fe-hmm for the wave equationfe-hmm for the wave equation marcus grote university of basel,...

FE-HMM for the Wave Equation

Marcus Grote

University of Basel, Switzerland

May 11, 2012

joint work with:

Assyr Abdulle, EPF Lausanne

Christian Stohrer, University of Basel

Introduction

The wave equation

Homogenization

A FE-HMM method

Convergence

Numerical examples

FE-HMM for long time

Concluding remarks

Waves in heterogeneous media

The wave equation

For T > 0 consider in Ω ∈ Rn the model problem

∂2

∂t2uε −∇ · (aε(x)∇uε) = F (x , t) in Ω× (0,T ) ,

uε(x , t) = 0 on ∂Ω× (0,T ) ,uε(x , 0) = f (x) in Ω ,

∂

∂tuε(x , 0) = g(x) in Ω ,

where aε(x) is uniformly coercive, bounded and symmetric.

Here ε� 1 denotes a small, microscopic length scale in the(periodic, random, etc.) medium.

GOAL: Capture macroscopic behavior of uε.

Variational formulation

For 0 < t < T , find uε such that

d2

dt2(uε(t, .), v) + Bε(uε(t, .), v) = (F (t, .), v), ∀v ∈ H10 (Ω),

where (·, ·) denotes the standard scalar product and

Bε(v ,w) =

∫Ω

aε(x)∇v · ∇w dx .

Numerical approximation:

I Galerkin projection onto VH ⊂ H10(Ω), dim(VH)

Standard FEM

I Typical convergence rates are of order O((hε )

l).

I Therefore h� ε is needed.I Since ε� 1 we cannot afford resolving the finest scales.

Computational domain Ω with periodic media aε.

Standard FEM


l).

I Therefore h� ε is needed.

I Since ε� 1 we cannot afford resolving the finest scales.


Standard FEM


l).

I Therefore h� ε is needed.I Since ε� 1 we cannot afford resolving the finest scales.


Idea

Instead consider effective (upscaled or homogenized) problem forε→ 0.

I aε → a0?I uε → u0?I Equation for u0?

Idea


I aε → a0?

I uε → u0?I Equation for u0?

Idea


I aε → a0?I uε → u0?

I Equation for u0?

Idea


I aε → a0?I uε → u0?I Equation for u0?

Classical homogenization

Assume that aε(x) = a(x , x/ε) = a(x , y) and that a(x , y) is1-periodic in y , y ∈ Y = (0, 1)n. Then

uε ⇀ u0 weakly∗ in L∞(0,T ; H10 (Ω)),

(see e.g. [1], [2], [3], [4])

where u0 solves the “homogenized waveequation”:

∂2

∂t2u0 −∇ ·

(a0(x)∇u0

)= F (x , t) in Ω× (0,T )

Here a0(x) is the homogenized tensor; it is also elliptic andsymmetric, but contains no small scale behavior.

[1] Bensoussan, Lions, Papanicolaou; 1978

[2] Jukov, Kozlov, Oleinik; 1991

[3] B.-Otsmane, Francfort, Murat; J. Math. Pures Appl.; 1992

[4] Cioranescu, Donato; 1999

Classical homogenization

Assume that aε(x) = a(x , x/ε) = a(x , y) and that a(x , y) is1-periodic in y , y ∈ Y = (0, 1)n. Then

uε ⇀ u0 weakly∗ in L∞(0,T ; H10 (Ω)),

(see e.g. [1], [2], [3], [4]) where u0 solves the “homogenized waveequation”:

∂2

∂t2u0 −∇ ·

(a0(x)∇u0

)= F (x , t) in Ω× (0,T )

Here a0(x) is the homogenized tensor; it is also elliptic andsymmetric, but contains no small scale behavior.

[1] Bensoussan, Lions, Papanicolaou; 1978

[2] Jukov, Kozlov, Oleinik; 1991

[3] B.-Otsmane, Francfort, Murat; J. Math. Pures Appl.; 1992

[4] Cioranescu, Donato; 1999

The homogenized tensor a0

Assume that aε is periodic in the small scale y = x/ε.Then we have the following formula:

a0i ,j =

∫Y

(aεi ,j(x , y) +

n∑k=1

aεi ,k(x , y)χj

∂yk(x , y)

)dy ,

where χj(x , ·) are the solutions of the cell problems∫Y∇χjaε∇z dy = −

∫Y

(aεej)T∇z dy , ∀z ∈W 1per (Y ),

ej are the canonical basis vectors of Rn and

W 1per (Y ) = {v ∈ H1per (Y ) :∫

Yv dx = 0}.

Variational formulation (homogenized)

If we can solve all the cell problems and know a0(x) explicitly, weimmediately have for 0 < t < T :

d2

dt2(u0(t, .), v) + B0(u0(t, .), v) = (F (t, .), v), ∀v ∈ H10 (Ω),

where

B0(v ,w) =

∫Ω

a0(x)∇v · ∇w dx .

No small scale behaviour. Thus we can apply a standard FEM tothe homogenized problem.

FEM for homogenized problem

For 0 < t < T , find u0,H such that

d2

dt2(u0,H(t, .), vH) + B0,H(u0,H(t, .), vH) = (F (t, .), vH),

for all

vH ∈ S`0(Ω, TH) = {vH ∈ H10 (Ω); vH |K ∈ P`(K ), ∀K ∈ TH},

where

B0,H(vH ,wH) =∑

K∈TH

J∑j=1

ωj ,Ka0(xj ,K )∇vH · ∇wH

and (xj ,K , ωj ,K )1≤j≤J are the quadrature points and weights.

Visualization

Homogen-−−−−−−→

ization

PROBLEM: a0(x) is not a “simple average” (e.g. arithmetic orharmonic) and can only rarely be computed analytically.

Heterogeneous multiscale methods (HMM)

GOAL: devise a numerical method to compute u0(x , t), which doesnot require the explicit knowledge of a0(x), at a cost independentof ε.

FE-HMM for parabolic and elliptic problems:I E, Engquist; Comm. Math. Sci.; 2003I Abdulle, E; J. Comput. Phys.; 2003I E, Ming, Zhang; J. Am. Math. Soc.; 2004I Abdulle, Schwab; Multiscale Model. Simul.; 2005I Abdulle; Multiscale Model. Simul.; 2005I Ming, Zhang; Math. Comput.; 2007

FD-HMM for the wave equation:I Engquist, Holst, Runborg; Commun. Math. Sci., 2011

FE-HMM for the wave equationWe wish to estimate B0,H(vH ,wH).

B0,H(vH ,wH) =∑

K∈TH

J∑j=1

ωj ,Ka0(xj ,K )∇vH(xj ,K ) · ∇wH(xj ,K )

B0,H(vH ,wH) =∑

K∈TH

J∑j=1

ωj ,K a0(xj ,K )∇vH(xj ,K ) · ∇wH(xj ,K )︸︷︷︸

to be estimated!

a0(xj ,K )∇vH(xj ,K ) · ∇wH(xj ,K ) ≈1

|Kδj,K |

∫Kδj,K

aε(x)∇v · ∇w dx ,

where Kδj,K is a cube of size δ centered at xj ,K , and v (resp. w) isthe solution of the following micro-problem:Find v on Kδj,K such that (v − vH,lin)(t, .) ∈W 1per (Kδj,K ) and∫

Kδj,K

aε(x)∇v · ∇z dx = 0 ∀z ∈W 1per (Kδj,K ),

FE-HMM for the wave equation

We wish to estimate B0,H(vH ,wH).

B0,H(vH ,wH) =∑

K∈TH

J∑j=1


to be estimated!

a0(xj ,K )∇vH(xj ,K ) · ∇wH(xj ,K ) ≈1

|Kδj,K |

∫Kδj,K

aε(x)∇v · ∇w dx ,

where Kδj,K is a cube of size δ centered at xj ,K , and v (resp. w) isthe solution of the following micro-problem:Find v on Kδj,K such that (v − vH,lin)(t, .) ∈W 1per (Kδj,K ) and∫

Kδj,K

aε(x)∇v · ∇z dx = 0 ∀z ∈W 1per (Kδj,K ),


Consider the macro-FE space S`0(Ω, TH). Inside each macroelement K ∈ TH pick integration points xj ,K , with samplingdomains Kδj,K of size δ ≥ ε, centered at xj ,K .


From now on we write Kδj for Kδj,K . The FE-HMM reads asfollows:Find ūH ∈ [0,T ]× S`0(Ω, TH)→ R such that ∀vH ∈ S`0(Ω, TH)

d2

dt2(ūH , vH) +

∑K∈TH

J∑j=1

ωj ,K|Kδj |

∫Kδj

aε(x)∇u · ∇v dx︸︷︷︸B̄H(ūH ,vH)

= (F (t, .), vH)

where u and v are the exact solutions of the micro problems.

Micro Solver

The solution of the micro-problem must be approximated, too!Again a standard FEM-method with smaller mesh size h.

Micro-solvers: Find uh (and resp. vh) on Kδj , such that(uh − uH,lin)(t, .) ∈ Sh(Kδj , Th) and∫

Kδj

aε(x)∇uh · ∇zhdx = 0 ∀zh ∈ Sh(Kδj , Th),

where Sh(Kδj , Th) = {vh ∈W 1per (Kδj ); vh|K ∈ Pq(K ), ∀K ∈ Th}.

Replacing the exact solutions u and v by uh and vh (inside eachKδj ), we obtain the fully discrete FE-HMM.


Macro-solver: Find uH ∈ [0,T ]× S`0(Ω, TH)→ R such that∀vH ∈ S`0(Ω, TH)

d2

dt2(uH , vH)+

∑K∈TH

J∑j=1

ωj ,K|Kδj |

∫Kδj

aε(x)∇uh · ∇vh dx︸︷︷︸BH(uH ,vH)

= (F (t, .), vH).

Micro-solvers: Find uh (and resp. vh) on Kδj , such that(uh − uH,lin)(t, .) ∈ Sh(Kδj , Th) and∫

Kδj

aε(x)∇uh · ∇zh dx = 0 ∀zh ∈ Sh(Kδj , Th),

where Sh(Kδj , Th) = {vh ∈W 1per (Kδj ); vh|K ∈ Pq(K ), ∀K ∈ Th},

Visualization

Convergence

Theorem: The FE-HMM solution uH(x , t) satisfies the errorestimates:

‖∂tu0 − ∂tuH‖L∞(0,T ;L2(Ω)) + ‖u0 − uH‖L∞(0,T ;H1(Ω))

≤ C

(H` + ε+

(h

ε

)2q)

and

‖u0 − uH‖L∞(0,T ;L2(Ω)) ≤ C

(H`+1 + ε+

(h

ε

)2q)

where C = C (u0,T ) is independent of ε, H, and h.

Introduction

The wave equation

Homogenization

A FE-HMM method

Convergence

Numerical examplesConvergence study1D Example, macro and micro scale dependence2D example, micro and macro scale dependence2D example, triangular meshLong time behavior

FE-HMM for long time

Concluding remarks

1D model problem

Let Ω be an interval , T > 0 and ε > 0. For the first threeexamples we consider the following one-dimensional modelproblem:

∂2

∂t2uε −

∂

∂x

(aε(x)

∂

∂xuε

)= 0 in Ω× (0,T ) ,


∂

∂tuε(x , 0) = g(x) in Ω ,

where aε(x) = a(x , x/ε).

In the first 1D examples shown here, we use P1-elements both forthe macro- and the microproblems, with a periodic couplingcondition. For the time-discretization we use the leap-frog method.

Example 1: convergence study

aε(x) =

√17

4+

1

4sin(

2πx

ε

), ε =

4

10000

In this special case, the homogenized tensor a0 can easily becomputed:

a0 =

(1

ε

∫ ε0

1

aε(x)

)−1= 1

(see e.g. [1],[2])

We set Ω = [0, 1], f (x) = sin(πx) and g = 0. Hence,

u0(x , t) = sin(πx) cos(πt).

[1] Bensoussan, Lions, Papanicolaou; 1978 (chapter 1 1.3)

[2] Cioranescu, Donato; 1999 (chapter 5.3 pp. 95)

Example 1 (continued):

I error betweenHMM-solution uH andhomogenized solution u0.

I δ = ε

I h/H = constant

I second order O(∆t2,H2)L2-error vs. H at T = 2.

Example 1 (continued):

Only macromesh refinement, h fixed.

L2-error vs. H at T = 2.

We achieve the expected second-order convergence, but only if themacro- and the micromesh are refined simultaneously.

Example 2: macro and micro scale dependence

aε(x) =

{√2 + sin 2π xε , x < 0 or x ∈ (k , k + 0.5), k ∈ N0√2 + sin 2π xε + 2 , x ∈ (k + 0.5, k + 1), k ∈ N0

We set ε = δ = 1/1000. The tensor for the computational domainis show below, with a zoom about x = 4.5.

Example 2 (continued)

I right-moving Gaussianpulse

I ε = δ = 10−3

I macro meshsize H = 10−2

I micro meshsize h = 10−4

I time step ∆t = 10−3

I reference solution uεI comparison with naive

piecewise averaged medium

0

0.5

u(x,

0)−3 −2 −1 0 1 2 3 4 50

5

xa ε

(x)

Initial condition


T = 1

0

0.5

u(x,

1)

naive averagereference

−3 −2 −1 0 1 2 3 4 50

5

x

a ε(x

)


T = 2

0

0.5

u(x,

1)


−3 −2 −1 0 1 2 3 4 50

5

x

a ε(x

)


T = 3

0

0.5

u(x,

1)


−3 −2 −1 0 1 2 3 4 50

5

x

a ε(x

)


T = 1

0

0.5

u(x,

1)

HMMreference

−3 −2 −1 0 1 2 3 4 50

5

x

a ε(x

)


T = 2

0

0.5

u(x,

2)

HMMreference

−3 −2 −1 0 1 2 3 4 50

5

x

a ε(x

)


T = 3

0

0.5

u(x,

3)

HMMreference

−3 −2 −1 0 1 2 3 4 50

5

x

a ε(x

)

2D model problem

Let Ω ∈ R2, T > 0 and ε > 0 and consider

∂2

∂t2uε −∇ · (aε(x)∇uε) = 0 in Ω× (0,T ) ,


∂

∂tuε(x , 0) = g(x) in Ω ,

where aε(x) = a(x , x/ε) is a 2× 2 tensor.

In the following two examples, we set δ = ε.For the time-discretization we use the leap-frog method.

Example 3: dependence on both scales

aε(x) =

(1.1 + 12

(sin 2πx1 + sin 2π

x1ε

)0

0 1.1 + 12(sin 2πx1 + sin 2π

x1ε

)) ε = 1300

Cross-section (fixed y = y0) through material.

(cf. [1] section 4.3.2)

[1] Engquist, Holst, Runborg; Commun. Math. Sci., 2011


The homogenized tensor a0 is given by.

a0 =

(√(1.1 + 0.5 sin 2πx1)2 − 0.52 0

0 1.1 + 0.5 sin 2πx1

).

Let Ω be [0, 4]× [0, 4]. The inital conditions are given by

f (x) = exp(−‖x − xM‖22)/σ2, xM = (2, 2) and σ = 0.1,g(x) = 0

The discretization parameters are

H =1

100, δ = � =

1

300, h =

1

3000, ∆t =

1

1000


T = 0

HMM-solution homogenized solution


T = 1



T = 2



T = 3


Example 4: 2D triangular mesh

Ω = [0, 2]× [−1, 1] ⊂ R2 T = 2

The computational domain is divided into four distinct subdomains.

Ω1

Ω2Ω4

Ω3

At (1,−0.5) a measurement point is situated.


The material tensor aε(x) differs in every subdomain. We set

aε(x) =

I2×2 for x ∈ Ω1(√

2 + sin(2π x2ε

))I2×2 for x ∈ Ω2(√

2 + 12 sin (2πx2) + sin(2π x2ε

))I2×2 for x ∈ Ω3

2I2×2 for x ∈ Ω4


We will use P1 elements on a mesh that respects the innerinterfaces.

Here we show only a coarse mesh (with 646 elements) forvisualization. The mesh we use has 63’498 elements.


I initial condition: down moving Gaussian plane wave

I homogeneous Neumann boundary conditions

The discretisation parameters are

δ = � =1

1000, h =

1

7000, ∆t =

1

1000

If we would use a fully resolved triangular mesh with the fine meshsize h we would have almost 400 millions elements (compare to63’498 elements that we actually need).


T = 0

HMM homogenized average


T = 0.415



T = 0.83



T = 1.245



T = 1.66



T = 2



Measurement at x = (1,−0.5)

Example 5: Long Time Effects

I aε(x) =√

2 + sin 2πx

ε, ε =

1

50(Here we have a0(x) = 1.)

I Gaussian pulse with zero initial velocity

I periodic boundary condition

I cubic FEM for macro and micro solver

I H = 0.0133 and h = 0.001


T = 0

I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.


T = 0.25



T = 0.5



T = 1



T = 1.25



T = 1.5



T = 1.75



T = 2



T = 4



T = 10



T = 20



T = 40



T = 60



T = 80



T = 100



T = 100 (50 rev.)


Heterogeneous multiscale methods (HMM) (revisited)

GOAL: capture dispersive behaviour for long times T = O(ε−2)

.

1D Homogenization for long times:I Symes, Santosa; SIAM J. Appl. Math.; 1991I Fish, Chen and Nagai; Inter. J. Num. Meth. Engrg.; 2002I Lamacz; Math. Models Methods Appl. Sci.; 2011

FD-HMM for the wave equation:I Engquist, Holst, Runborg; arXiv; 2011I Engquist, Holst, Runborg; Proceedings of a Winter Workshop at BIRS 2009; 2012

Heterogeneous multiscale methods (HMM) for long times

GOAL: capture dispersive behaviour for long times T = O(ε−2)

.

1D Homogenization for long times:I Symes, Santosa; SIAM J. Appl. Math.; 1991I Fish, Chen and Nagai; Inter. J. Num. Meth. Engrg.; 2002I Lamacz; Math. Models Methods Appl. Sci.; 2011

FD-HMM for the wave equation:I Engquist, Holst, Runborg; arXiv; 2011I Engquist, Holst, Runborg; Proceedings of a Winter Workshop at BIRS 2009; 2012

Effective dispersive equation (Lamacz)For 1-D periodic medium only:

∂2t ueff − a0∂2xueff − ε2

b

a0∂2x∂

2t u

eff = 0

T = 100

FE-HMM for the wave equation (revisited)


B0,H(vH ,wH) =∑

K∈TH

J∑j=1


to be estimated!

IDEA: Estimate the effective flux F0(xj ,K , vH) to obtainB0,H(vH ,wH).

FE-HMM for the wave equation (revisited)


B0,H(vH ,wH) =∑

K∈TH

J∑j=1

ωj ,K a0(xj ,K )∇vH(xj ,K )︸︷︷︸

=:F0(xj,K ,vH)

·∇wH(xj ,K )

︸︷︷︸to be estimated!

IDEA: Estimate the effective flux F0(xj ,K , vH) to obtainB0,H(vH ,wH).

Estimating the flux

Following Engquist, Holst and Runborg (2011), we estimate F0 as

F0(xj ,K , vH) ≈∫ τ−τ

∫ η−η

kτ (t)kη(x) aε(x + xj ,K )∇v(x , t)︸︷︷︸

=Fε

dx dt ,

where kη, kτ are kernels and v is the solution of the following timedependent hyperbolic micro-problem:

vtt(x , t)−∇ · (aε(x)∇v(x , t)) = 0 on Kδ × (−τ, τ),v(x , 0) = ∇vH(xj ,K ) · (x − xj ,K ),vt(x , 0) = 0, (⇒ v time symmetric)v(x , t)− v(x , 0) Kδ-periodic.

Kernel space Kp,q

k ∈ Kp,q ⇔

k ∈ Cqc (R) and supp k = [−1, 1]∫R

k(t)tr dt =

{1 if r = 0,

0 if 1 ≤ r ≤ p.

For k ∈ Kp,q we denote the scaled kernel by

kη(x) =1

ηk

(x

η

).

(Engquist, Tsai (2005))

We use symmetric polynomial kernels.

Kernel space Kp,q

Examples for averaging kernels k ∈ Kp,q

Example 5 Hyperbolic FE-HMM

I aε(x) =√

2 + sin 2πx

ε, ε =

1

50I Gaussian pulse with zero initial velocity, periodic boundary

condition

I cubic FEM for macro and micro solver

I H = 0.0133 and h = 0.001

I Kernel space: kτ , kη ∈ K9,9 with τ = η = 10ε.I Sampling domain Kδ is chosen such that boundary effects

have no influence.

Example 5 (continued)T = 100

I Hyperbolic FE-HMM recovers the homogenized solution u0,too

I Yet no improvement from hyperbolic micro-problems withlinear coupling.

Capturing long time effects (in 1D)

To improve coupling between the macro and the micro problems,we now use a third order approximation of vH as initial data for themicro problems.

Time dependent microproblem:

vtt(x , t)− ∂x(aε(x)∂xv(x , t)) = 0 on Kδ × (−τ, τ),v(x , 0) = p(x − xj ,K ),

+ q(x − xj ,K )2 + r(x − xj ,K )3,

vt(x , 0) = 0,

v(x , t)− v(x , 0) Kδ-periodic,

where

p = ∂xvH(xj ,K ).

q =∂2xvH(xj ,K )

2, r =

∂3X vH(xj ,K )

6.

Remark: We must use at least P3-elements at the macro scale.

Capturing long time effects (in 1D)

To improve coupling between the macro and the micro problems,we now use a third order approximation of vH as initial data for themicro problems.

Time dependent microproblem:

vtt(x , t)− ∂x(aε(x)∂xv(x , t)) = 0 on Kδ × (−τ, τ),v(x , 0) = p(x − xj ,K ) + q(x − xj ,K )2 + r(x − xj ,K )3,vt(x , 0) = 0,

v(x , t)− v(x , 0) Kδ-periodic,

where

p = ∂xvH(xj ,K ), q =∂2xvH(xj ,K )

2, r =

∂3X vH(xj ,K )

6.

Remark: We must use at least P3-elements at the macro scale.


T = 100

I Long time dispersive behavior is now captured.

Concluding remarks (short time)

I We have proposed a FE-HMM method for the wave equation.

I We have proved optimal convergence rates in the L2 and theenergy norm on a fixed time interval, when the macro- andmicro-mesh are refined simultaneously.

I BH computed only once.

I Fine mesh used only inside small sampling domains

I Total work and memory requirement independent of ε.

I Permits coarser mesh and larger time steps due to CFLcondition ∆t ≤ CH.

⇒ significant memory and CPU time reduction

Finite Element Heterogeneous Multiscale Method for the WaveEquation, SIAM MMS 9, 2011

Concluding remarks (short time)

I We have proposed a FE-HMM method for the wave equation.

I We have proved optimal convergence rates in the L2 and theenergy norm on a fixed time interval, when the macro- andmicro-mesh are refined simultaneously.

I BH computed only once.

I Fine mesh used only inside small sampling domains

I Total work and memory requirement independent of ε.

I Permits coarser mesh and larger time steps due to CFLcondition ∆t ≤ CH.⇒ significant memory and CPU time reduction

Finite Element Heterogeneous Multiscale Method for the WaveEquation, SIAM MMS 9, 2011

Concluding remarks (long time)

I Proposed alternative hyperbolic FE-HMM

I Elliptic and hyperbolic FE-HMM identical at short times

I Hyperbolic FE-HMM captures long-time dispersive effects

I Promising numerical results

I Current work:I Extension to higher dimensionsI Convergence proof for long times

Thank you for your attention !





I Promising numerical resultsI Current work:

I Extension to higher dimensionsI Convergence proof for long times






I Promising numerical resultsI Current work:

I Extension to higher dimensionsI Convergence proof for long times


IntroductionThe wave equationHomogenizationA FE-HMM methodConvergenceNumerical examplesConvergence study1D Example, macro and micro scale dependence2D example, micro and macro scale dependence2D example, triangular meshLong time behavior

FE-HMM for long timeConcluding remarks

fe-hmm for the wave equationfe-hmm for the wave equation marcus grote university of basel,...

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