fe-hmm for the wave equationfe-hmm for the wave equation marcus grote university of basel,...
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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.
-
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)
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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
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.
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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.
-
Idea
Instead consider effective (upscaled or homogenized) problem forε→ 0.
I aε → a0?I uε → u0?I Equation for u0?
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Idea
Instead consider effective (upscaled or homogenized) problem forε→ 0.
I aε → a0?
I uε → u0?I Equation for u0?
-
Idea
Instead consider effective (upscaled or homogenized) problem forε→ 0.
I aε → a0?I uε → u0?
I Equation for u0?
-
Idea
Instead consider effective (upscaled or homogenized) problem forε→ 0.
I aε → a0?I uε → u0?I Equation for u0?
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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
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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
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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}.
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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.
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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.
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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.
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Visualization
Homogen-−−−−−−→
ization
PROBLEM: a0(x) is not a “simple average” (e.g. arithmetic orharmonic) and can only rarely be computed analytically.
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Visualization
Homogen-−−−−−−→
ization
PROBLEM: a0(x) is not a “simple average” (e.g. arithmetic orharmonic) and can only rarely be computed analytically.
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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
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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
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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 ),
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FE-HMM for the wave equation
We wish to estimate B0,H(vH ,wH).
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 ),
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FE-HMM for the wave equation
We wish to estimate B0,H(vH ,wH).
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 ),
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FE-HMM for the wave equation
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 .
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FE-HMM for the wave equation
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 .
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FE-HMM for the wave equation
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.
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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.
-
FE-HMM for the wave equation
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},
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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.
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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
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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 ) ,
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) = 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.
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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)
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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
-
Example 2 (continued)
T = 1
0
0.5
u(x,
1)
naive averagereference
−3 −2 −1 0 1 2 3 4 50
5
x
a ε(x
)
-
Example 2 (continued)
T = 2
0
0.5
u(x,
1)
naive averagereference
−3 −2 −1 0 1 2 3 4 50
5
x
a ε(x
)
-
Example 2 (continued)
T = 3
0
0.5
u(x,
1)
naive averagereference
−3 −2 −1 0 1 2 3 4 50
5
x
a ε(x
)
-
Example 2 (continued)
T = 1
0
0.5
u(x,
1)
HMMreference
−3 −2 −1 0 1 2 3 4 50
5
x
a ε(x
)
-
Example 2 (continued)
T = 2
0
0.5
u(x,
2)
HMMreference
−3 −2 −1 0 1 2 3 4 50
5
x
a ε(x
)
-
Example 2 (continued)
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 ) ,
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) = 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
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Example 3 (continued)
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
-
Example 3 (continued)
T = 0
HMM-solution homogenized solution
-
Example 3 (continued)
T = 1
HMM-solution homogenized solution
-
Example 3 (continued)
T = 2
HMM-solution homogenized solution
-
Example 3 (continued)
T = 3
HMM-solution homogenized solution
-
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.
-
Example 4 (continued)
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
-
Example 4 (continued)
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.
-
Example 4 (continued)
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).
-
Example 4 (continued)
T = 0
HMM homogenized average
-
Example 4 (continued)
T = 0.415
HMM homogenized average
-
Example 4 (continued)
T = 0.83
HMM homogenized average
-
Example 4 (continued)
T = 1.245
HMM homogenized average
-
Example 4 (continued)
T = 1.66
HMM homogenized average
-
Example 4 (continued)
T = 2
HMM homogenized average
-
Example 4 (continued)
Measurement at x = (1,−0.5)
-
Example 4 (continued)
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
-
Example 5 (continued)
T = 0
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 0.25
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 0.5
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 1
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 1.25
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 1.5
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 1.75
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 2
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 4
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 10
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 20
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 40
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 60
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 80
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 100
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
Example 5 (continued)
T = 100 (50 rev.)
I Reference solution shows dispersive behaviour.I Homogenized solution does not show the dispersionI FE-HMM does not capture this effect.
-
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)
We wish to estimate B0,H(vH ,wH).
B0,H(vH ,wH) =∑
K∈TH
J∑j=1
ωj ,K a0(xj ,K )∇vH(xj ,K ) · ∇wH(xj ,K )︸ ︷︷ ︸
to be estimated!
IDEA: Estimate the effective flux F0(xj ,K , vH) to obtainB0,H(vH ,wH).
-
FE-HMM for the wave equation (revisited)
We wish to estimate B0,H(vH ,wH).
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.
-
Example 5 (continued)
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 (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 !
-
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 resultsI Current work:
I Extension to higher dimensionsI Convergence proof for long times
Thank you for your attention !
-
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 resultsI Current work:
I Extension to higher dimensionsI Convergence proof for long times
Thank you for your attention !
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