homogenization wave equation

8/8/2019 Homogenization Wave Equation

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Numerical Homogenization of the Acoustic Wave

Equations with a Continuum of Scales.

Houman Owhadiand Lei Zhang

November 12, 2007

Abstract

In this paper, we consider numerical homogenization of acoustic waveequations with heterogeneous coefficient, namely, when the bulk modulus

and the density of the medium are only bounded. We show that undera Cordes type condition the second order derivatives of the solution withrespect to harmonic coordinates are L2 (instead H1 with respect to Eu-clidean coordinates) and the solution itself is in L(0, T ,H 2()) (insteadofL(0, T ,H 1()) with respect to Euclidean coordinates). Then, we pro-pose an implicit time stepping method to solve the resulted linear system oncoarse spatial scales, and present error estimates of the method. It followsthat by pre-computing the associated harmonic coordinates, it is possibleto numerically homogenize the wave equation without assumptions of scaleseparation or ergodicity.

1 Introduction and main results

Let be a bounded and convex domain of class C2 ofRn. Let T > 0. Considerthe following acoustic wave equation

AMS 2000 Subject Classification. Primary 35L05, 35B27; secondary 65M15, 86-08, 74Q15.Key words and phrases. multiscale problem, compensation, numerical homogenization, up-

scaling, acoustic wave equation.California Institute of Technology Applied & Computational Mathematics, Control & Dy-

namical systems, MC 217-50 Pasadena , CA 91125, [email protected] Institute of Technology Applied & Computational Mathematics MC 217-50

Pasadena , CA 91125, [email protected]

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K1(x)2t u = div(1(x)u(x, t)) + g in (0, T).

u(x, t) = 0 for (x, t) (0, T).u(x, t) = u(x, 0) for (x, t) {t = 0}.tu(x, t) = ut(x, 0) for (x, t) {t = 0}.

(1.1)

Write T :=

(0, T) and a := 1. We assume that a is a n

n uniformlyelliptic matrix on the closure of whose entries are in L(). K is assumed tobe scalar such that K and K1 belong to L(). Let Kmax := KL() andKmin := (K1L())1. We assume that g belongs to L2(T).

Equation (1.1) can be used to model wave propagation in heterogeneous media,and therefore it is important in many applications such as geophysics, seismology,and electromagnetics [31, 9, 33, 5]. In geophysical and seismic prospecting, Kstands for the bulk modulus, the density and u the unknown pressure. Thevelocity c and acoustic impedance are given by c =

K/ and =

K.

Wave propagation in heterogeneous media involves many different spatialscales. Therefore, a direct simulation of the highly heterogeneous media is dif-

ficult if not impossible even with modern state-of-the-art supercomputers. That iswhy we want to use multiscale methods to solve (1.1) on the coarse spatial scales.More precisely, we want to know how to transfer information from fine scales tocoarse scales, and how to use the information obtained to solve the coarse scaleproblem with much fewer degrees of freedom. We often refer this procedure asnumerical homogenization or numerical upscaling.

The idea of using oscillating tests functions in relation to homogenizationcan be backtracked to the work of Murat and Tartar on homogenization and H-convergence, in particular we refer to [32] and [26] (recall also that the frameworkof H-convergence is independent from ergodicity or scale separation assumptions).The numerical implementation and practical application of oscillating test func-tions in finite element numerical homogenization have been called multi-scale finiteelement methods (MsFEM) and have been studied by numerous authors [6], [11],[27], [19], [16], [14], [15], [2]. In particular, numerical schemes have been developedto solve the acoustic wave equation with discontinuous coefficients, for examplein [9] by nonconforming finite element method and in [5] by domain decomposi-tion. Numerical homogenization or numerical upscaling methods such as [33] areproposed for wave equation with heterogeneous coefficients recently.

The finite element method proposed in this paper is closer in spirit to thework of Hou and Wu [19] and Allaire and Brizzi [2]. It is based on a technique firstintroduced in [29] for elliptic equations and extended in [28] to parabolic equations

characterised by a continuum of scales in space and time. The main difference liesin the fact that, instead of solving a local cell problem to get a basis function asin MsFEM or to calculate effective media property as in upscaling method [13],

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we use a global change of coordinates. The global change of coordinates allowsto avoid the so called cell resonance problem and obtain a scheme convergingin situations where the medium has no separation between scales. This makesour method amenable to problems with strongly non-local medias, such as highconductivity channels.

We use a composition rule to construct the finite element space. Allaire andBrizzi [2] have introduced the composition rule in the multi-scale finite elementformulation, and have observed that a multi-scale finite element method withhigher order Lagrange polynomials has a higher accuracy. In fact, I. Babuska etal. introduced the so called change of variable technique [6] in the general settingof partition of unity method (PUM) with p-version of finite elements. Throughthe change of variable, the original problem is mapped into a problem which canbe better approximated.

The main difference with parabolic equations [28] lies in the fact that withhyperbolic equations, energy is conserved and after homogenization there is nohope of recovering the energy (or information) lying in the highest frequencies.However when the medium is highly heterogeneous the eigenfunctions associated

to the highest frequencies are localized, thus energy is mainly transported by thelowest frequencies. That is why, when one is only interested in the large scaletransport of energy it is natural to approximate the solutions of (1.1) by thesolutions of an homogenized operator. For localization of waves in heterogeneousmedia, we refer to [30, 3, 22, 23, 24].

This paper is organized as three parts. In the next section, we present theformulation of the mathematical problem and numerical methods, and also showmain results. In Section 3, we will give the detailed proof and explanation ofthe results in Section 2. In Section 4, we present several numerical examples andconclusions.

2 Main Results

In general, the approximation power of finite element method is subject to thebest approximation for an exact solution with respect to the finite element space.Therefore, we require some smoothness of the solution to prove convergence the-orems. That is one of the reasons why standard methods are not applicablefor problems with heterogeneous media. For example, in (1.1), we only haveu L(0, T , H 1()), and we can not gain anything if we approximate the so-lution with usual C0 or C1 finite element basis. However, as in [29], we can findharmonic coordinates which the solution of the wave equation is smoothly depen-dent on. We call this compensation phenomena.

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2.1 Compensation Phenomena

Let us focus on space dimension n = 2. The extension to higher dimension isstraightforward conditioned on the stability of . Let F := (F1, F2) be the har-monic coordinates satisfying

div aF = 0 in ,F(x) = x on . (2.1)Let := tF aF and

:= esssupx

max(x)min

(x)

. (2.2)We assume that satisfies Cordes type condition < and (Trace[])1

L().We write L2

0, T; H10()

the Sobolev space associated to the norm

v2L2(0,T;H10 ()) := T0

v(., t)2H10 () dt (2.3)We also write

vL(0,T,H2()) = esssup0tT

i,j

ijv(x, t)

2dx 1

2

. (2.4)

We require the right hand side g, initial value u(x, 0) and ut(x, 0) to be smoothenough. For example, we can made the following assumptions,

Assumption A. Assume that the g satisfies tg

L2(T), g

L(0, T , L2()),

initial data u(x, 0) and tu(x, 0) satisfy tu(x, 0) H1() and a(x)u(x, 0) L2() or equivalently 2t u(x, 0) L2().

We have the following theorem,

Theorem 2.1. Suppose that Assumption A holds, then u F1 L(0, T , H 2()) and

u F1L(0,T,H2()) CgL(0,T,L2()) + tgL2(T) + tu(x, 0)H1()

+ 2t u(x, 0)L2()

.(2.5)

The constant C can be written

C = C(n, , Kmin, Kmax, min(a), max(a))

(1 1

2 )

(Trace[])1L(T)

. (2.6)

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Remark 2.1. We have gained one more order of integrability in the harmonic co-ordinates since u L(0, T , H 1()). The condition g L2(T) is sufficient toobtain Theorem 2.1 and the following theorems. For the sake of clarity we havepreferred to restrict ourselves to g L(0, T , L2()).

2.2 Numerical Homogenization in Space

Suppose we have a quasi-uniform mesh. Let Xh be a finite dimensional subspaceof H10 () W1,() with the following approximation properties: There exists aconstant CX such that

Interpolation property, i.e., for all f H2() H10()infvXh

f vH10() CXhfH2(). (2.7)

Inverse Sobolev inequality, i.e., for all v Xh,

vH2

() CXh

1

vH

1

0(), (2.8)

andvH1

0() CXh1vL2(). (2.9)

These properties are known to be satisfied when Xh is a C1 finite element space.One possibility is to use weighted extended B-splines (WEB) method developedby K. Hollig in [17, 18], these elements are in general C1-continuous. They are ob-tained from tensor products of one dimensional elements. The Dirichlet boundarycondition is satisfied using a smooth weight function , such that = 0 at theboundary.

WriteVh :=

F(x) : Xh. (2.10)

For v H10 () write Rhv the projection ofv on Vh with respect to the bilinearoperator a[, ], i.e., the unique element of Vh such that for all w Vh,

a[w, v Rhv] = 0. (2.11)Define YhT the subspace of L

2

0, T; H10()

as

YhT := {v L2

0, T; H10()

: v(x, t) Vh, t [0, T]}. (2.12)We use the following notation

a[v, w] :=

tv(x, t)a(x)w(x, t) dx. (2.13)

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Write uh the solution in YhT of the following system of ordinary differential

equations:

(K1, 2t uh)L2() + a[, uh] = (, g)L2() for all t (0, T) and Vh,uh(x, 0) = Rhu(x, 0),tuh(x, 0) = Rhtu(x, 0).

(2.14)The following theorem shows the accuracy of the semidiscrete solution. We

need an improved assumption on the regularity of the forcing term g and theinitial data, which guarantees O(h) convergence of the scheme (2.14). On theother hand, we can see that even if g and all the initial data are smooth, underthe assumption a(x) L, we can at most expect u L(0, T , H 1()) instead ofthe improved regularity L(0, T , H 2()) in the harmonic coordinates.

Assumption B. Assume that the forcing term g satisfies 2t g L2(T), tg L(0, T , L2()), initial value u(x, 0) and tu(x, 0) satisfy

2t u(x, 0) H1() and

a(x)

tu(x, 0)

L2() or equivalently 3

t

u(x, 0)

L2().From now on we will always suppose without explicitly mentioning that As-

sumption B is satisfied in the discussion of numerical homogenization method.

Theorem 2.2. Suppose that Assumption B holds, we havet(u uh)(., T)L2() + (u uh)(., T)H10()

ChtgL(0,T,L2()) + 2t gL2(T)+2t u(x, 0)H1() + 3t u(x, 0)L2()

.

(2.15)

The constant C depends on CX, n, , , Kmin, Kmax, min(a), max(a), and(Trace[])1L(T).2.3 Numerical Homogenization in Time and Space

Let M N, (tn = n TM)0nM is a discretization of [0, T]. (i) is a basis of Xh.Write trial space ZhT the subspace of Y

hT such that

ZhT = {w YhT : w(x, t) =i

ci(t)i

F(x, t)

, ci(t) are linear on (tn, tn+1]}.(2.16)

Write test space UhT the subspace of YhT such that

UhT = { YhT : (x, t) =i

dii

F(x, t)

, di are linear (on [0, T]).}. (2.17)

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Write vh the solution in ZhT of the following system of implicit weak formulation:

for n {0, . . . , M 1} and UhT,

(K1, tvh)(tn+1) (K1, tvh)(tn) =tn+1tn

(K1t, tvh)dt

tn+1

tn

a[, vh]dt + tn+1

tn

(, g)dt.

(2.18)

In equation (2.18), tvh(t) stands for lim0(vh(t)vh(t ))/. The unknownsare tvh(t), once we know the values of them at tn, we can use the following relationto obtain vh(tn+1),

tvh(tn+1) =vh(tn+1) vh(tn)

tn+1 tn . (2.19)

The trial function space ZhT and test function space UhT introduced in this

section are of different degrees of freedom, therefore we are solving a least squareproblem at each time step.

Denote mass matrix M by Mij = i(F(x)), j(F(x) and stiffness matrix Kby Kij = a[i(F(x)), j(F(x))], explicit calculation shows that (2.18) is equivalentto a linear equation

Ce = f, (2.20)

with the matrix C

C =

M + t

2

2K

tnM +t2(2tn+1+tn)

6K

. (2.21)

Since M and K are positive definite, the least square problem has a uniquesolution, which also gives the existence and uniqueness of vh. The computationalcost of solving the least square problem only depends the degrees of freedom of

the coarse mesh.The following Theorem 2.3 shows the stability of the implicit scheme (2.18):

Theorem 2.3.

tvh(., T)L2()+vh(., T)H10() C

gL2(T) + tu(x, 0)L2()+ u(x, 0)H1()

+ Ch

tgL(0,T,L2()) + 2t gL2(T)+ 2t u(x, 0)H1() + 3t u(x, 0)L2()

.

(2.22)

The constant C depends on min(a), max(a), Kmin, Kmax, and T.

The following Theorems 2.4 and 2.5 give us the accuracy of the implicit scheme(2.18).

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Theorem 2.4.(tuh tvh)(., T)L2() + (uh vh)(., T)H1() Ct(1 + h1)tgL(0,T,L2()) + 2t gL2(T) + 2t u(x, 0)H1() + 3t u(x, 0)L2(). (2.23)The constant C depends on CX , T, min(a), max(a), Kmin, Kmax, , and

(Trace[])1L(T).Theorem 2.5.(uh vh)(., T)L2() +

T0

(uh vh)(., t) dtH10()

Cth

(t)1

2tgL(0,T,L2()) + 2t gL2(T) + 2t u(x, 0)H1() + 3t u(x, 0)L2(). (2.24)The constant C depends on CX , T, min(a), max(a), Kmin, Kmax, ,(Trace[])1

L(T), andtvhL(0,T,L2()) (which is bounded by Theorem 2.3) .

3 Proofs

The proofs are organized into three subsections corresponding to the three subsec-tions of section 2.

3.1 Compensation Phenomena: Proof of Theorem 2.1

Lemma 3.1. We have

2t u2L2()(T) + a[tu](T) C(T,KmaxKmin

, Kmax)

a[tu](0)

+

2t u(x, 0)

2L2() +

tg

2L2(T).

(3.1)

Proof. In case a is smooth, differentiating (1.1) with respect to t, we have

K13t u div atu = tg. (3.2)multiplying by 2t u, and integrating over , we obtain that

1

2

d

dtK 122t u2L2() +

1

2

d

dta[tu] = (tg,

2t u)L2(). (3.3)

Integrating the latter equation with respect to t and using Cauchy-Schwartzinequality we obtain that

K 12 2t u2L2()(T) + a[tu](T) K122t u2L2()(0) + a[tu](0)

+ tgL2(T)2t uL2(T).(3.4)

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Consider the following differential inequality,

X(t) A(t) + B(t) t0

X(s) ds 12 . (3.5)

Write Y(t) = sups[0,t] X(s), one has

X(t) A(t) + B(t)t1

2Y(t) 12 . (3.6)Let t be a time such that Y(t) = X(t), then

Y(t) 2A(t) + tB(t)2. (3.7)It follows that

2t u2L2()(T) + a[tu](T) C(T,KmaxKmin

, Kmax)

a[tu](0)

+ 2t u2L2()(0) + tg2L2(T).(3.8)

In the case where a is nonsmooth we use Galerkin approximations ofu in (1.1)and then pass to limit. This technique is standard and we refer to [12, Section7.3.2.c] for a reminder.

Lemma 3.2.

tu2L2()(T) + a[u](T) C(T,KmaxKmin

, Kmax)

a[u](0)

+ tu2L2()(0) + g2L2(T)

.(3.9)

Proof. Multiplying 1.1 by tu, and integrating over , we obtain that1

2

d

dtK12tu2L2() +

1

2

d

dta[u] = (g, tu)L2(). (3.10)

The remaining part of the proof is similar to the proof of Lemma 3.1.

We now need a variation of Campanatos result [10] on non-divergence formelliptic operators. Let us write for a symmetric matrix M,

M :=Trace(M)

Trace(tMM). (3.11)

We consider the following Dirichlet problem:

LMv = f (3.12)

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with LM :=2

i,j=1 Mij(x)ij . The following Theorem (3.1) is an adaptationof theorem 1.2.1 of [25]. They are proven in [25] under the assumption that Mis bounded and elliptic. It is easy to check that the conditions M < 1 andM L() are sufficient for the validity of those theorems. We refer to [29] and[28] for that adaptation.

Theorem 3.1. Assume that M < 1, M

L() and is convex. If f

L2()

the Dirichlet problem (3.12) has a unique solution satisfying

vH2() C1

1

2

M

MfL2(). (3.13)

Remark 3.1. M is the Cordes parameter associated to M defined by

betaM := esssup

n (Trace[M])2

Trace[tMM]

(3.14)

Let us now prove the compensation theorem Theorem 2.1. Choose

M := | det(F)| 12 F1. (3.15)

It is easy to check that (3.15) is well defined. Moreover observe that M = and

M2L(T) C

min(a)

(Trace[])12L(T)

. (3.16)

Fix t [0, T]. Choose

f :=(K12t u g)

|det(

F)|1

2

F1. (3.17)

Observe that by the change of variable y = F(x) one obtains that

f2L2() K1min2t uL2() + g2L2(). (3.18)

It follows from Theorem 3.1 that there exists a unique v W2,2() satisfyingthe equation

i,j

(F1(y))

i,j

ijv(y, t) = K1(y)2t u(y, t) g(y, t), (3.19)

and v2H2() CM2L(T)

(1 1

2 )2

K1min2t uL2() + g2L2()

. (3.20)

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Use the notation K := K F1, g := g F1, and u := u F1. By changeof variable y = F(x) and the identity div aF = 0 we deduce that (3.19) can bewritten as

div

a(v F) = K12t u g. (3.21)If 2t u

L2() and g(., t)

L2() we can use the uniqueness property for

the solution of a divergence form elliptic equation with homogeneous Dirichletboundary condition.

div

aw = K12t u g. (3.22)to obtain that v F = u. Thus we have proven Theorem 2.1.

3.2 Numerical Homogenization in Space: Proof of Theo-

rem 2.2.

In the following sections we will prove the convergence of semidiscrete and fully

discrete numerical homogenization formulation (2.14) and (2.18).

Lemma 3.3. We have

2t uh2L2()(T) + a[tuh](T) C(T,KmaxKmin

, Kmax)

a[tuh](0)

+ 2t uh(x, 0)2L2() + tg2L2(T)

.(3.23)

Proof. The proof is similar to the proof of Lemma 3.1.

Lemma 3.4.

tuh2L2()(T) + a[uh](T) C(T, KmaxKmin , Kmax)a[uh](0)

+ tuh2L2()(0) + g2L2()T

.

(3.24)

Proof. The proof is similar to the proof of Lemma 3.2.

Write Rh the projection operator mapping L2(0, T; H10()) onto YhT , for allv YhT :

AT[v, uRhu] = 0 (3.25)let := uRhu and := Rhu uh.

For t

[0, T] and v

H10 (), we writeR

h,tv(., t) the solution of:

ta(x)(v Rh,tv(x, t)) dx = 0 for all Vh (3.26)

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It is obvious that Rhu(., t) = Rh,tu(., t). For example, we can choose a seriesof test functions which is separable in space and time, v(x, t) = T(t)X(x), T(t) issmooth in t and has (t) function as its weak limit.

We need the following lemma:

Lemma 3.5. For v H10 () we have

a[v Rhv] 12 Chmax(a) 12 14vW2,2D

(3.27)

Proof. Using the change of coordinates y = F(x) we obtain that (write v :=v F1)

a[v] = Q[v] (3.28)

with

Q[w] :=

tw(y)Q(y)w(y) dy (3.29)

andQ(y) :=

det(F) F1. (3.30)

Using the definition ofRhv we derive that

Q[v Rhv F1] = infXh

Q[v ]. (3.31)

Using interpolation property (2.7) it follows,

Q[v Rhv F1] max(Q)C2Xh2v2W2,2D

(). (3.32)

It is easy to obtain that

max(Q) Cmax(a)1

2 (3.33)

which finishes the proof.

We will use the Lemmas 3.5, 3.6, 3.7, 3.8 and 3.9 to obtain the approximationproperty of the projection operator Rh.Lemma 3.6. t(u F1) L(0, T , H 2()) and

t(u F1)L(0,T,H2()) CtgL(0,T,L2()) + 2t gL2(T)

+

3t u(x, 0)

L2() +

2t u(x, 0)

H1().

(3.34)

Remark 3.2. The constant C is the one given in Theorem 2.1.

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Proof. The proof follows from the differentiation of (1.1) with respect to t and issimilar to the proof of Theorem 2.1.

Lemma 3.7. We haveAT[t] 12 ChtgL(0,T,L2()) + 2t gL2(T)+

3t u(x, 0)

L2() +

2t u(x, 0)

H1().

(3.35)

Proof. The proof is a straightforward application of Lemma 3.6 and Lemma 3.5.

Lemma 3.8. We have

tL2(T) Ch2tgL(0,T,L2()) + 2t gL2(T)

+3t u(x, 0)L2() + 2t u(x, 0)H1()

.(3.36)

The constant C in Lemma 3.7 and 3.8 depends on CX , n, , , min(a),max(a), Kmin, Kmax, and

(Trace[])1

L(T)

. If n 3 it also depends on

Trace[]L(T).Proof. The proof follows from standard Aubin-Nitsche duality techniques [4]. Wecan deduce that, AT[v Rhv] 12 ChtL2(T). (3.37)

SincetL2(T) Ch

AT[t] 12 . (3.38)We deduce the lemma by applying Lemma 3.7 to bound AT[t].

Lemma 3.9. We have the following estimates for initial data,

t(uh(x, 0) u(x, 0))L2

() Ch2tg(x, 0)L2() + 2t u(x, 0)H1() + 3t u(x, 0)L2()uh(x, 0) u(x, 0)H1

0() Ch

tg(x, 0)L2() + 2t u(x, 0)H1() + 3t u(x, 0)L2()(3.39)

Proof. We can estimate tL2() similar to Lemma 3.8 and apply Lemma 3.5 toget the second inequality.

Lemma 3.10. We have

t(u uh)2L2()(T) + a[u uh](T) C(Kmin, Kmax, T)t(u uh)2L2()(0)

+a[u uh](0) + tL2(T)2t (u uh)L2(T) + AT[t].(3.40)

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Proof. We have(K1, 2t (u uh)) + a[, u uh] = 0. (3.41)

Let = t = t(u uh) t, it results1

2

d

dtK12t(u uh)2L2() +

1

2

d

dta[u uh] = (K1t, 2t (u uh)) + a[t, u uh].

(3.42)Integrate with respect to t, using Cauchy-Schwartz inequality, we have

1

2K 12t(u uh)2L2()(T)

1

2K 12t(u uh)2L2()(0) +

1

2a[u uh](T)

12

a[u uh](0) T0

K1mintL2()2t (u uh)L2()dt +AT[t]AT[u uh] 12 .

(3.43)

The remaining part of the proof is similar to the proof of Lemma 3.1.

Theorem 2.2 is a straightforward application of Lemma 3.1, Lemma 3.3, Lemma

3.7, Lemma 3.8, Lemma 3.9, and Lemma 3.10.

3.3 Numerical Homogenization in Space and Time: Proof

of Theorem 2.32.5

Stability Choose UhT in equation (2.18) such that (x, t) = tvh(x, t) fort (tn, tn+1]. We obtain that

K12tvh2L2()(tn+1) (K1tvh(tn+1), tvh(tn))L2() = tn+1tn

a[tvh, vh]dt

+tn+1tn

(tvh, g)L2()dt.

(3.44)

Observing thattn+1tn

a[tvh, vh]dt =1

2a[vh](tn+1) 1

2a[vh](tn). (3.45)

using Cauchy-Schwartz inequality we obtain

K

1

2tvh

2(tn+1

) + a[vh

](tn+1

)

K1

2 tvh

2(tn

) + a[vh

](tn

)

+2

tn+1tn

(tvh, g)L2()(t) dt.(3.46)

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Summing from 0 to M,

K 12tvh2(T) + a[vh](T) K 12tvh2(0) + a[vh](0) + 2T0

(tvh, g)L2() dt.

(3.47)We conclude the proof of Theorem 2.3 in a way similar to the proof of Lemma

3.1.

H1 Error Estimate We derive from equations (2.18) and (2.14) that

(K1, tuh tvh)(tn+1) (K1, tuh tvh)(tn)

tn+1tn

(K1t, tuh tvh)dt +tn+1tn

a[, uh vh]dt = 0.(3.48)

Let = tuh tvh where uh is the linear interpolation of uh over ZhT. Writeyh = uh vh and wh = uh uh, it follows that

(K1tyh, tyh)(tn+1) + (K1twh, tyh)(tn+1) (K1tyh, tyh)(tn)

(K1

twh, tyh)(tn) + tn+1

tna[tyh, yh]dt +

tn+1

tna[twh, yh]dt = 0.

(3.49)

Observingtn+1tn

twh(x, t) dt = 0 we need the following lemma:

Lemma 3.11. Iftn+1tn

u(s) ds = 0, then

u2 14

t

tn+1tn

u(s)2 ds. (3.50)

Proof. We have

u(t)2 = (u(t) 1t

tn+1

tn

u(s)ds)2

=1

t2(

tn+1tn

(

ts

u()d)ds)2

1t2

(

tn+1tn

12ds)(

tn+1tn

(

ts

(u())d)2ds)

=1

t

tn+1tn

(

ts

(u())d)2ds

1t

tn+1tn

|ts

12d| |ts

(u())2d|ds

=t

4tn+1tn

(u

(s))2

ds.

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Since 2t wh(x, t) = 2t uh(x, t) in (tn, tn+1], by Lemma 3.11 we have

|twh(x, t)|2 dxdt 14

t

tn+1tn

|2t uh(x, t)|2 dxdt, (3.51)

and

tn+1tn

|twh(x, t)|2 dxdt

1

4t2 tn+1

tn

|2t uh(x, t)|2 dxdt. (3.52)

Using the inverse Sobolev inequality (2.9) we obtain from equation (3.52) thattn+1tn

a[twh] dxdt Ct2

h2

tn+1tn

|2t uh(x, t)|2 dxdt. (3.53)

Summing (3.49) over n, notice yh(0) = 0, tyh(0) = 0 we obtain that

(K1tyh, tyh)L2()(T) +1

2a[yh(., T)] =

T

0

a[twh, yh]dt (K1(twh, tyh)L2()(T).(3.54)

Theorem 2.4 is a straightforward consequence of (3.54), the estimates (3.51),(3.53), Lemma 3.3 and Lemma 3.9.

L2 Error Estimate Following [8], write y(t) :=Tt

(uhvh)ds and (t) the linearinterpolation ofy(t) on ZhT. Write z(t) = y(t)(t). Using the test function in(3.48) we obtain that

(K1(y z), t(uh vh))(tn+1) (K1(y z), t(uh vh))(tn)

+ tn+1tn

(K1(uh vh + tz), t(uh vh))dt + tn+1tn

a[y z, uh vh)]dt = 0(3.55)

Observe thattn+1tn

(K1(uhvh), t(uhvh))dt = 12K12 (uhvh)2(tn+1)1

2K 12 (uhvh)2(tn).

(3.56)Moreover

tn+1

tn

a[y, uh

vh]dt =

tn+1

tn

1

2

d

dta[y]dt

= 12

a[y](tn+1) +1

2a[y](tn).

(3.57)

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Write

In+1 := (K1(yz), t(uhvh))(tn+1)+1

2K12 (uhvh)2(tn+1)1

2a[

Ttn+1

(uhvh)ds].(3.58)

We have

In+1 = In tn+1tn

(K1tz, t(uh vh))dt + tn+1tn

a[z, uh vh)]dt.

It follows by induction that

In+1 = I0 tn+10

(K1tz, t(uh vh))dt J1

+

tn+10

a[z, uh vh)]dt J2

. (3.59)

Choose T = tn+1. Observe that vh(0) = uh(0), tvh(0) = tuh(0), I0 =

12

a[

T

0(uh vh)ds], and In+1 = 12K

1

2 (uh vh)2(T). Moreover using Theorems2.3, 2.4 and Lemma 3.4, it follows that

tzL

2

()is bounded by t2(1 + h1)

and t(uh vh)L2() is bounded by t(1 + h1), hence we can deduce thatJ1 Ct3/h2.

Similarly we obtain that

J2 T0

a[z]1

2 a[uh vh] 12 dt

Cth

supt[0,T]

uh vhH1()T0

uh vhH1()dt

Ch2

t3.

(3.60)

This concludes the proof of Theorem 2.5.

4 Numerical Experiments

In this section, we will first give the numerical algorithm. Several examples willbe presented. The computational domain is the unit square in dimension two.Equation (1.1) is solved on a fine tessellation characterized by 16129 interior nodes(degrees of freedom). Three different coarse tessellations with 9, 49, and 225degrees of freedom (dof) are considered.

We use web extended B-spline based finite element [17] to be the space Xh

introduced in subsection 2.2. For all the numerical examples, we compute thesolutions up to time T = 1. The initial condition is u(x, 0) = 0 and ut(x, 0) = 0.

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The fine mesh solver for the wave equation is Matlab routine hyperbolic, whichuses linear finite element basis in space and adaptive ODE integrator in time. Thefine mesh solver for F is Matlab routine assempde.

Algorithm 4.1 (Algorithm for Numerical Homogenization).

1. Compute F on fine mesh.

2. Construct multi-scale finite element basis = F, compute stiffness matrix K andmass matrix M ( is piecewise linear on the fine mesh).3. Time march (2.18) and (2.19) on coarse mesh. Repeat 3 if we have multiple right hand

sides.

Example 4.1. Time independent site percolation

In this example we consider the site percolating medium associated to figure1. Equation (1.1) is solved with g = 1, g = sin(2.4x 1.8y + 2t), and a Gaussiansource function given by

g(x, y) =1

22exp

x2 + (y 0.15)222 (4.1)

with = 0.05. Notice that as 0, the source function will become moresingular. We have used following numerical homogenization methods,

LFEM: A multi-scale finite element where F is computed locally (insteadof globally) on each triangle K of the coarse mesh as the solution of a cellproblem with boundary condition F(x) = x on K. This method has beenimplemented in order to understand the effect of the removal of global infor-mation in the structure of the metric induced by F.

FEM lin: The Galerkin scheme using the finite elements i = i F, wherei are the piecewise linear nodal basis elements.

FEM sp: The Galerkin scheme using the finite element i = i F, wherei are weighted cubic B-spline elements.

Figure 2 shows u computed on 16129 interior nodes and uh computed on 9 interiornodes in the case g = 1 at time 1 using method FEM sp. Numerical errors aregiven in Tables 16.

Tables 1 and 2 compare permance of different numerical homogenization meth-ods with coarse mesh degree of freedom 49. We observe that the methods usingglobal F have much better performance, and FEM sp is better than FEM lin.

Tables 3 and 4 present the error for time dependent forcing term using method

FEM sp. Tables 5 and 6 present the error for relatively singular forcing termusing FEM sp. Generally speaking, we have reasonable errors if forcing terms aresmooth.

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Figure 1: Site percolation

Table 6: Fine mesh error: Time independent site percolation with Gaussian sourceg

dof L1 L L2 H1

9 0.0977 0.4595 0.1192 0.485749 0.0927 0.4144 0.1102 0.3857

225 0.0866 0.2030 0.1098 0.3802

(a) u. (b) uh.

Figure 2: u computed on 16129 interior nodes and uh computed on 9 interior nodes

at time 1

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Table 7: Coarse mesh error (dof 49): High conductivity channel with differentaspect ratio

A L1 L L2 H1

10 0.0041 0.0197 0.0083 0.0208100 0.0080 0.0459 0.0126 0.0492

1000 0.0349 0.0934 0.0484 0.1051

Example 4.2. Time independent high conductivity channel

High conductivity channel is an interesting test problem in the petroleum ap-plications because of its strong non-local effects. In this example a is characterizedby a narrow and long ranged high conductivity channel. We choose a(x) = A 1,ifx is in the channel, and a(x) = O(1) and random, ifx is not in the channel. Themedia is illustrated in figure 3.

Figure 3: High conductivity channel superposed on a random medium

Tables 7 and 8 give the coarse and fine meshes errors for g = 1 with fixed coarsemesh (dof 49) and A = 101, 102, 103 respectively. From the table we can see thatthe errors grow with A increasing, but the growth is moderate and the numericalbehavior of the method is stable. The fine mesh and coarse mesh errors for timedependent right hand side g = sin(2.4x 1.8y + 2t) with A = 102 are also givenin tables 9 and 10.

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Table 11: Coarse mesh error: Neumann boundary condition, fine mesh dof 16129dof L1 L L2 H1

9 0.0468 0.893 0.0506 0.132049 0.0138 0.0713 0.0166 0.1353

225 0.0094 0.0436 0.0115 0.1778

Example 4.3. Time dependent source with Neumann boundary condition

In this example we again consider the site percolating medium, but with Neu-mann boundary condition and a more realistic forcing term. The source term isgiven by g(x, t) = T(t)X(x, y), X(x, y) is the Gaussian source function describedby

X(x, y) =1

22exp

x2 + y222

, (4.2)

with = 0.05, T(t) = T1(t)T2(t)

T1(t) =101

21 (1)k

ksin(2kt), (4.3)

and T2(t) = erfc(8(t 0.5)), erfc is the complementary error function. We usethis source term to emulate a source acting around the origin before t = 0.5, thensuddenly decays. See figure 4 for T(t) in (0, 1).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0.5

0

0.5

1

1.5

2

2.5

Figure 4: t g(0, t)

Errors are given in tables 11 and 12 for fine mesh with dof 16129, in tables 13 and14 for fine mesh with dof 65025. Figure 5 shows u computed on 16129 interiornodes and uh computed on 9 interior nodes at time 1.

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5 Conclusion and Further Remarks

From above analysis and numerical examples, for accoustic wave equation withheterogeneous coefficients, we can get reasonable approximation with much fewerdegrees of freedom. Compared with the multiscale finite element method whichcompute the basis locally, our method has much better accuracy, especially forproblems with strong non-local effects.

Actually, as it has been done in [29], once one understand that the key ideafor the homogenization of (1.1) lies in its higher regularity properties with respectto harmonic coordinates one can homogenize (1.1) through a different numericalmethod (such as a finite volume method).

Moreover, it could be observed that one could use any set of n linearly inde-pendent solutions of (1.1) instead of the harmonic coordinates. The key propertyallowing the homogenization of (1.1) lies in the fact that if the data (right handside and initial values) has enough integrability then the space of solutions is atsmall scales close in H1 norm to a space of dimension n. Thus once one has ob-served at least n linearly independent solutions of (1.1), one has seen all of them

at small scales. Let us further explain this in the following:Write L := a. L1 maps H1() into H10(), it also maps L2() into Va sub-vector space ofH10(). The elements of V is close in H

1 norm to a space ofdimension n (the dimension of the physical space ) in the following sense.

Let Th be a triangulation of Rn of resolution h (where 0 < h < diam()).Let set of mappings from Th into the unit sphere ofRn+1 (if then isconstant on each triangle K Th and (K) = 1), then

supv1,v2,...,vn+1V

inf

n+1i=1 iviH10 ()

n+1i=1 aviL2()

Ch (5.1)

Equation (5.1) is saying that any n+1 elements ofV are (at an h approximationin H1 norm) linearly dependent. Recall that n + 1 vectors are linearly dependentin a linear combination (with non null coefficients) of these vectors in the nullvector. In (5.1) the linear combination of the n + 1 vectors is at relative distanceof order h (resolution of the triangulation) from 0.

We notice that some recent results using global information [1, 21, 20] areformulated in a partition of unity framework [7]. In this case, {1, F1, , Fn} canbe used to construct the local approximation space.

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