

k |u|3,K + |u|4,K + hK |u|5,K)∀0 ≤ p ≤ 2

infvh


h

k|Ψ(λ)|3,Ω +

h

k2|Ψ(λ)|4,Ω +

h2

k2|Ψ(λ)|5,Ω

)Computational cost reduced by more than50%!


Documents

Analytical and numerical aspects of a stable DG method for ......Analytical and numerical aspects of a stable DG method for Helmholtz problems MagdalenaStrugaru Basque Center for Applied