high dimensional finite elements for multiscale wave equations

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

MULTISCALE MODEL. SIMUL. c© 2014 Society for Industrial and Applied MathematicsVol. 12, No. 4, pp. 1622–1666

HIGH DIMENSIONAL FINITE ELEMENTS FOR MULTISCALEWAVE EQUATIONS∗

BINGXING XIA† AND VIET HA HOANG†

Abstract. For locally periodic multiscale wave equations in Rd that depend on a macroscopicscale and n microscopic separated scales, we solve the high dimensional limiting multiscale homog-enized problem that is posed in (n + 1)d dimensions and is obtained by multiscale convergence.We consider the full and sparse tensor product finite element methods, and analyze both the spa-tial semidiscrete and the fully (both temporal and spatial) discrete approximating problems. Withsufficient regularity, the sparse tensor product approximation achieves a convergence rate essentiallyequal to that for the full tensor product approximation, but requires only an essentially equal numberof degrees of freedom as for solving an equation in Rd for the same level of accuracy. For the initialcondition u(0, x) = 0, we construct a numerical corrector from the finite element solution. In thecase of two scales, we derive an explicit homogenization error which, together with the finite elementerror, produces an explicit rate of convergence for the numerical corrector. Numerical examples fortwo- and three-scale problems in one or two dimensions confirm our analysis.

Key words. multiscale wave equations, homogenization, finite element, high dimension, sparsetensor product, corrector

AMS subject classifications. 35B27, 35L20, 65N30, 74Q10

DOI. 10.1137/120902409

1. Introduction. Wave propagation in heterogeneous media arises in many im-portant practical and engineering areas. As for other multiscale problems, a directnumerical simulation requires resolving all the fine scales and needs a mesh size thatis at most of the order of the smallest scale; therefore it is prohibitively expensive.The theory of homogenization studies the limit when the fine scales converge to zerowhich approximates the solution to the original multiscale problem macroscopically.To establish the limiting homogenized equation, we need the solutions of local cellproblems (see, for example, Benssousan, Lions, and Papanicolaou [4]). The totalnumber of degrees of freedom and floating point operations needed for solving thesecell problems can be exceedingly large, especially when the multiscale problem is onlylocally periodic and when it depends on many microscopic scales.

For multiscale wave equations, there have been some attempts to address thisissue though when comparing to elliptic and parabolic equations, the literature is stillquite limited. Owhadi and Zhang [22] proposed a general method for d dimensionalproblems that solves d multiscale elliptic problems and uses their solutions to trans-form a macroscopic scale basis to a multiscale one. Though general, the complexity ofestablishing this basis is comparable to that for directly solving the multiscale prob-lem. In [18], Jiang, Efendiev, and Ginting studied a numerical procedure based on theMultiscale Finite Element (FE) method [17] to solve wave equations with continuumspatial scales assuming limited global information. The method is quite general, butthe complexity of constructing the multiscale FE basis can be rather high. Using the

∗Received by the editors December 12, 2012; accepted for publication (in revised form) March 10,2014; published electronically November 6, 2014. BXX is supported by an NTU graduate scholarship,VHH is supported by the AcRF Tier 1 research grant RG69/10, the Singapore A*Star SERC grant122-PSF-0007, and the AcRF Tier 2 grant MOE 2013-T2-1-095, ARC 44/13.

http://www.siam.org/journals/mms/12-4/90240.html†Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang

Technological University, Singapore 637371 ([email protected], [email protected]).

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mailto:[email protected]

mailto:[email protected]


HIGH DIMENSIONAL FEs FOR MULTISCALE WAVE EQUATIONS 1623

Heterogeneous Multiscale Method [9], Engquist, Holst, and Runborg [10] developeda finite difference method for multiscale wave equations. They also performed com-putation for large time that shows dispersive behavior similar to that in Santosa andSymes [23]. The Heterogeneous Multiscale Method for multiscale wave equations wasdeveloped by Abdulle and Grote in [1] to compute the solution of the homogenizedequation. They show that the number of degrees of freedom required is independentof the scale but it grows superlinearly when the discretizing mesh size decreases.

In this paper, we develop the high dimensional approach for locally periodic waveequations that depend on multiple separated scales. The method is based on thesparse high dimensional finite element method developed in Hoang and Schwab [15]for elliptic equations (see also [14] and [13]) which solves the multiscale homogenizedproblems obtained via multiscale convergence [21, 2, 3]. These problems are posed ina tensorized domain and depend on n+1 d dimensional variables when the multiscaleproblems are posed in R

d and depend on one macroscopic scale and n microscopicscales. In comparison to the methods mentioned above, the number of degrees offreedom required grows only log-linearly when the discretizing mesh decreases whilethe rate of convergence is essentially equal to that for the full tensor product finiteelements, whose complexity grows superlinearly. Apart from the log term, the rateof convergence and the complexity are uniform with respect to the number of scales.Further, as shown in [15], the resulting linear system at each time step can be solvedby the conjugate gradient method that requires only a log-linear number of floatingpoint operations and a log-linear number of memory units with respect to the numberof degrees of freedom. We obtain the solution of the homogenized equation which de-scribes the multiscale solution macroscopically, and the scale interacting terms whichprovide the microscopic behavior. Restricting our consideration to the case wherethe initial condition g0 in (2.5) equals 0, we construct a numerical corrector fromthe finite element approximations of these scale interacting terms for the multiscaleproblem. The method thus provides a feasible approach to compute numerically boththe solution to the homogenized equation and the corrector. For two-scale problems,we deduce a homogenization rate of convergence, which, together with the finite el-ement rate of convergence, implies an explicit rate of convergence for the numericalcorrector. For general initial conditions, as shown in [5], it is more complicated toconstruct correctors for multiscale wave equations than for elliptic equations as theenergy of the multiscale wave equations does not always converge to the energy of thehomogenized equations. However, these scale interacting terms always form a part ofthe corrector, albeit another term which requires that solving a separate multiscaleproblem is needed (see [5]).

This paper is organized as follows. In the next section, we define the multi-scale wave equation and recall the concept of multiscale convergence applying totime dependent problems. We then deduce the high dimensional limiting multiscalehomogenized equation of the wave equation. We study the approximation of themultiscale homogenized equation by using general finite element spaces in section 3where we consider both the spatial semidiscrete and the fully discrete problems. Forthe convergence of the general approximating schemes in section 3, and to obtain theconvergence rates for the particular cases of full and sparse tensor FE approximationsin section 5, we require sufficient regularity of the solution of the high dimensionalmultiscale homogenized problem. We show that the regularity can be achieved withsufficiently regular coefficients and domains in section 4. In section 5, we consider thefull tensor product and the sparse tensor product approximations. When the solutionis sufficiently regular, we derive explicit rates of convergence for the finite element

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1624 BINGXING XIA AND VIET HA HOANG

approximations. In particular, we show that while the number of degrees of freedomfor the sparse tensor product approximation is essentially (within a log term) equalto that for solving an equation that depends only on the slow variable, it achievesessentially equal accuracy to that of the full tensor product approximation. To getthe rate of convergence for the numerical corrector constructed in section 6, we need ahomogenization rate of convergence for (2.5). For the two-scale case, when the initialcondition g0 in (2.5) equals 0, adapting the procedure of Jikov, Kozlov, and Oleınik[19] for elliptic equations we deduce the O(ε1/2) homogenization rate of convergencein terms of the microscopic scale ε, although we need to overcome some technicaldifficulty associated with wave equations. To the best of our knowledge, such a ratehas not been established for the homogenization of wave equations. For multiscaleproblems, a homogenization rate is not available. However, in section 6 we can stilldeduce a general corrector for the multiscale homogenized problem. In section 7, weconstruct a numerical corrector using the finite element solution of the high dimen-sional multiscale homogenized problem, and for the case of two scales, we deduce anexplicit rate of convergence for the corrector in terms of both the homogenization rateof convergence and the finite element error. Finally, in section 8, we present somenumerical examples for two- and three-scale problems in one and two dimensions thatconfirm our analysis. We then summarize the paper with some conclusions on thecomplexity of the method. This paper is concluded with some appendices that presentthe long proofs of and remarks on some results in the previous sections.

Throughout this paper, by c we denote a generic constant that does not dependon the scales and the approximating parameters. For a Banach space X , we denoteby Lp(0, T ;X) the Bochner spaces of p-summable functions from (0, T ) to X . By ∇,without specifying explicitly the variable, we denote the gradient with respect to x.For a function q(t, x) that depends on t and x, when we only wish to emphasize thet dependence, we write it as q(t). The symbol # denotes spaces of periodic functionswith respect to the unit cube Y in R

d; in particular, Hk#(Y ) denotes the spaces of

Sobolev spaces of periodic functions in Y and Ck#(Y ) denotes the spaces of periodic

functions that are k times continuously differentiable.

2. Multiscale wave equation. We now introduce the multiscale wave equationfor which we study the homogenization limit via multiscale convergence. We firstpresent the setting up for the problem. We then review the concept of multiscaleconvergence and apply it to our equation.

2.1. Problem setting. Let D ∈ Rd be a bounded domain. Let Y = (0, 1)d be

the unit cube in Rd. For an integer n that denotes the number of microscopic scales

that the problem depends on, let Y1, Y2, . . . , Yn be n copies of Y ; and by Y the spaceof all nd dimensional vectors y = (y1, y2, . . . , yn) such that yi ∈ Yi for i = 1, . . . , n.We denote by A a mapping from the product space D × Y to the space R

d×dsym of

real symmetric matrices such that for all x ∈ D and y ∈ Y, A(x,y) satisfies theboundedness condition

(2.1) |A(x,y)ξ · ζ| ≤ β|ξ||ζ|

and the coerciveness condition

(2.2) α|ξ|2 ≤ A(x,y)ξ · ξ

for all vectors ξ, ζ ∈ Rd where α and β are positive constants. We assume that the

coefficient A is periodic with respect to yi with the period being Yi (i = 1, . . . , n) and

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is sufficiently smooth:

(2.3) A(x,y) ∈ C1(D, C1#(Y1, . . . , C

1#(Yn) . . .)),

where C1#(Yi) denotes the space of continuously differentiable functions that are peri-

odic in Yi. Let ε > 0 be a small quantity. Let ε1(ε), . . . , εn(ε) be n positive functionsof ε that denote the n microscopic scales that the problem depends on. These scalesconverge to 0 when ε→ 0. We assume that they satisfy the scale separation assump-tion, i.e., for i = 1, . . . , n− 1,

(2.4) limε→0

εi+1(ε)

εi(ε)= 0.

Without loss of generality, we let ε1 = ε. The multiscale coefficient Aε : D → Rd×dsym

is defined as

Aε(x) = A

(x,

x

ε1, . . . ,

x

εn

).

We denote V = H10 (D) and H = L2(D). Let g0 ∈ V and g1 ∈ H . For T > 0, let

f(t, x) ∈ L2(0, T ;H). We consider the wave equation in D with Dirichlet boundarycondition: Find uε(t, x) ∈ L2(0, T ;V ) ∩H1(0, T ;H) such that

(2.5)

∂2uε

∂t2−∇ · (Aε(x)∇uε) = f,

uε(0, ·) = g0,∂uε

∂t(0, ·) = g1,

uε(t, x) = 0, for x ∈ ∂D.

Denoting by (·, ·)H the inner product in H , extended to the duality pairing 〈V ′, V 〉,we consider the weak form of (2.5) as

(2.6)

(∂2uε

∂t2, φ

)H

+

∫D

Aε(x)∇uε(t, x) · ∇φ(x)dx =

∫D

f(t, x)φ(x)dx ∀φ ∈ V.

From conditions (2.1) and (2.2), we have the following existence and uniquenessresult.

Proposition 2.1. Problem (2.6) has a unique solution uε ∈ L2(0, T ;V ) ∩H1(0, T ;H) such that

(2.7) ‖uε‖L2(0,T ;V )∩H1(0,T ;H) ≤ c(‖f‖L2(0,T ;H) + ‖g0‖V + ‖g1‖H),

where the constant c only depends on α, β and T .The proof of this proposition is standard; see, for example, Wloka [24]. Indeed,

it can be shown that uε ∈ C([0, T ];V ) ∩ C([0, T ];H). In this paper, we study theasymptotic behavior of uε when ε → 0 via multiscale convergence method. We firstextend the standard notions of Nguetseng [21], Allaire [2], and Allaire and Briane [3]to functions that depend on time variable t.

2.2. Multiscale convergence. We first formulate the concept of multiscaleconvergence for functions that depend on t. The definition is a simple extension ofthat by Allaire and Briane [3].

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Definition 2.2. A sequence of functions uε ∈ L2(0, T ;H) is said to (n+1)-scaleconverge to a function u0 ∈ L2(0, T ;D×Y) if for all smooth functions φ(t, x,y) whichare Yi periodic with respect to yi,

limε→0

∫ T

0

∫D

uε(t, x)φ

(t, x,

x

ε1, . . . ,

x

εn

)dxdt =

∫ T

0

∫D

∫Y

u0(t, x,y)φ(t, x,y)dydxdt.

This definition makes sense because of the following result.Proposition 2.3. From a bounded sequence uε in L2(0, T ;H), there is a subse-

quence (not renumbered) that (n+ 1)-scale converges to a function u0 ∈ L2(0, T ;D×Y).

For a bounded sequence {uε}ε ⊂ L2(0, T ;V ) we have the following proposition.Proposition 2.4. Let {uε}ε ⊂ L2(0, T ;V ) be a bounded sequence. There is

a subsequence (not renumbered), a function u0 ∈ L2(0, T ;V ), and n functions ui ∈L2((0, T )×D × Y1 × . . .× Yi−1, H

1#(Yi)/R) such that uε ⇀ u0 in L2(0, T ;V ) and

∇uε (n+1)−scale−→ ∇u0 +∇y1u1 + . . .+∇ynun.

The proofs of Propositions 2.3 and 2.4 are almost identical to those in Allaire [2]and Allaire and Briane [3].

2.3. Multiscale limit of the wave equation. We study the multiscale con-vergence of the solution to the multiscale wave equation (2.6). For conciseness, fori = 1, . . . , n, we denote by

Vi := L2(D × Y1 × · · · × Yi−1, H1#(Yi)/R),

and by

V := V × V1 × . . .× Vn.

The space V is equipped with the norm: for v = (v0, v1, . . . , vn) ∈ V,

‖v‖V = ‖v0‖V +

n∑i=1

‖vi‖Vi = ‖∇v0‖H +

n∑i=1

‖∇yivi‖L2(D×Y1×···×Yi).

We then have

L2(0, T ;Vi) = L2((0, T )×D × Y1 × · · · × Yi−1, H1#(Yi)/R),

and

L2(0, T ;V) = L2(0, T ;V )×n∏

i=1

L2((0, T )×D × Y1 × . . .× Yi−1, H1#(Yi)/R).

With the bilinear form B : V ×V → R defined by

B(w,v) =

∫D

∫Y

A(x,y)(∇w0+∇y1w1+· · ·+∇ynwn)·(∇v0+∇y1v1+· · ·+∇ynvn)dydx

for all w = (w0, w1, . . . , wn),v = (v0, v1, . . . , vn) in V, we have the following result.

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Theorem 2.5. The solution uε of (2.6) converges weakly to u0 in L2(0, T ;V )and ∇uε (n+1)-scale converges to ∇u0+∇y1u1+ · · ·+∇ynun where ui ∈ L2(0, T ;Vi);u = (u0, u1, . . . , un) is the unique solution of the problem

(2.8)(∂2u0(t)

∂t2, v0

)H+B(u,v) =

∫D

f(t, x)v0(x)dx

for all v = (v0, v1, . . . , vn) ∈ V, with the initial condition

(2.9) u0(0, x) = g0(x),∂u0∂t

(0, x) = g1(x).

Proof. From (2.7), we deduce that there exists u ∈ V such that uε ⇀ u0 inL2(0, T ;V ) and ∇uε (n + 1)-scale converges to ∇u0 + ∇y1u1 + · · · + ∇ynun. Letψ0 ∈ C∞

0 (D); and for i = 1, . . . , n, ψi ∈ C∞0 (D,C∞

# (Y1, . . . C∞# (Yi−1, C

∞# (Yi))) · · · )

and q(t) ∈ C∞0 ((0, T )). Let the test function φ in (2.6) be

φ(t, x) =

(ψ0(x) +

n∑i=1

εiψi

(x,

x

ε1, . . . ,

x

εi

))q(t).

We then obtain∫ T

0

(u0(t), ψ0 +

n∑i=1

εiψi

(·, ·ε1, . . . ,

·εi

))H

d2q(t)

dt2dt

+

∫ T

0

∫D

Aε(x)∇uε(x)

·(∇ψ0 +

n∑i=1

εi

(∇xψi

(x,

x

ε1, . . . ,

x

εi

)+

i∑j=1

1

εj∇yjψi

(x,

x

ε1, . . . ,

x

εi

)))q(t)dxdt

=

∫ T

0

∫D

f(t, x)

(ψ0(x) +

n∑i=1

εiψi

(x,x

ε1, . . . ,

x

εi

))q(t)dxdt.

From (2.4), in the limit ε→ 0, we obtain∫ T

0

(u0(t, ·), ψ0(·)

)H

d2q(t)

dt2dt+

∫ T

0

B(u(t),ψ)q(t)dt =

∫ T

0

∫D

f(t, x)ψ0(x)q(t)dxdt.

As the previous equations holds for all q(t) ∈ C∞0 (0, T ), we deduce(

∂2u0∂t2

(t), ψ0

)H

+B(u(t),ψ) =

∫D

f(t, x)ψ0(x)dx

in the sense of distribution, where

ψ = (ψ0, ψ1, . . . , ψn).

Using a density argument, we deduce (2.8). Initial conditions (2.9) come from thefact that

∂uε

∂t⇀

∂u0∂t

in L2(0, T ;H) and∂2uε

∂t2⇀

∂2u0∂t2

in L2(0, T ;V ′).

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We now show that problem (2.8) has a unique solution, i.e., problem (2.8) withzero initial conditions and with f = 0 only has zero solution. Following the sameprocedure as in Wloka [24, Theorem 29.1], fixing s ∈ (0, T ), we define for each i =0, 1, . . . , n

wi(t) =

{−∫ s

tui(σ)dσ for t < s,

0 for t ≥ s.

Let w = (w0, w1, . . . , wn). We have

d

dt

(∂u0(t)

∂t, w0(t)

)H

=

(∂2u0(t)

∂t2, w0(t)

)H

+

(∂u0(t)

∂t,∂w0(t)

∂t

)H

.

Letting v = w in (2.8), we have∫ s

0

(B(u(t),w(t)) +

d

dt

(∂u0(t)

∂t, w0(t)

)H

−(∂u0(t)

∂t,∂w0(t)

∂t

)H

)dt = 0.

As

B(u(t),w(t)) =1

2

d

dtB(w(t),w(t)) and

(∂u0(t)

∂t,∂w0(t)

∂t

)H

=1

2

d

dt(u0(t), u0(t))H ,

from the fact that ∂u0/∂t(0) = 0, u0(0) = 0, and w(s) = 0, we have

B(w(0),w(0)) + ‖u0(s)‖2H = 0.

This implies that u0(s) = 0 and∫ s

0u(σ)dσ = 0 for all s. From this, we deduce that

u(s) = 0.To illustrate how the solution uε of the multiscale problem (2.5) can be approxi-

mated by the solution u of problem (2.8), we mention the following result for the caseof two scales (n = 1). The proof for it can be found, for example, in [5].

Proposition 2.6. For n = 1, assume that the initial condition g0 = 0. Assumefurther that the solution wi(x, y) of cell problems (6.1) are in C(D,W 1,∞(Y )). Thefollowing corrector result holds

limε→0

∥∥∥∥∥∇uε(·, ·)−(∇u0(·, ·) +∇yu1

(·, ·, ·

ε

))∥∥∥∥∥L∞(0,T ;H)

= 0.

We will study in more details correctors for the homogenization problem (2.8) insection 6.

3. Finite element discretization. In this section, we consider finite elementdiscretization of the high dimensional limiting problem (2.8). To study the generalframework, we assume a general nested sequence of finite element spaces for approx-imating u ∈ V without specifying these spaces explicitly. We will study particularfinite element spaces in the later sections. To approximate u0 ∈ V , we consider ahierarchy of finite element subspaces of V ,

V 1 ⊂ V 2 ⊂ · · · ⊂ V

such that⋃∞

L=1 VL is dense in V ; and for i = 1, . . . , n, to approximate ui, we consider

the finite element subspaces of Vi

V 1i ⊂ V 2

i ⊂ · · · ⊂ Vi

such that⋃∞

L=1 VLi is dense in Vi. We first consider the spatial semidiscrete problem

of (2.8).

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3.1. Spatial semidiscrete problem. With the spaces V L ⊂ V and V Li ⊂ Vi

above, we define by

VL = V L × V L1 × · · · × V L

n ⊂ V.

We consider the spatial semidiscrete approximating problem: Find uL = (uL0 , uL1 , . . . ,

uLn) ∈ VL such that

(3.1)

(∂2uL0∂t2

, vL0

)H

+B(uL,vL) = (f, vL0 )H

for all vL = (vL0 , vL1 , . . . , v

Ln ) ∈ VL, with the initial condition uL0 (0, ·) = gL0 ∈ V L

and∂uL

0

∂t (0, ·) = gL1 ∈ V L. The initial conditions gL0 and gL1 are chosen so that theyapproximate g0 and g1 in V and H , respectively, e.g., as the orthogonal projectionsof g0 and g1 to V L.

Proposition 3.1. Problem (3.1) has a unique solution.Proof. Let M be the Gram matrix for the basis of V L. For the bilinear form

B(uL,vL), let A00 be the matrix describing the interaction of the basis functions ofV L, let A01 be the matrix describing the interaction of the basis functions of V L withthe basis functions of V L

1 × V L2 × · · · × V L

n , and let A11 be the matrix describing theinteraction of the basis functions of V L

1 × V L2 × · · · × V L

n with themselves. Let F bethe column matrix describing the interaction of f and the basis functions of V L. Letc0 be the coefficient vector in the expansion of uL0 with respect to the basis of V L.Let c1 be the coefficient vector in the expansion of (uL1 , . . . , u

Ln) with respect to the

basis of V L1 × · · · × V L

n . We have

Md2c0dt2

+A00c0 +A01c1 = F,

A�01c0 +A11c1 = 0.

As A11 is positively definite, we have

Md2c0dt2

+ (A00 −A01A−111 A

�01)c0 = F.

Since detM = 0, this equation has a unique solution.We analyze problem (3.1) following the approach of Dupont [8]. For each t ∈

[0, T ], let wL(t) = (wL0 , w

L1 , . . . , w

Ln ) ∈ VL be the solution of the problem

B(wL(t)− u(t),vL) = 0(3.2)

for all vL = (vL0 , . . . , vLn ) ∈ VL.

For i = 0, . . . , n, let

(3.3) ηLi = wLi − ui.

We denote by ηL = (ηL0 , ηL1 , . . . , η

Ln ) ∈ V. We then have from (2.8) and (3.2)

(3.4) B(wL,vL) =

(f − ∂2u0

∂t2, vL0

)H

in the distribution sense for all vL = (vL0 , vL1 , . . . , v

Ln ) ∈ VL.

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We have the following estimates for ‖η‖V.Lemma 3.2. For the solution wL of (3.2),

(3.5) ‖ηL(t)‖V ≤ c infvL∈VL

‖u(t)− vL‖V.

Proof. From (3.2), we have

B(wL − u,wL − u) = B(wL − u,vL − u)

for all vL ∈ VL. From (2.1) and (2.2), we get the conclusion.For the development of the finite element convergence in the later sections, we

develop the following estimates for the time derivatives of wL.Lemma 3.3. If ∂u

∂t ∈ C([0, T ];V), then∥∥∥∥∂ηL

∂t

∥∥∥∥L∞(0,T ;V)

≤ c supt∈[0,T ]

infvL∈VL

∥∥∥∥∂u∂t (t)− vL∥∥∥∥V

.

If ∂2u∂t2 ∈ L2(0, T ;V), then∥∥∥∥∂2ηL

∂t2

∥∥∥∥L2(0,T ;V)

≤ c infvL∈L2(0,T ;VL)

∥∥∥∥∂2u∂t2 − vL∥∥∥∥L2(0,T ;V)

.

Proof. If ∂u∂t ∈ C([0, T ];V), from (3.2) we have

(3.6) B

(∂(wL − u)

∂t,vL

)= 0

for all vL ∈ VL. We then proceed as in the proof of Lemma 3.2 to show the firstinequality. The proof for the second inequality is similar.

Let

(3.7) ζL = uL −wL.

We have from (3.1) and (3.4)

(3.8)

(∂2ζL0∂t2

, vL0

)H

+B(ζL,vL) = −(∂2ηL0∂t2

, vL0

)H

for all vL = (vL0 , vL1 , . . . , v

Ln ) ∈ VL. We then have the following estimate.

Lemma 3.4. Assume that u ∈ H1(0, T ;V) and∂2ηL

0

∂t2 ∈ L2(0, T ;H). Then∥∥∥∥∂ζL0∂t∥∥∥∥2L∞(0,T ;H)

+ ‖ζL‖2L∞(0,T ;V)(3.9)

≤ c

[‖∇ζL0 (0)‖2H +

∥∥∥∥∂ζL0∂t (0)

∥∥∥∥2H

+

∥∥∥∥∂2ηL0∂t2

∥∥∥∥2L2(0,T ;H)

].

Proof. As u ∈ H1(0, T ;V), ∂u∂t is well defined and so are ∂wL

∂t and ∂ζL

∂t . Let

vL = ∂ζL

∂t in (3.8),(∂2ζL0∂t2

,∂ζL0∂t

)H

+B

(ζL,

∂ζL

∂t

)= −

(∂2ηL0∂t2

,∂ζL0∂t

)H

,

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then

1

2

d

dt

[(∂ζL0∂t

,∂ζL0∂t

)H

+B(ζL, ζL)

]= −

(∂2ηL0∂t2

,∂ζL0∂t

)H

≤ 1

2γ

∥∥∥∥∂2ηL0∂t2

∥∥∥∥2H

+γ

2

∥∥∥∥∂ζL0∂t∥∥∥∥2H

,

where γ > 0 is chosen to be sufficiently small. Integrating on (0, t), from (2.1) and(2.2) we have

supt∈(0,T )

{∥∥∥∥∂ζL0∂t (t)

∥∥∥∥2H

+ α‖ζL(t)‖V

}

≤ 1

2γ

∥∥∥∥∂2ηL0∂t2

∥∥∥∥2L2(0,T ;H)

+γT

2sup

t∈(0,T )

∥∥∥∥∂ζL0∂t (t)

∥∥∥∥2H

+ β‖ζL(0)‖2V +

∥∥∥∥∂ζL0∂t (0)

∥∥∥∥2H

.

For t = 0, let vL0 = 0 and vLi = ζLi (0) in (3.8). We then have∫D

∫Y

a(x,y)(∇xζL0 (0) +∇y1ζ

L1 (0) + · · ·+∇ynζ

Ln (0))

· (∇y1ζL1 (0) + · · ·+∇ynζ

Ln (0))dydx = 0.

From this, (2.1), and (2.2), we deduce that

α‖∇y1ζL1 (0) + · · ·+∇ynζ

Ln (0)‖L2(D×Y) ≤ β‖∇ζL0 (0)‖H .

Letting γ be sufficiently small, we get the conclusion.We then have the following convergence result for the semidiscrete problem (3.1).

Proposition 3.5. Assume that ∂2u∂t2 ∈ L2(0, T ;V), and that

(3.10) limL→∞

‖gL0 − g0‖V = 0 and limL→∞

‖gL1 − g1‖H = 0.

Then

limL→∞

{∥∥∥∥∂(uL0 − u0)

∂t

∥∥∥∥L∞(0,T ;H)

+ ‖uL − u‖L∞(0,T ;V)

}= 0.

Proof. From Lemma 3.4, as uL − u = ζL + ηL, we have∥∥∥∥∂(uL0 − u0)

∂t

∥∥∥∥2L∞(0,T ;H)

+ ‖uL − u‖2L∞(0,T ;V)

≤ c

[‖∇ζL0 (0)‖2H +

∥∥∥∥∂ζL0∂t (0)

∥∥∥∥2H

+

∥∥∥∥∂2ηL0∂t2

∥∥∥∥2L2(0,T ;H)

]+

∥∥∥∥∂ηL0∂t∥∥∥∥2L∞(0,T ;H)

+ c‖ηL‖2L∞(0,T ;V).(3.11)

We show that limL→∞ ‖ηL‖L∞(0,T ;V) = 0. As u ∈ C([0, T ],V) u is uniformly con-tinuous as a function from [0, T ] to V. For ε > 0, there is a piecewise constant (withrespect to t) function u ∈ L∞(0, T ;V) such that ‖u − u‖L∞(0,T ;V) < ε. As u(t)

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obtains only a finite number of V-values, there is an L and vL ∈ L∞(0, T ;VL) suchthat ‖u− vL‖L∞(0,T ;V) < ε. Thus

limL→∞

supt∈(0,T )

infvL∈VL

‖u(t)− vL‖V = 0.

We then apply Lemma 3.2.Similarly, we have

limL→∞

∥∥∥∥∂ηL0∂t∥∥∥∥L2(0,T ;H)

= 0 and limL→∞

∥∥∥∥∂2ηL0∂t2

∥∥∥∥2L2(0,T ;H)

= 0.

Further, we have that

‖∇ζL0 (0)‖H ≤ ‖∇uL0 −∇u0‖H + ‖∇u0 −∇wL0 ‖H ,

which converges to 0 when L→ ∞ due to (3.10) and Lemma 3.2. Similarly, we have

limL→∞

∥∥∥∥∂ζL0∂t (0)

∥∥∥∥H

= 0.

From these, we get the conclusion.

3.2. Fully discrete problems. In this subsection we consider time discretiza-tion of the Galerkin approximating problem (3.1). We follow the approach and nota-tion of Dupont [8] for a stable time dicretizing scheme.

Let M be a positive integer, and let Δt = T/M . Let tm = mΔt. For a Ba-nach space X , for any functions r(t, x) ∈ C([0, T ];X), we denote by rm = r(tm, ·).Following Dupont [8], we employ the following notation:

rm+1/2 = (1/2)(rm+1 + rm), rm,θ = θrm+1 + (1 − 2θ)rm + θrm−1,

∂trm+1/2 = (rm+1 − rm)/Δt, ∂2t rm = (rm+1 − 2rm + rm−1)/(Δt)2,

δtrm = (rm+1 − rm−1)/(2Δt).

Assuming that f ∈ C([0, T ];H), for problem (2.8), with the finite element spaceVL in the previous subsection, we define the following fully discrete problem: Form = 1, . . . ,M , find uL

m = (uL0,m, uL1,m, . . . , u

Ln,m) ∈ VL such that form = 1, . . . ,M−1,

(3.12)(∂2t u

L0,m, v

L0

)H+B(uL

m,1/4,vL) = (fm,1/4, v

L0 )H

for all vL = (vL0 , vL1 , . . . , v

Ln ) ∈ VL.

As in [8], we choose θ = 1/4 for the scheme to be stable and achieve the minimaltime truncation error when the solution is sufficiently smooth.

For a continuous function r : [0, T ] → X , we denote by

‖r‖L∞(0,T ;X) = max0≤m<M

∥∥rm+1/2

∥∥X.

We also denote by

‖∂tr‖L∞(0,T ;X) = max0≤m<M

‖∂trm+1/2‖X .

For the solution of problem (3.12), we define ζLm as

(3.13) ζLm = uLm −wL

m,

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where wL is defined in (3.2). We then have the following result.Lemma 3.6. Assume that u ∈ H1(0, T ;V) and ∂2ηL0 /∂t

2 ∈ L2(0, T ;H). If∂3u0/∂t

3 ∈ L2(0, T ;H), then there exists a constant c (independent of Δt and u)such that∥∥∂tζL0 ∥∥L∞(0,T ;H)

+ ‖ζL‖L∞(0,T ;V)

≤ c

[‖ζL1/2‖V + ‖∂tζL0,1/2‖H +

∥∥∥∥∂2ηL0∂t2

∥∥∥∥L2(0,T ;H)

+

∥∥∥∥∂3u0∂t3

∥∥∥∥L2(0,T ;H)

Δt

],

and if ∂4u0/∂t4 ∈ L2(0, T ;H), then there exists a constant c such that∥∥∂tζL0 ∥∥L∞(0,T ;H)

+ ‖ζL‖L∞(0,T ;V)

≤ c

[‖ζL1/2‖V + ‖∂tζL0,1/2‖H +

∥∥∥∥∂2ηL0∂t2

∥∥∥∥L2(0,T ;H)

+

∥∥∥∥∂4u0∂t4

∥∥∥∥L2(0,T ;H)

(Δt)2

].

The proof of this lemma is long; we present it in Appendix A.Lemma 3.7. There is a constant c independent of the finite element spaces VL

and Δt such that

(3.14) ||ηL||L∞(0,T ;V) ≤ c sup0≤m<M

infvL∈VL

‖um+1/2 − vL‖V.

Further, if ∂u0/∂t ∈ L∞(0, T ;V ), then there exists a constant c such that

‖∂tηL0 ‖L∞(0,T ;V ) ≤ c sup0≤m<M

infvL∈VL

‖∂tum+1/2 − vL‖V.

Proof. Averaging (3.2) at tm+1 and tm with weights 1/2 and 1/2, we obtain

B(wLm+1/2 − um+1/2,v

L) = 0

for all vL ∈ VL. We, therefore, have

B(wLm+1/2 − um+1/2,w

Lm+1/2 − um+1/2) = B(wL

m+1/2 − um+1/2,vL − um+1/2)

for all vL ∈ VL. Using (2.1) and (2.2), we have

‖wLm+1/2 − um+1/2‖V ≤ c‖vL − um+1/2‖V,

for all vL ∈ VL. From this, we get the conclusion. The proof for the estimate of‖∂tηL0 ‖L∞(0,T ;V ) is similar.

Proposition 3.8. Assume that u ∈ H1(0, T ;V) and ∂2ηL0 /∂t2 ∈ L2(0, T ;H).

If ∂3u0/∂t3 ∈ L2(0, T ;H), then there exists a constant c such that

‖∂tuL0 − ∂tu0‖L∞(0,T ;H) + ‖uL − u‖L∞(0,T ;V)

≤ c

[Δt

∥∥∥∥∂3u0∂t3

∥∥∥∥L2(0,T ;H)

+

∥∥∥∥∂2ηL0∂t2

∥∥∥∥L2(0,T ;H)

+ ‖∂tζL0,1/2‖H + ‖ζL1/2‖V

]

+‖∂tηL0 ‖L∞(0,T ;H) + ‖ηL‖L∞(0,T ;V).(3.15)

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If ∂4u0/∂t4 ∈ L2(0, T ;H), then there exists a constant c such that

‖∂tuL0 − ∂tu0‖L∞(0,T ;H) + ‖uL − u‖L∞(0,T ;V)

≤ c

[(Δt)

2

∥∥∥∥∂4u0∂t4

∥∥∥∥L2(0,T ;H)

+

∥∥∥∥∂2ηL0∂t2

∥∥∥∥L2(0,T ;H)

+ ‖∂tζL0,1/2‖H + ‖ζL1/2‖V

]

+‖∂tηL0 ‖L∞(0,T ;H) + ‖ηL‖L∞(0,T ;V).(3.16)

In particular, if ∂2u/∂t2 ∈ L2(0, T ;V) and ∂3u0/∂t3 ∈ L2(0, T ;H), when uL

0 and uL1

are chosen so that

limL→∞

‖∂tζL0,1/2‖H = 0, limL→∞

‖ζL1/2‖V = 0,

then

(3.17) limL→∞Δt→0

‖∂tuL0 − ∂tu0‖L∞(0,T ;H) + ‖uL − u‖L∞(0,T ;V) = 0.

Proof. The proof of (3.15) and (3.16) follows from the definition of ζL and wL

and Lemma 3.6. The limit (3.17) then follows from Lemmas 3.2 and 3.7. We notethat as u ∈ H2(0, T ;V), from Lemma 3.3

limL→∞

∥∥∥∥∂2ηL0∂t2

∥∥∥∥L2(0,T ;H)

= 0.

Further, as ∂u∂t ∈ C([0, T ];V) we have

sup0≤m<M

infvL∈VL

‖∂tum+1/2 − vL‖V ≤ sup0≤t≤T

infvL∈VL

∥∥∥∥∂u∂t − vL∥∥∥∥V

→ 0 when L→ ∞.

The proof for this uses the fact that ∂u∂t is uniformly continuous. The argument is

similar to that in the proof of Proposition 3.5.

4. Regularity for the solution u of problem (2.8). To obtain the estimatesand convergence in the previous section and the rates of convergence in section 5, itis essential that the solution u of (2.8) possesses the sufficient temporal and spatialregularity. We thus establish in this section the smoothness condition on the datathat guarantees the required regularity.

We first deduce the homogenized problem for u0 from (2.8). We have from (2.8)that

un = wnl

(∂u0∂xl

+∂u1∂y1l

+ · · ·+ ∂un−1

∂y(n−1)l

),

where wnl ∈ L2(D × Y1 × · · · × Yn−1, H1#(Yn)/R) satisfies the cell problem∫

D

∫Y1

. . .

∫Yn

A(el +∇ynwnl) · ∇ynφndydx = 0

for all φn ∈ L2(D×Y1× · · ·×Yn−1, H1#(Yn)/R), where el is the lth unit vector in R

d.Using this, we have from (2.8) that∫

D

∫Y1

. . .

∫Yn−1

∫Yn

A(I +∇ynwn)

(∇xu0 +

n−1∑k=1

∇ykuk

)· ∇yn−1φn−1dy1 . . . dyn−1dyndx = 0,

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where wn denotes the vector function (wn1, . . . , wnd). Denoting by

An−1(x, y1, . . . , yn−1) =

∫Yn

A(I+∇ynwn)dyn =

∫Yn

A(I+∇ynwn) · (I+∇ynwn)dyn,

we have that

un−1 = w(n−1)l

(∂u0∂xl

+∂u1∂y1l

+ · · ·+ ∂un−2

∂y(n−2)l

),

where w(n−1)l ∈ L2(D × Y1 × · · · × Yn−1, H1#(Yn)/R) satisfies the problem∫

D

∫Y1

. . .

∫Yn−1

An−1(el +∇yn−1w(n−1)l) · ∇yn−1φn−1dy1 . . . dyn−1dx = 0,

for all φn−1 ∈ L2(D× Y1 × · · · × Yn−2, H1#(Yn−1)/R). Letting A

n = A, we then have,recursively,

ui = wil

(∂u0∂xl

+ · · ·+ ∂ui−1

∂y(i−1)l

),

where wil ∈ L2(D × Y1 × · · · × Yi−1, H1#(Yi)/R) satisfies the problem

(4.1)

∫D

∫Y1

. . .

∫Yi

Ai(el +∇yiwil) · ∇yiφidxdy1 . . . dyi = 0

for all φi ∈ L2(D × Y1 × · · · × Yi−1, H1#(Yi)/R). The matrix Ai is defined as

(4.2) Ai(x, y1, . . . , yi) =

∫Yi+1

Ai+1(I +∇yi+1wi+1) · (I +∇yi+1wi+1)dyi+1

for i < n. Continuing this process, we finally get the homogenized coefficient A0(x)as

A0(x) =

∫Y1

A1(x, y1)(I +∇y1w1) · (I +∇y1w1)dy1.

The homogenized problem is then

(4.3)∂2u0∂t2

−∇ · (A0(x)∇u0) = f(t, x), u0(0, ·) = g0,∂u0∂t

(0, ·) = g1.

The solution u of (2.8) is written in terms of u0 as

(4.4) ui = wi · (I +∇yi−1wi−1) · · · (I +∇y1w1)∇u0.

As A ∈ C1(D, C1#(Y1, (. . . C

1#(Yn) . . .))), we deduce that wil ∈ C1(D, C1

#, (Y1 . . .H1#

(Yi) . . .)). From which we have that Ai ∈ C1(D, C1#(Y1, . . . , C

1#(Yi) . . .)) and A0 ∈

C1(D). For the convergence in Propositions 3.5 and 3.8, we need regularity of u withrespect to t. We now establish these.

Proposition 4.1. Assume that

(4.5)

f ∈ H2(0, T ;H),g1 ∈ V,f(0)−∇ · (A0(x)∇g0) ∈ V,∂f∂t (0)−∇ · (A0(x)∇g1) ∈ H,

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then

(4.6)∂2u0∂t2

∈ L2(0, T ;V ) and∂3u0∂t3

∈ L2(0, T ;H).

Assume

(4.7)

f ∈ H3(0, T ;H),g1 ∈ V,f(0)−∇ · (A0(x)∇g0) ∈ V,∂f∂t (0)−∇ · (A0(x)∇g1) ∈ V,∂2f∂t2 (0) +∇ · (A0(x)∇(f(0) −∇ · (A0(x)∇g0))) ∈ H,

then

(4.8)∂3u0∂t3

∈ L2(0, T ;V ) and∂4u0∂t4

∈ L2(0, T ;H).

Proof. The proof follows directly from the regularity of the wave equation (4.3)where (4.5) and (4.7) are the compatibility conditions that guarantee the regularities(4.6) and (4.8) (see, e.g., Wloka [24, Chapter 5]). From (4.5), we have that

(4.9)∂2

∂t2

(∂u0∂t

)−∇ ·

(A0∇∂u0

∂t

)=∂f

∂t,

with the compatibility initial conditions

∂u0∂t

(0, ·) = g1(·) ∈ V,∂

∂t

∂u0∂t

(0, ·) = f(0, ·)−∇(A0(·)∇g0(·)) ∈ H.

We have further that

(4.10)∂2

∂t2

(∂2u0∂t2

)−∇ · (A0∇∂2u0

∂t2) =

∂2f

∂t2,

with the compatibility initial conditions

∂2u0∂t2

(0, ·) = f(0, ·)−∇(A0(·)∇g0(·)) ∈ V

and

∂

∂t

∂2u

∂t2(0, ·) = ∂f

∂t(0, ·)−∇(A0(·)∇g1(·)) ∈ H.

We then deduce (4.6). Regularity condition (4.8) is proved similarly.Remark 4.2. In section 6, where we establish the corrector for the homogenized

problem (4.3), we restrict our consideration to the case where g0 = 0. In this case,condition (4.5c) holds when f(0) ∈ V and (4.5d) holds when g1 ∈ V ∩H2(D). Con-dition (4.7d) holds when g1 ∈ H3

0 (D) and ∂f/∂t(0) ∈ V , and condition (4.7e) holdswhen ∂2f/∂t2(0) ∈ H and f(0) ∈ H2(D).

For the finite element rates of convergence in section 5, we need spatial regularityof u. To obtain explicit rates of convergence, we define the regularity spaces H andH as follows.

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We define by Hi the spaces of functions w(x, y1, . . . , yi) ∈ L2(D × Y1 × · · · ×Yi−1, H

2#(Yi)/R) such that w ∈ L2(Y1×· · ·×Yi, H1(D)) and w ∈ L2(D×

∏j �=k Yj , H

1#

(Yk)/R) for all k = 1, . . . , i− 1. The space Hi is equipped with the norm

‖w‖Hi= ‖w‖L2(D×Y1×···×Yi−1,H2

#(Yi)/R) + ‖w‖L2(Y1×···×Yi,H1(D))

+

i−1∑k=1

‖w‖L2(D×∏j �=k Yj ,H1

#(Yk)/R).

We define by H the space

(4.11) H = {w = (w0, w1, . . . , wn) ∈ V : w0 ∈ H2(D), wi ∈ Hi, i = 1, . . . , n},

which is equipped with the norm

‖w‖H = ‖w0‖H2(D) +

n∑i=1

‖wi‖Hi.

For the sparse tensor product approximation in subsection 5.3, we define theregularity space H ⊂ H. First, we define the spaces Hi of functions w(x, y1, . . . , yi)that are Yj-periodic with respect to yj for j = 1, . . . , i such that, for all (α0, . . . , αi) ∈(Rd)i+1 with 0 ≤ |αj | ≤ 1 for j = 0, . . . , i− 1 and |αi| ≤ 2,

∂|α0|+···+|αi|w∂α0x∂α1y1 . . . ∂αiyi

∈ L2(D × Y1 × · · · × Yi).

We equip the space Hi with the norm

‖w‖Hi=

∑1≤|αi|≤2

0≤|αj |≤1(0≤j≤i−1)

∥∥∥∥ ∂|α0|+...+|αi|w∂α0x∂α1y1 . . . ∂αiyi

∥∥∥∥L2(D×Y1×···×Yi)

.

In other words, it is the norm of the spaceH1(D,H1#(Y1, . . . , H

1#(Yi−1, H

2#(Yi)/R) . . .)).

We define the space

(4.12) H = {w = (w0, w1, . . . , wn) ∈ V : w0 ∈ H2(D), wi ∈ Hi, i = 1, . . . , n},

which is equipped with the norm

‖w‖H = ‖w0‖H2(D) +

n∑i=1

‖wi‖Hi.

We have the following results.Proposition 4.3. Assume that f ∈ H1(0, T ;H), g1 ∈ V , g0 ∈ V ∩H2(D), and

the domain D is convex. Then u ∈ L∞(0, T ; H).Proof. As A(x,y) ∈ C1(D, C1

#(Y1, . . . , C1#(Yn) . . . )), A

0(·) ∈ C1(D). From (4.9),we deduce that

∂u0∂t

∈ L∞(0, T ;V ) and∂2u0∂t2

∈ L∞(0, T ;H).

Thus

(4.13) −∇ · (A0∇u0) = f − ∂2u0∂t2

∈ L∞(0, T ;H).

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Due to the convexity of the domain D, from Theorem 3.2.1.2 of Grisvard [12], wededuce that u0 ∈ L∞(0, T ;H2(D)). From (4.1), we have that

−∇yi(A(x, y1, . . . , yi)∇yiwil) = ∇yi(A(x, y1, . . . , yi)ei).

From (2.3), we then have

(4.14) wil ∈ C1(D, C1(Y1, . . . , H2#(Yi)/R . . .)).

Therefore, from (4.4) we deduce that ui ∈ Hi.

5. Full and sparse tensor finite element methods. As ui is posed in theproduct domain D × Y1 × · · · × Yi, it is natural to define the finite element approx-imating spaces V L

i in the previous section as tensor product spaces. In particular,we use full tensor product spaces and sparse tensor product spaces that are con-structed from increment or detailed spaces of a nested sequence. We, therefore, firstdefine a hierarchy of finite element spaces from which the tensor product spaces areconstructed.

5.1. Hierarchical finite element spaces. Let D be a polyhedron with planesides. We divide D into simplices recursively. The set of simplices T l of mesh sizehl = O(2−l) is obtained by dividing each simplex in T l−1 into four congruent trianglesfor d = 2 or 8 tedrahedra for d = 3. The cube Y is divided into sets of simplices T l

#

which are periodically distributed in a similar manner. We define the following finiteelement spaces:

V l = {u ∈ H1(D) : u|K ∈ P1(K) ∀K ∈ T l},V l0 = {u ∈ H1

0 (D) : u|K ∈ P1(K) ∀K ∈ T l},V l# = {u ∈ H1

#(Y ) : u|K ∈ P1(K) ∀K ∈ T l#},

where P1(K) denotes the set of linear polynomials in K. We recall the followingapproximating properties for functions with sufficient regularity (see, e.g., Ciarlet[6]). For s ≥ 0

infv∈V l

‖u− v‖L2(D) ≤ ch1+sl ‖u‖H1+s(D)

for all u ∈ H1+s(D), and

infv∈V l

0

‖u− v‖H1(D) ≤ chsl ‖u‖H1+s(D)

for u ∈ H1+s(D)⋂V . For periodic functions on Y we have

infv∈V l

#/R‖u− v‖H1

#(Y )/R ≤ chsl ‖u‖H1+s# (Y )/R,

infv∈V l

#

‖u− v‖L2(Y ) ≤ ch1+sl ‖u‖H1+s

# (Y )

for all periodic functions u ∈ H1+s# (Y ).

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5.2. Full tensor product finite elements. As ui ∈ L2(D×Y1×· · ·×Yi−1, H1#

(Yi)/R) ∼= L2(D)⊗ L2(Y1)⊗ · · · ⊗ L2(Yi−1)⊗H1#(Yi)/R, we choose V L

i as

V Li = V L ⊗ V L

# ⊗ . . .⊗ V L#︸︷︷︸

i− 1 times

⊗V L# /R

to approximate ui. We define

VL = {(vL0 , {vLi }) : vL0 ∈ V L0 , v

Li ∈ V L

i }.

For these spaces, when w is in the regularity space Hi defined in section 4, we havethe following approximating properties.

Lemma 5.1. For w ∈ Hi,

infvL∈V L

i

‖w − vL‖L2(D×Y1×···×Yi−1,H1#(Yi)/R) ≤ chL‖w‖Hi

.

We refer to Hoang and Schwab [15] for a proof. For the space H defined in (4.11),we deduce that for w ∈ H,

infvL∈VL

‖w − vL‖V ≤ chL‖w‖H.

The spatial semidiscrete problem (3.1) is now written as follows: Find uL = (uL0 , uL1 , . . . ,

uLn) such that

(5.1)

(∂2uL0∂t2

, vL0

)H

+B(uL, vL) = (f, vL0 )H ∀ vL ∈ VL

with uL0 (0) = gL0 and∂uL

0

∂t (0) = gL1 . Equation (3.7) now becomes

ζL = uL −wL.

We then have the following result for the continuous time approximation.Proposition 5.2. Assume that condition (4.5) holds and the domain D is convex

and that gL0 and gL1 are chosen so that

(5.2) ‖gL0 − g0‖V ≤ chL and ‖gL1 − g1‖H ≤ chL,

then ∥∥∥∥∂(uL0 − u0)

∂t

∥∥∥∥L∞(0,T ;H)

+ ‖uL − u‖L∞(0,T ;V) ≤ chL.

Proof. We show that the right-hand side of (3.11) is smaller than ch2L where

c is independent of L. With condition (4.5), from (4.10), we deduce that ∂2u0

∂t2 ∈L2(0, T ;V ). From (4.4) and (4.14), we have ∂2u

∂t2 ∈ L2(0, T ;V). From (4.13), we havethat u0 ∈ L∞(0, T ;H2(D)) so u ∈ L∞(0, T ; H).

From Lemmas 3.2 and 5.1, we deduce that for all t ∈ [0, T ]

‖ηL(t)‖V ≤ chL‖u(t)‖H.

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‖∇ζL0 (0)‖H ≤ ‖∇uL0 (0)−∇u0(0)‖H + ‖∇u0(0)−∇wL0 (0)‖H ≤ chL,

due to (5.2) and ‖η(0)‖V ≤ chL. We now show that for all t ∈ [0, T ]∥∥∥∥∂ηL0∂t∥∥∥∥L∞(0,T ;H)

=

∥∥∥∥∂u0∂t − ∂wL0

∂t

∥∥∥∥L∞(0,T ;H)

≤ chL,

where c is independent of L. We consider φ(t) ∈ V as the unique solution of theelliptic problem

(5.3) B(φ(t),v) =

∫D

∂(u0 − wL0 )

∂t(t, x)v0(x)dx

for all v ∈ V. This problem has a unique solution as∂(u0−wL

0 )∂t (t, ·) ∈ H . We have

that φ0 is the solution of the elliptic problem

−∇ · (A0(x)∇φ0(t, x)) =∂(u0 − wL

0 )

∂t.

As D is convex,

‖φ0(t)‖H2(D) ≤ c

∥∥∥∥∂(u0 − wL0 )

∂t(t)

∥∥∥∥H

.

Further, similarly to (4.4), we have

φi(t) = wi · (I +∇yi−1wi−1) . . . (I +∇y1w1)∇φ0(t),

which implies

‖φ(t)‖H ≤ c

∥∥∥∥∂(u0 − wL0 )

∂t(t, ·)

∥∥∥∥H

.

Let v = ∂(u−wL)∂t (t) ∈ V. We have

B

(φ(t),

∂(u−wL)

∂t(t)

)=

∥∥∥∥∂(u0 − wL0 )

∂t(t)

∥∥∥∥2H

.

From (3.6), due to the symmetry of B, for all vL ∈ V L

B

(φ(t)− vL, ∂(u−wL)

∂t(t)

)=

∥∥∥∥∂(u0 − wL0 )

∂t(t)

∥∥∥∥2H

.

We, therefore, have∥∥∥∥∂(u0 − wL0 )

∂t(t)

∥∥∥∥2H

≤ β infvL∈VL

‖φ(t)− vL‖V∥∥∥∥∂(u−wL)

∂t(t)

∥∥∥∥V

≤ chL‖φ(t)‖H∥∥∥∥∂(u−wL)

∂t(t)

∥∥∥∥V

≤ chL

∥∥∥∥∂(u0 − wL0 )

∂t(t)

∥∥∥∥H

∥∥∥∥∂(u−wL)

∂t(t)

∥∥∥∥V

.

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From Lemma 3.3, as ∂u∂t ∈ L∞(0, T ;V), we deduce that ‖∂(u−wL)

∂t (t)‖V is boundeduniformly for all t. Hence ∥∥∥∥∂(u0 − wL

0 )

∂t(t)

∥∥∥∥H

≤ chL.

In particular, for t = 0, we have∥∥∥∥∂ζL0∂t (0, ·)∥∥∥∥H

≤ ‖gL1 − g1‖H +

∥∥∥∥∂u0∂t (0)− ∂wL0

∂t(0)

∥∥∥∥H

≤ chL.

Similarly, by considering the problem

B(φ(t),v) =

∫D

∂2(u0 − wL0 )

∂t2(t, x)v0(x)dx,

using the fact that ‖∂2(u−wL)∂t2 ‖L2(0,T ;V) is bounded with respect to L, we deduce that∥∥∥∥∂2(u0 − wL

0 )

∂t2

∥∥∥∥L2(0,T ;H)

≤ chL.

From this, we get the conclusion.The fully discrete problem is now written as follows: Find uL

m = (uL0,m, uL1,m, . . . ,

uLn,m) ∈ VL such that

(5.4) (∂2t uL0,m, v

L0 )H +B(uL

m,1/4, vL) = (fm,1/4, v

L0 )H

for all vL = (vL0 , vL1 , . . . , v

Ln ) ∈ VL.

We have the following result.Proposition 5.3. Assume that condition (4.5) holds and that D is a convex

domain. If

‖∂tζL0,1/2‖H ≤ c(Δt+ hL), ‖ζL1/2‖V ≤ c(Δt+ hL),

then

‖∂tuL0 − ∂tu0‖L∞(0,T ;H) + ‖uL − u‖L∞(0,T ;V) ≤ c(Δt+ hL).

If (4.7) holds, and

‖∂tζL0,1/2‖H ≤ c((Δt)2 + hL), ‖ζL1/2‖V ≤ c((Δt)2 + hL),

then

‖∂tuL0 − ∂tu0‖L∞(0,T ;H) + ‖uL − u‖L∞(0,T ;V) ≤ c((Δt)2 + hL).

Proof. We use (3.15) and (3.16). As shown in the proof of Proposition 5.2, wehave ∥∥∥∥∂2ηL0∂t2

∥∥∥∥L2(0,T ;H)

≤ chL.

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From Lemma 3.2 and the regularity u ∈ L∞(0, T ; H), we have

‖ηL‖L∞(0,T ;V) ≤ chL.

It remains to show that ‖∂tηL0 ‖L∞(0,T ;H) ≤ chL. We proceed similarly as for the proofof Proposition 5.2. We consider φm ∈ V as the solution of the problem

B (φm,v) =

∫D

ηL0,m+1 − ηL0,mΔt

v0dx

for all v = (v0, v1, . . . , vn) ∈ V. AsηL0,m+1−ηL

0,m

Δt ∈ H , we deduce that φm ∈ H andthat

‖φm‖H ≤ c

∥∥∥∥∥ηL0,m+1 − ηL0,m

Δt

∥∥∥∥∥H

.

Let v =ηL

m+1−ηLm

Δt ∈ V, we have that

B

(φm,

ηLm+1 − ηL

m

Δt

)=

∥∥∥∥∥ηL0,m+1 − ηL0,m

Δt

∥∥∥∥∥2

.

From (3.2), we have

B

(φm − vL,

ηLm+1 − ηLm

Δt

)=

∥∥∥∥∥ηL0,m+1 − ηL0,m

Δt

∥∥∥∥∥2

for all vL ∈ VL. We then deduce∥∥∥∥∥ηL0,m+1 − ηL0,m

Δt

∥∥∥∥∥2

H

≤ β infvL∈VL

‖φm − vL‖V∥∥∥∥ηL

m+1 − ηLm

Δt

∥∥∥∥V

≤ chL

∥∥∥∥∥ηL0,m+1 − ηL0,m

Δt

∥∥∥∥∥H

∥∥∥∥ηLm+1 − ηLm

Δt

∥∥∥∥V

.

As u ∈ H2(0, T ;V) so ηL ∈ H2(0, T ;V). We have that

∥∥∥∥ηLm+1 − ηL

m

Δt

∥∥∥∥V

≤ c supt∈(0,T )

∥∥∥∥∂ηL

∂t(t)

∥∥∥∥V

,

which is uniformly bounded for all Δt due to Lemma 3.3. From this we have∥∥∥∥∥ηL0,m+1 − ηL0,m

Δt

∥∥∥∥∥H

≤ chL.

We then get the conclusion.

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5.3. Sparse tensor product finite elements. The dimension of the full tensorproduct space VL is O(2ndL) which is prohibitively large when n and L are largefor solving problems (5.1) and (5.4). In this section, we develop the sparse tensorproduct FE spaces with much reduced dimension but approximate both the spatialsemidiscrete problem and the fully discrete problem with essentially equal accuracyas the full tensor product FEs. We first define the following orthogonal projectionsin the norm of L2(D), L2(Y ), and H1

#(Y ):

P l0 : L2(D) → V l, P l0# : L2(Y ) → V l

#, P l1# : H1

#(Y )/R → V l#/R.

The increment spaces are defined as

W l = (P l0 − P (l−1)0)V l, W l0# = (P l0

# − P(l−1)0# )V l

#, W l1# = (P l1

# − P(l−1)1# )V l

#/R

with the convention that P−10 = 0, P−10# = 0, and P−11

# = 0. We then have

V l =⊕0≤i≤l

W i, V l# =

⊕0≤i≤l

W i0# , V l

#/R =⊕0≤i≤l

W i1# .

The full tensor product space V Li can be written as

V Li =

⊕0≤jk≤L

k=0,1,...,i

W j0 ⊗W j10# ⊗ · · · ⊗W

ji−10# ⊗W ji1

# .

We define the sparse tensor product spaces as

V Li =

⊕0≤j0+j1+···+ji≤L

W j0 ⊗W j10# ⊗ · · · ⊗W

ji−10# ⊗W ji1

# ,

and the space

VL = {(uL0 , {uLi }) : uL0 ∈ V L0 , u

Li ∈ V L

i }.

For functions in the regularity spaces Hi defined in section 4, we have the followingestimate.

Lemma 5.4. For w ∈ Hi,

infvL∈V L

i

‖w − vL‖Vi ≤ cLi/2hL‖w‖Hi.

For w ∈ H, we then have

infvL∈VL

‖w − vL‖V ≤ cLn/2hL‖w‖H.

The proof of this lemma can be found in Hoang and Schwab [15].The spatial semidiscrete problem (3.1) now becomes the following: find uL =

(uL0 , uL1 , . . . , u

Ln) such that

(5.5)

(∂2uL0∂t2

, vL0

)H

+B(uL, vL) = (f, vL0 )H ∀ vL ∈ VL

with uL0 = gL0 and∂uL

0

∂t = gL1 . We then have the following result for the semidiscreteproblem.

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Proposition 5.5. Assume that condition (4.5) holds and the domain D is convexand that gL0 and gL1 are chosen so that

(5.6) ‖gL0 − g0‖V ≤ cLn/2hL and ‖gL1 − g1‖H ≤ cLn/2hL,

then ∥∥∥∥∂(uL0 − u0)

∂t

∥∥∥∥L∞(0,T ;H)

+ ‖uL − u‖L∞(0,T ;V) ≤ cLn/2hL.

The proof of the proposition is identical to that for Proposition 5.2. Indeed, theregularity φ0 ∈ H2(D) implies that φ(t) defined in (5.3) is in H and that

‖φ(t)‖H ≤ c

∥∥∥∥∂(u0 − wL0 )

∂t(t)

∥∥∥∥H

.

From this, by a similar argument as in the proof of Proposition 5.2, we get∥∥∥∥∂(u0 − wL0 )

∂t(t)

∥∥∥∥H

≤ chLLn/2.

The fully discrete problem (3.12) becomes the following: Find uLm = (uL0,m, u

L1,m, . . . ,

uLn,m) ∈ VL such that

(5.7) (∂2t uL0,m, v

L0 )H +B(uL

m,1/4, vL) = (fm,1/4, v

L0 )H

for all vL = (vL0 , vL1 , . . . , v

Ln ) ∈ VL.

We have the following result.Proposition 5.6. Assume that condition (4.5) holds and that D is a convex

domain. Assume further that

‖∂tζL0,1/2‖H ≤ c(Δt+ hLLn/2), ‖ζL1/2‖V ≤ c(Δt+ hLL

n/2).

Then

‖∂tuL0 − ∂tu0‖L∞(0,T ;H) + ‖uL − u‖L∞(0,T ;V) ≤ c(Δt+ hLLn/2).

If condition (4.7) holds and

‖∂tζL0,1/2‖H ≤ c((Δt)2 + hLLn/2), ‖ζL1/2‖V ≤ c((Δt)2 + hLL

n/2),

then

‖∂tuL0 − ∂tu0‖L∞(0,T ;H) + ‖uL − u‖L∞(0,T ;V) ≤ c((Δt)2 + hLLn/2).

The proof of this proposition is similar to that for Proposition 5.3.The dimension of the sparse tensor product space VL is O(h−d

L Ln), which is muchless than the dimension of VL. The error we obtain in Proposition 5.6 is essentiallyequal to that obtained in Proposition 5.3. We will make some remarks on the choiceof uL

0 and uL1 in Appendix B.

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6. Corrector for the homogenization problem (2.8). For two-scale ellipticproblems it is well known that with sufficient regularity for the solutions wi(x, y) ofcell problems (6.1), there is always a corrector for the weak convergence uε ⇀ u0(see, e.g., [4]). However, a similar corrector result for the wave equation (2.5) doesnot always hold as the energy of the multiscale problem (2.5) only converges to theenergy of the homogenized equation when the initial condition g0 satisfies specialconditions. Brahim-Otsmane, Francfort, and Murat [5] proved a corrector result forwave equation (2.5) with another term vε which satisfies the multiscale wave equation(2.5) with zero right-hand side and special initial conditions.

In this section, we restrict our attention to the case where the initial conditionis g0 = 0. This allows us to deduce corrector results similar to those for ellipticequations, and to deduce a homogenization rate of convergence for the two-scale case.These results will be used to deduce the convergence of the finite element solution uL

in the physical variable x.

6.1. Corrector for two-scale problem. For two-scale problems (n = 1), wededuce in this section a homogenization rate of convergence. We denote the coefficientA(x,y) in this case as A(x, y).

For i = 1, . . . , d, we let wi(x, y) ∈ L2(D;H1#(Y )/R) be the solution of the cell

problem

(6.1) −∇y · (A(x, y)(ei +∇ywi(x, y))) = 0,

where ei is the ith unit vector in Rd. The homogenized coefficient A0(x) is defined as

(6.2) A0(x) =

∫Y

A(x, y)(I +∇yw) · (I+∇yw)dy,

where I is the identity matrix and w = (w1, w2, . . . , wd) ∈ (L2(D,H1#(Y )/R))d.

Since A(x, y) ∈ C1(D, C1#(Y )), wi(x, y) ∈ C1(D,H1

#(Y )), and, therefore, A0(x) ∈(C1(D))d×d

sym. We then have the following result on the homogenization rate of con-vergence.

Proposition 6.1. For two-scale problems, assume that

g0 = 0, g1 ∈ H2(D) ∩ V, f ∈ H2(0, T ;H), f(0) ∈ V

and D is a convex domain. There exists a constant c that does not depend on ε suchthat∥∥∥∥∂uε∂t − ∂u0

∂t

∥∥∥∥L∞(0,T ;H)

+

∥∥∥∥∇uε −[∇u0 +∇yu1

(·, ·, ·

ε

)]∥∥∥∥L∞(0,T ;H)

≤ cε1/2.

We present the proof of this proposition in Appendix C.

6.2. Corrector for multiscale problem. For general multiscale problems, weare not aware of any analytic homogenization rates of convergence, even for ellipticproblems. However, for the case where the scales are such that εi/εi+1 is an integerfor all i = 1, . . . , n − 1, a corrector for the elliptic problem is deduced in Hoang andSchwab [15] following the earlier work by Cioranescu, Damlamian, and Griso [7]. Wenow deduce a corresponding corrector for the general multiscale wave equation (2.5).We will use the following operator.

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Definition 6.2. The (n+1)-scale “unfolding” operator T εn : L1(D) → L1(D×Y)

is defined as

T εn (φ)(x,y) = φ

(ε1

[ xε1

]+ ε2

[ y1ε2/ε1

]+ · · ·+ εn

[ yn−1

εn/εn−1

]+ εnyn

),

where the function φ ∈ L1(D × Y ) is understood as 0 outside D; [·] denotes the“integer” part with respect to Y .

Let Dε be the 2ε neighborhood of D, we have

(6.3)

∫D

φdx =

∫Dε

∫Y

T εn (φ)dydx .

A simple proof for this can be found in, e.g., [7]. It can be shown that

(6.4) T εn (∇uε)⇀ ∇u0 +∇y1u1 + . . .+∇ynun in L2(D ×Y).

Following [7], we define the “folding” operator Uεn as follows.

Definition 6.3. For Φ ∈ L1(D ×Y) (understood to be zero when x /∈ D) andfor ε > 0 sufficiently small, the “folding” operator Uε

n(Φ) ∈ L1(D) is defined as

Uεn(Φ)(x) =

∫Y1

. . .

∫Yn

Φ(ε1

[ xε1

]+ ε1t1,

ε2ε1

[ε1ε2

{ xε1

}]+ε2ε1t2, . . . ,

εnεn−1

[εn−1

εn

{ x

εn−1

}]+

εnεn−1

tn,{ x

εn

})dtn . . . dt1(6.5)

where {·} = · − [·].For Φ ∈ L1(D ×Y) which are understood as 0 when x /∈ D,∫

Dε

Uεn(Φ)(x)dx =

∫D

∫Y

Φdydx

(see [7]). We have the following results.Proposition 6.4. For the multiscale wave equation (2.5), assuming that g0 = 0,

g1 ∈ V⋂H2(D), and f ∈ H1(0, T ;H), we have

limε→0

∥∥∥∥∂uε∂t − ∂u0∂t

∥∥∥∥L∞(0,T ;H)

+ ‖∇uε(t)− Uεn(∇u0(t) +∇y1u1(t) + · · ·+∇ynun(t))‖L∞(0,T ;H) = 0.

We present the proof of this proposition in Appendix D.

7. Convergence in the physical variable x. We now derive numerical correc-tors from the finite element approximation of problem (2.8) for the multiscale solutionuε.

7.1. Convergence in the physical variable for two-scale problem. In thissection, we establish the convergence in the physical variables of the solutions ofproblems (3.1), (3.12) for both the full and sparse tensor approximations. In the twoscale case, the operator Uε

n defined in (6.5) becomes

(7.1) Uε(Φ)(x) =

∫Y

Φ(ε[xε

]+ εz,

{xε

})dz.

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Let Dε be a 2ε neighborhood of D. Regarding Φ as zero when x is outside D, wehave

(7.2)

∫Dε

Uε(Φ)(x)dx =

∫D

∫Y

Φ(x, y)dxdy.

We then have the following result.Lemma 7.1. Assume that g0 = 0, g1 ∈ V , f ∈ H1(0, T ;H), and the domain D

is convex. Then

supt∈(0,T )

∫D

∣∣∣∣∇yu1

(t, x,

x

ε

)− Uε(∇yu1(t, ·, ·))(x)

∣∣∣∣2 dx ≤ cε2.

Proof. We note from (4.9) that ∂2u0/∂t2 ∈ L∞(0, T ;H). Therefore, from (4.3),

we have

−∇ · (A0∇u0) = f − ∂2u0∂t2

∈ L∞(0, T ;H).

Thus u0 ∈ L∞(0, T ;H2(D)). Since A ∈ C1(D, C1#(Y )), wl(x, y) ∈ C1(D, C1

#(Y )).The proof then follows from that of Lemma 5.5 of Hoang and Schwab [16].

We then have the following result on the semidiscrete approximation (3.1).Theorem 7.2. Assume that f ∈ H1(0, T ;H), g0 = 0, g1 ∈ V

⋂H2(D), f(0) ∈

V , the domain D is convex, and the initial conditions are chosen so that gL0 = 0 and‖gL1 − g1‖H ≤ chL. Then the solution of the semidiscrete problem (5.1) for the fulltensor product FE spaces satisfies∥∥∥∥∂uε∂t − ∂uL0

∂t

∥∥∥∥L∞(0,T ;H)

+ ‖∇uε − [∇uL0 + Uε(∇yuL1 )]‖L∞(0,T ;H) ≤ c(hL + ε1/2),

and the solution of problem (5.5) for the sparse tensor product FE spaces satisfies∥∥∥∥∂uε∂t − ∂uL0∂t

∥∥∥∥L∞(0,T ;H)

+ ‖∇uε − [∇uL0 + Uε(∇y uL1 )]‖L∞(0,T ;H) ≤ c(hLL

n/2 + ε1/2).

Proof. With the hypothesis of the theorem, condition (4.5) holds. The regularityrequired for Propositions 5.2, 5.5, and 6.1 hold. For all Φ ∈ L2(D × Y ), we have

(Uε(Φ)(x))2 ≤ Uε(Φ2)(x).

Therefore, for t ∈ [0, T ]

‖Uε(∇yu1(t))− Uε(∇yuL1 (t))‖H ≤ ‖∇yu1(t)−∇yu

L1 (t)‖L2(D×Y ).

From Propositions 5.2, 5.5, and 6.1 we get the conclusion for the full tensor productapproximation. The result for sparse tensor product approximation can be derivedsimilarly.

For the fully discrete problems (5.4) and (5.7) we have the following result.Theorem 7.3. Assume that g0 = 0 and the domain D is convex. If

(7.3) f ∈ H2(0, T ;H), g1 ∈ V ∩H2(D), f(0) ∈ V,

then when

‖∂tζL0,1/2‖H ≤ c(Δt+ hL), ‖ζL1/2‖V ≤ c(Δt+ hL)

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for the full tensor product approximating problem (5.4) we have

max0≤m<M

∥∥∥∥∂uε∂t (tm)− ∂tuL0,m+1/2

∥∥∥∥H

+ ‖∇uε − [∇uL0 + Uε(∇yuL1 )]‖L∞(0,T ;H)(7.4)

≤ c(Δt+ hL + ε1/2)

and when

‖∂tζL0,1/2‖H ≤ c(Δt+ hLL1/2), ‖ζL1/2‖V ≤ c(Δt+ hLL

1/2),

for the sparse tensor product approximating problem (5.7) we have

max0≤m<M

∥∥∥∥∂uε∂t (tm)− ∂tuL0,m+1/2

∥∥∥∥H

+ ‖∇uε − [∇uL0 + Uε(∇yuL1 )]‖L∞(0,T ;H)(7.5)

≤ c(Δt+ hLL1/2 + ε1/2).

If we assume further that

(7.6) f ∈ H3(0, T ;H), g1 ∈ H30 (D), and f(0) ∈ H2(D),

then when

‖∂tζL0,1/2‖H ≤ c((Δt)2 + hL), ‖ζL1/2‖V ≤ c((Δt)2 + hL),

for the full tensor approximating problem (5.4) we have

Δt max0≤m<M

∥∥∥∥∂uε∂t (tm)− ∂tuL0,m+1/2

∥∥∥∥H

+ ‖∇uε − [∇uL0 + Uε(∇yuL1 )]‖L∞(0,T ;H)(7.7)

≤ c((Δt)2 + hL + ε1/2)

and when

‖∂tζL0,1/2‖H ≤ c((Δt)2 + hLL1/2), ‖ζL1/2‖V ≤ c((Δt)2 + hLL

1/2),

for the sparse tensor product approximating problem (5.7)

Δt max0≤m<M

∥∥∥∥∂uε∂t (tm)− ∂tuL0,m+1/2

∥∥∥∥H

+ ‖∇uε − [∇uL0 + Uε(∇yuL1 )]‖L∞(0,T ;H)(7.8)

≤ c((Δt)2 + hLL1/2 + ε1/2).

Proof. With A(x, y) ∈ C1(D, C1#(Y )), g0 = 0 and the domain D is convex, and

with either (7.3) or (7.6), the conditions of Proposition 6.1 holds. We note that

1

Δt(u0,m+1 − u0,m)− ∂u0

∂t(tm) =

∂u0∂t

(τ)− ∂u0∂t

(tm) =

∫ τ

tm

∂2u0∂t2

(τ ′)dτ ′

for tm ≤ τ ≤ tm+1. From (4.9), we deduce that ∂2u0

∂t2 ∈ L∞(0, T ;H). Therefore,

sup0≤m<M

∥∥∥∥∂tu0,m+1/2 −∂u0∂t

(tm)

∥∥∥∥H

≤ cΔt.

With (7.3), (4.5) holds. From Propositions 5.3 and 5.6, we deduce (7.4) and (7.5).The two estimates (7.7) and (7.8) are deduced similarly as (7.6) leads to (4.7).

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7.2. Convergence in the physical variable in the multiscale case. Forgeneral multiscale problems, we deduce in this section the convergence of the solutionof both the semidiscrete problem (3.1) and the fully discrete problem (3.12). As wedo not have an explicit homogenization rate of convergence we will not distinguishthe two cases of full and sparse tensor products. We thus use our general notation forgeneral finite element spaces in section 3. The results in this section hold for both fulland sparse tensor product spaces. For problem (3.1), we have the following result.

Theorem 7.4. Assume that g0 = 0, g1 ∈ V , f ∈ H2(0, T ;H), f(0) ∈ V , andgL0 = 0, ‖gL1 − g1‖H → 0 when L → ∞. Then for the solution of the semidiscreteproblem (3.1) we have

limL→∞

∥∥∥∥∂uε∂t − ∂uL0∂t

∥∥∥∥L∞(0,T ;H)

+‖∇uε−Uεn(∇uL0 +∇y1u

L1 +· · ·+∇ynu

Ln)‖L∞(0,T ;H) = 0.

Proof. With the hypothesis of the theorem, condition (4.5) holds. We, therefore,

have ∂2u0

∂t2 ∈ L2(0, T ;V ). Thus from (4.4), ∂2u∂t2 ∈ L2(0, T ;V).


‖Uεn(∇u0(t) + ∇y1u1(t) + · · ·+∇ynun(t))− Uε

n(∇uL0 (t)+ ∇y1u

L1 (t) + · · ·+∇ynu

Ln(t))‖H

≤ ‖u(t)− uL(t)‖V.

The result then follows from Propositions 3.5 and 6.4.Similarly, we have the following result for the fully discrete problem (3.12).Theorem 7.5. Assume that g0 = 0, g1 ∈ V , f ∈ H2(0, T ;H) and f(0) ∈ V , and

gL0 = 0. If uL0 and uL

1 are chosen so that

limL→∞

‖∂tζL0,1/2‖H = 0, limL→∞

‖ζL1/2‖V = 0,

then

limL→∞Δt→0

sup0≤m<M

∥∥∥∥∂uε∂t (tm)− ∂tuL0,m+1/2

∥∥∥∥H

+ ‖∇uε − Uεn(∇uL0 +∇y1u

L1 + · · ·+∇ynu

Ln)‖L∞(0,T ;H) = 0.

Proof. From the hypothesis of the theorem, conditions (4.5) hold. Thus conver-gence (3.17) holds. From Propositions 3.8 and 6.4 we get the result.

8. Numerical results. We present in this section some numerical examples fortwo- and three-scale problems that confirm our analysis above. We solve the fullydiscretized Galerkin problem (3.12). At each time step m, we solve for uL

m+1 in fullor sparse tensor finite element product spaces as developed in subsections 5.2 and5.3. The discretized problem is formulated as a linear system of equations for cm thatrepresents components of uL

m in the basis of VL. The linear system we solve is(1

(Δt)2M+

1

4S

)cm+1 =

1

4Fm+1 +

1

2Fm +

1

4Fm−1

+2

(Δt)2Mc0,m − 1

(Δt)2Mc0,m−1 −

1

2Scm − 1

4Scm−1.

Here c0,m denotes the dimVL × 1 column vector whose first dimV L componentsare those of uL0,m in the basis of V L and all the other components are 0, M is the

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Gram matrix for the basis functions of V L in the H norm extending to 0 to form adimVL × dimVL matrix. The stiffness matrix S is derived from A00, A01, A11 inProposition 3.1. On the right-hand side, Fm is the time dependent column vectorwhose first dimV L components form the column vector that describes the interactionof f(tm) and the basis functions of V L in the H norm and all other components are 0.The problem is then solved by conjugate gradient method with full tensor FE basisin subsection 5.2 and sparse tensor FE basis in subsection 5.3.

To construct the sparse tensor finite element product spaces, it is convenient touse the wavelet preconditioning. We thus briefly present the assumptions on the finiteelement spaces on D and Y in the next subsection.

8.1. Multilevel FE spaces and preconditioning. To construct the incrementspaces W l, W l0

# , and W l1# in section 5.3, we assume the following assumptions.

Assumption 8.1. (i) For each j ∈ Nd0, there exists a set of indices Ij ⊂ N

d0 and

a set of basis functions φjk ∈ H1(D), k ∈ Ij, such that V l = span{φjk : |j|∞ ≤ l

}.

There are constants c2 > c1 > 0 such that if φ =∑

|j|∞≤l,k∈Ij φjkcjk ∈ V l, then thenorm equivalence holds:

c1∑

|j|∞≤l

k∈Ij

22s|j|∞ |cjk|2 ≤ ‖φ‖2Hs(D) ≤ c2∑

|j|∞≤l

k∈Ij

22s|j|∞ |cjk|2, s = 0, 1.

(ii) For the space L2(Y ), for each j ∈ Nd0, there exists a set of indices Ij0 ⊂ N

d0 and

a set of basis functions φjk0 ∈ L2(Y ), k ∈ Ij0 , such that V l# = span{φjk0 : |j|∞ ≤ l}.


|j|∞≤l,k∈Ij0φjk0 cjk ∈ V l, then

c3∑

|j|∞≤l

k∈Ij0

|cjk|2 ≤ ‖φ‖2L2(Y ) ≤ c4∑

|j|∞≤l

k∈Ij0

|cjk|2.

(iii) For each j ∈ Nd0, there exists a set of indices Ij1 ⊂ N

d0 and a set of basis

functions φjk1 ∈ H1#(Y )/R, k ∈ Ij1 , such that {φjk1 : |j|∞ ≤ l} is a basis of V l

#/R.


|j|∞≤l,k∈Ij1φjk1 cjk, then

c5∑

|j|∞≤l

k∈Ij1

|cjk|2 ≤ ‖φ‖2H1#(Y )/R ≤ c6

∑|j|∞≤l

k∈Ij1

|cjk|2.

We consider the following examples of hierarchical FE bases in one dimension.Example. (i) For D = (0, 1) we define the hierarchical base for L2(D) as follows.

At level 0, we choose three piecewise linear functions: φ01 takes obtains values (1, 0)at (0, 1/2) and is 0 in (1/2, 1), φ02 is continuous piecewise linear and obtains values(0, 1, 0) at (0, 1/2, 1), and φ03 obtains values (0, 1) at (1/2, 1) and is 0 in (0, 1/2). Foreach level, we choose piecewise wavelet function φ that takes values (0,−1, 2,−1, 0)at (0, 1/2, 1, 3/2, 2). The left boundary function φleft takes values (−2, 2,−1, 0) at(0, 1/2, 1, 3/2), and the right boundary function φright takes values (0,−1, 2,−2) at(1/2, 1, 3/2, 2). For levels j ≥ 1, the index set Ij = {1, 2, . . . , 2j}. The waveletfunctions are defined as φj1(x) = 2−j/2φleft(2jx), φjk(x) = 2−j/2φ(2jx− k+3/2) for

k = 2, . . . , 2j − 1 and φj2j

= φright(2jx− 2j + 2). This base satisfies Assumption 8.1(i).

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10−2

10−1

100

10−5

10−4

10−3

10−2

mesh size h

ener

gy e

rror

sparse tensor product

full tensor product

Fig. 1. The full and sparse tensor energy error for a one-dimensional two-scale problem.

(ii) For Y = (0, 1), we define the base of H1#(Y )/R as in (i) but exclude φ01,

φ03. At other levels, all the functions φleft and φright are replaced by the piecewiselinear functions that take values (0, 2,−1, 0) at (0, 1/2, 1, 3/2) and values (0,−1, 2, 0)at (1/2, 1, 3/2, 2), respectively.

When D = (0, 1)d, the basis functions can be constructed by taking the tensorproducts of the basis functions in (0, 1). They satisfy Assumption 8.1 after appropriatescaling; see [11].

Alternatively we can use generating systems, also called multilevel frames fordiscretization, as considered in [11] or [13].

8.2. One-dimensional two-scale problem. We first consider a one-dimensionaltwo-scale problem where A(x, y) = (2/3)(1 + x)(1 + cos2(2πy)) in the domain D =(0, 1). The two-scale problem is

∂2uε

∂t2− ∂

∂x

(2

3(1 + x)

(1 + cos2

(2πx

ε

)) ∂uε∂x

)= t3 + 9t

(x− log(1 + x)

log 2

)in D,

uε(t, 0) = uε(t, 1) = 0, uε(0, x) = 0,∂uε

∂t(0, x) = 0.

The two-scale limiting equation has the exact homogenized solution

u0(t, x) =3

2√2

(x− log(1 + x)

log 2

)t3

and the scale interaction term u1(x, y) (with an additive constant C),

u1(t, x, y) =3

2√2

(1− 1

(1 + x) log 2

)(1

2πtan−1

(tan 2πy√

2

)− y + C

)t3.

In Figure 1, we plot the error ‖u − uL‖V and the error ‖u − uL‖V versus themesh size hL for t = 1 for the full and sparse tensor product approximations. We

choose Δt = h1/2L /2 or the closest value so that 1/Δt is an integer. In particular, we

plot for (hL,Δt) = (1/4, 1/4), (1/8, 1/6), (1/16, 1/8), (1/32, 1/12), and (1/64, 1/16).

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We observe that the full and sparse tensor product FE errors are essentiallyequal despite the fact that the number of degrees of freedom we used for the sparsetensor approximation is much less than that for the full tensor product spaces. Thisconfirms the results of Proposition 5.6 that the sparse tensor FE method provides asolution with essentially equal accuracy as the full tensor product FEs, using a muchreduced number of degrees of freedom. Figure 1 shows clearly that both the errorsof sparse and full FEs behave as O(hL). In the log− log coordinates the multiplyingfactor | loghL|1/2 in the error of the sparse tensor FE does not show any effect. As

we choose the time step Δt to be of the order h1/2L , Figure 1 also shows that we

achieve the error O((Δt)2) with respect to t when the solution is sufficiently smooth,as established in Proposition 5.6.

8.3. One-dimensional three-scale problem. We then consider a one-dimensionalthree-scale problem where A(x, y1, y2) = (4/9)(1+x)(1+cos2(2πy1))(1+cos2(2πy2)).The problem is

∂2uε

∂t2− ∂

∂x

(4

9(1 + x)

(1 + cos2

(2π

x

ε1

))(1 + cos2

(2π

x

ε2

))∂uε

∂x

)

= t3 +27t

4

(x− log(1 + x)

log 2

)in D,

uε(t, 0) = uε(t, 1) = 0, uε(0, x) = 0,∂uε

∂t(0, x) = 0.

The three-scale limiting problem has the exact homogenized solution

u0(t, x) =9

8

(x− log(1 + x)

log 2

)t3

and the scale interaction terms u1(x, y1) and u2(x, y1, y2) (C1 and C2 are constants)

u1(t, x, y1) =9

8

(1− 1

(1 + x) log 2

)(1

2πtan−1

(tan 2πy1√

2

)− y1 + C1

)t3,

u2(t, x, y1, y2)

=9

8

(1− 1

(1 + x) log 2

)(1

2πtan−1

(tan 2πy2√

2

)− y2 + C2

) √2

1 + cos2 2πy1t3.

In Figure 2 the error ‖u − uL‖V is plotted versus the mesh size for the sparsetensor product for t = 1. The mesh size h and the time step Δt are chosen asfor the two-scale problem above. This example confirms our theoretical results inProposition 5.6 for three scale problems. In terms of the finite element mesh hL, inthe log− log coordinates, the error ‖u − uL‖V versus the mesh size hL is a straightline of slope 1, i.e., the O(hL) behavior is dominant. Although the theoretical resultof Proposition 5.6 states that the error is now of the order L3/2hL, i.e., the effect ofthe log multiplying factor should be stronger than in the previous two-scale example,

it still does not show any significant effects in this figure. As we choose Δt = O(h1/2L ),

the example shows that the error O((Δt)2) is achieved with respect to the time stepwhen the solution is sufficiently regular.

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10−2

10−1

100

10−5

10−4

10−3

10−2

10−1

mesh size h

ener

gy e

rror

Fig. 2. The sparse tensor energy error for a one-dimensional three-scale problem.

8.4. Two-dimensional two-scale problem. We consider in D = (0, 1)×(0, 1),the two-scale problem

∂2uε

∂t2−∇x · (aε(x)∇xu

ε) = f(t, x) in D,

uε(t, ·) = 0 on ∂D, uε(0, x) = 0,∂uε

∂t(0, x) = 0,

where

aε(x) = (1 + x1)(1 + x2)(1 + cos2

(2πx1ε

))(1 + cos2

(2πx2ε

)),

f(t, x) = 6t(x1 − x21)(x2 − x22)−∇ ·(

3√2(1 + x1)(1 + x2)∇

((x1 − x21)(x2 − x22)

))t3.

Following Jikov, Kozlov, and Oleınik [19, page 17], the homogenized coefficient canbe computed exactly. This problem has the exact homogenized solution

u0(t, x) = (x1 − x21)(x2 − x22)t3

and the scale interaction terms

u1(t, x, y) = (1 − 2x1)(x2 − x22)

(1

2πtan−1

(tan 2πy1√

2

)− y1 + C1

)t3

+ (x1 − x21)(1− 2x2)

(1

2πtan−1

(tan 2πy2√

2

)− y2 + C2

)t3.

In Figure 3 the error ‖u − uL‖V is plotted versus the mesh size for the sparsetensor product for (Δt, h) = (1/4, 1/4), (1/8, 1/6), and (1/16, 1/8) for t = 1. Thestraight line of slope 1 once more confirms the theoretical results in Proposition 5.6.The predominant behavior of the error is O(hL) in terms of the finite element mesh

and is O((Δt)2) in terms of the time step, as we choose Δt = O(h1/2L ). This is achieved

as the solution is sufficiently regular.

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10−2

10−1

100

10−4

10−3

10−2

mesh sizeh

ener

gy e

rror

Fig. 3. The sparse tensor energy error for a two-dimensional two-scale problem.

Fig. 4. The analytical and numerical approximation to ∂uε

∂x2.

For this problem, we plot in Figure 4 the finite element approximation for ∂uε

∂x2in

Theorem 7.3 at t = 1

∂uL0∂x2

+ Uε

(∂uL1∂y2

)and compare it to the analytical approximation

∂u0(x)

∂x2+∂u1∂y2

(x,x

ε

)in Proposition 6.1.

In Figure 4, the picture on the left is the analytical approximation ∂u0

∂x2(x) +

∂u1

∂y2(x, xε ) and the picture on the right is its numerical approximation

∂uL0

∂x2+Uε

(∂uL1

∂y2

).

We choose h = 1/8 and ε = 1/64. The surface of the right picture correctly representsthe one shown in the left, which is the analytical approximation of the derivative ∂uε

∂x1.

This shows that the numerical corrector constructed in section 7 for the two-scaleproblem, in particular in Theorem 7.3, using the operator Uε, provides an accuratedescription of the behavior of the multiscale solution uε in the H1(D) norm. We note

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00.05

0.10.15

0.20.25

0

0.05

0.1

0.15−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

00.05

0.10.15

0.20.25

0

0.05

0.1

0.15−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 5. Zoomed in Figure 4.

two features. In the numerical approximation∂uL

0

∂x2+Uε

(∂uL1

∂y2

), the piecewise constant

behavior of∂uL

0

∂x2is quite clear due to the piecewise linear finite element approximation

of the solution u0, although globally the numerical corrector still represents correctlythe exact solution. In Figure 5 we show a “zoomed in” portion of Figure 4 where theleft picture is the analytical approximation in Proposition 6.1 and the right picture isthe numerical approximation in Theorem 7.3. Both pictures show clearly the locallyoscillating behavior of the exact and the approximated solution constructed from theoperator Uε.

9. Conclusions. In this paper, we propose the full and sparse tensor productfinite element methods for multiscale wave equations. With sufficient regularity, weachieve the rate of convergence O(Δt)2 + O(hL| log hL|n/2) in Proposition 5.6 forthe sparse tensor approximation. The number of degrees of freedom used is NL =O(hdL| log hL|n). For solving problem (5.7), the term (∂2t u

L0,m, v

L0 ) only contributes a

low dimension matrix. Solving the system by conjugate gradient method, the maincost is for performing matrix-vector multiplication of the stiffness matrix of the bilin-ear form B. As shown in Hoang and Schwab [15], with an appropriate matrix vectormultiplication algorithm, the linear problem in each step can be solved with a total of

O(NL)(log NL)n+d(n+1) floating point operations. Letting Δt = O(h

1/2L ) the number

of time steps required is O(h−1/2L ), which is O(NL)

1/(2d)(log NL)−n/(2d). Thus the to-

tal number of floating point operations needed is O(N1/(2d)+1L )(logNL)

−n/(2d)+n+d(n+1).Apart from the log factor, the rate of convergence and the complexity involved areindependent of the number of scales. The numerical examples for two- and three-scaleproblems confirm the analysis. Our method not only produces the solution u0 of thehomogenized equation but also the full corrector for a class of problems and a part ofthe corrector for the general problems. Using them we can get the local behavior of uε

by computing an approximation for ∇uε. We have run a simulation to compute thisfor the whole two dimensional domain (0, 1)2, even though this is the local behaviorand only needs to be computed at points of interest.

Appendix A. In this appendix, we present the proof of Lemma 3.6.

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Proof. Averaging equation (3.4) at time steps tm+1, tm, and tm−1 with weights1/4, 1/2, and 1/4 we get

B(wLm,1/4,v

L) =

(fm,1/4 −

∂2u0,m,1/4

∂t2, vL0

)H

.

From this and (3.12) we have

(∂2t u

L0,m, v

L0

)H+B(ζLm,1/4,v

L) =

(∂2u0,m,1/4

∂t2, vL0

)H

.

Therefore,

(∂2t ζ

L0,m, v

L0

)H+B(ζLm,1/4,v

L) =

(−∂2twL

0,m +∂2u0,m,1/4

∂t2, vL0

)H

.

Since wL0 = u0 + ηL0 , we deduce that

(∂2t ζ

L0,m, v

L0

)H+B(ζLm,1/4,v

L) =

(−∂2t ηL0,m − ∂2t u0,m +

∂2u0,m,1/4

∂t2, vL0

)H

.

Choose vL = δtζLm, we then have

(∂2t ζ

L0,m, δtζ

L0,m

)H+B(ζLm,1/4, δtζ

Lm) =

(−∂2t ηL0,m − ∂2t u0,m +

∂2u0,m,1/4

∂t2, δtζ

L0,m

)H

.

We have the following relationships:

∂2t rm =1

Δt

(∂trm+1/2 − ∂trm−1/2

), rm,1/4 =

1

2(rm+1/2 + rm−1/2),

δtrm =1

2

(∂trm+1/2 + ∂trm−1/2

)=

1

Δt(rm+1/2 − rm−1/2).

Let sm =∂2u0,m,1/4

∂t2 − ∂2t u0,m, we get

1

2Δt

[(∂tζ

L0,m+1/2 − ∂tζ

L0,m−1/2, ∂tζ

L0,m+1/2 + ∂tζ

L0,m−1/2

)H

+B(ζLm+1/2 + ζLm−1/2, ζ

Lm+1/2 − ζLm−1/2)

]=

1

2

(sm − ∂2t η

L0,m, ∂tζ

L0,m+1/2 + ∂tζ

L0,m−1/2

)H.

As B(·, ·) is a symmetric bilinear form, we deduce

1

2Δt

[∥∥∥∂tζL0,m+1/2

∥∥∥2H−∥∥∥∂tζL0,m−1/2

∥∥∥2H+B(ζLm+1/2, ζ

Lm+1/2)−B(ζLm−1/2, ζ

Lm−1/2)

]≤ 1

2

∥∥sm − ∂2t ηL0,m

∥∥H

∥∥∥∂tζL0,m+1/2 + ∂tζL0,m−1/2

∥∥∥H

≤ 1

2γ

(‖sm‖2H +

∥∥∂2t ηL0,m∥∥2H)+γ

2

(∥∥∥∂tζL0,m+1/2

∥∥∥2H+∥∥∥∂tζL0,m−1/2

∥∥∥2H

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for all γ > 0. For 1 ≤ j < M , summing this up over m = 1, . . . , j, we deduce that[∥∥∥∂tζL0,j+1/2

∥∥∥2H−∥∥∥∂tζL0,1/2∥∥∥2

H+B(ζLj+1/2, ζ

Lj+1/2)−B(ζL1/2, ζ

L1/2)

]

≤ 1

γΔt

M∑m=1

(‖sm‖2H +

∥∥∂2t ηL0,m∥∥2H)+ 2γT max1≤m<M

∥∥∥∂tζL0,m+1/2

∥∥∥2H

+ γΔt‖∂tζL0,1/2‖2H .

Choosing γ sufficiently small, we deduce that

max1≤j<M

[∥∥∥∂tζL0,j+1/2

∥∥∥2H+B(ζLj+1/2, ζ

Lj+1/2)

]≤ cΔt

M∑m=1

(‖sm‖2H +

∥∥∂2t ηL0,m∥∥2H)+ c‖∂tζL0,1/2‖2H + c‖ζL1/2‖2V.

Following Dupont [8], using the integral formula of the remainder of Taylor expansion,we write

∂2t ηL0,m = (Δt)

−2∫ Δt

−Δt

(Δt− |τ |) ∂2ηL0∂t2

(tm + τ) dτ ;

this implies that

M∑m=1

‖∂2t ηL0,m‖2HΔt ≤ 4

3

∥∥∥∥∂2ηL0∂t2

∥∥∥∥2L2(0,T ;H)

.

For sm, we have

sm =1

4

∫ Δt

0

(1− 2

(1− |τ |

Δt

)2)∂3u0∂t3

(tm + τ)dτ

− 1

4

∫ 0

−Δt

(1− 2

(1− |τ |

Δt

)2)∂3u0∂t3

(tm + τ)dτ.

Therefore,

‖sm‖2H ≤ cΔt

∫ tm

tm−1

∥∥∥∥∂3u0∂t3(τ)

∥∥∥∥2H

dτ.

From (2.1) and (2.2), we get

∥∥∂tζL0 ∥∥2L∞(0,T ;H)+ ‖ζL‖2

L∞(0,T ;V)

≤ c

[(Δt)

2

∥∥∥∥∂3u0∂t3

∥∥∥∥2L2(0,T ;H)

+

∥∥∥∥∂2ηL0∂t2

∥∥∥∥2L2(0,T ;H)

+ ‖∂tζL0,1/2‖2H + ‖ζL1/2‖2V

].

If ∂4u0/∂t4 ∈ L2(0, T ;H), we have

sm =1

12

∫ Δt

−Δt

(Δt− |τ |)(3− 2

(1− |τ |

Δt

)2)∂4u0∂t4

(tm + τ)dτ.

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From this, we get

‖sm‖2H ≤ c(Δt)3∫ tm+1

tm−1

∥∥∥∥∂4u0∂t4

∥∥∥∥2H

dτ.

From (2.1) and (2.2) we get∥∥∂tζL0 ∥∥2L∞(0,T ;H)+ ‖ζL‖2

L∞(0,T ;V)

≤ c

[(Δt)

4

∥∥∥∥∂4u0∂t4

∥∥∥∥2L2(0,T ;H)

+

∥∥∥∥∂2ηL0∂t2

∥∥∥∥2L2(0,T ;H)

+ ‖∂tζL0,1/2‖2H + ‖ζL1/2‖2V

].

Appendix B. We make some remarks on the choice of uL0 and uL

1 so that theconditions of Proposition 5.6 are satisfied.

We assume conditions (4.7) and that the domain is convex. We find uL1 ∈V L1 , . . . , u

Ln ∈ V L

n such that∫D

∫Y

A(x,y)(∇g0 +∇y1 uL1 + · · ·+∇yn u

Ln) · (∇y1 v

L1 + · · ·+∇yn v

Ln )dxdy = 0

for all vL1 ∈ V L1 , . . . , v

Ln ∈ V L

n . We have

‖u1 − uL1 ‖V1 + · · ·+ ‖un − uLn‖Vn ≤ chLLn/2.

We find uL0 ∈ V L by projection such that ‖uL0−g0‖V ≤ chL and set uL0 = (uL0 , u

L1 , . . . , u

Ln).

The functions uL1t ∈ V L1 , . . . , u

Lnt ∈ V L

n are found by solving∫D

∫Y

A(x,y)(∇g1 +∇y1 uL1t + · · ·+∇yn u

Lnt) · (∇y1 v

L1 + · · ·+∇yn v

Ln )dxdy = 0

for all vL1 ∈ V L1 , . . . , v

Ln ∈ V L

n . From (4.7), ∂u0

∂t (0) ∈ H2(D) so ∂ui

∂t ∈ Hi. We thenhave

‖uL1t −∂u1∂t

(0)‖V1 + · · ·+ ‖uLnt −∂un∂t

(0)‖Vn ≤ chLLn/2.

Letting gL1 ∈ V L be such that ‖gL1 − g1‖V ≤ chL, we then set uL0t = gL1 , uL0t =

(uL0t, uL1t, . . . , u

Lnt). We find uL0tt ∈ V L such that ‖uL0tt − ∂2u0

∂t2 ‖H ≤ chL and ‖uL0tt‖V ≤‖∂2u0

∂t2 ‖V . We choose uL1 as

uL1 = uL

0 +ΔtuL0t +

1

2(Δt)2uL0tt.

We have

∂tζL0,1/2 =

1

Δt((uL0,1 − uL0,0 − (wL

0,1 − wL0,0))

= uL0t +1

2ΔtuL0tt −

∂wL0

∂t(0)− 1

2Δt

∂2wL0

∂t2(0) +X,

where ‖X‖H ≤ c(Δt)2. We note that∥∥∥∥uL0t − ∂wL0

∂t(0)

∥∥∥∥H

≤∥∥∥∥uL0t − ∂u0

∂t(0)

∥∥∥∥H

+

∥∥∥∥∂u0∂t (0)− ∂wL0

∂t(0)

∥∥∥∥H

≤ chLLn/2

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due to Lemma 3.3. Similarly, we have∥∥∥∥uL0tt − ∂2wL0

∂t2(0)

∥∥∥∥H

≤∥∥∥∥uL0tt − ∂2u0

∂t2(0)

∥∥∥∥H

+

∥∥∥∥∂2u0∂t2(0)− ∂2wL

0

∂t2(0)

∥∥∥∥H

≤ chLLn/2.

Therefore, ‖∂tζL0,1/2‖H ≤ c((Δt)2 + hLLn/2). Further,

‖ζL1/2‖V ≤ 1

2(‖ζL0 ‖V + ‖ζL1 ‖V)

≤ 1

2‖uL

0 − u0‖V +1

2‖u0 −wL

0 ‖V +1

2

∥∥∥∥uL0 +ΔtuL

0t − u0 −Δt∂u

∂t(0)

∥∥∥∥V

+1

2

∥∥∥∥u0 +Δt∂u

∂t(0)−wL

0 −Δt∂wL

∂t(0)

∥∥∥∥V

+ c(Δt)2

≤ c((Δt)2 + hLLn/2).

If only (4.5) holds, we require O(Δt) + hLLn/2 error and, therefore, do not need

to consider ∂2u0

∂t2 (0). Otherwise, to compute ∂2u0

∂t2 (0), we note that

∂2u0∂t2

(0) = f(0)−∇ ·∫Y

A(x,y)(∇u0(0) +∇y1u1(0) + · · ·+∇ynun(0))dy

= f(0)−∫Y

∇A(x,y)(∇u0(0) +∇y1u1(0) + · · ·+∇ynun(0))dy

−∫Y

A(x,y)∇((∇u0(0) +∇y1u1(0) + · · ·+∇ynun(0))dy.

The spatial derivative of ∂2u0

∂t2 (0) can be computed as follows. We note that for almostall x ∈ D, for all φ1 ∈ V1, . . . , φn ∈ Vn which does not depend on x, we have∫

Y

A(x,y)(∇u0(0) +∇y1u1(0) + · · ·+∇ynun(0)) · (∇y1φ1 + · · ·+∇ynφn)dy = 0.

Therefore,∫Y

A(x,y)∂

∂xk(∇u0(0) +∇y1u1(0) + · · ·+∇ynun(0)) · (∇y1φ1 + · · ·+∇ynφn)dy

= −∫Y

∂A(x,y)

∂xk(∇u0(0) +∇y1u1(0) + · · ·+∇ynun(0)) · (∇y1φ1 + · · ·+∇ynφn)dy.

Thus for all φ1 ∈ V1, . . . , φn ∈ Vn,∫D

∫Y

A(x,y)

(∇ ∂

∂xku0(0) +∇y1

∂

∂xku1(0) + · · ·+∇yn

∂

∂xkun(0)

)· (∇y1φ1 + · · ·+∇ynφn)dydx

= −∫D

∫Y

∂A(x,y)

∂xk(∇u0(0) +∇y1u1(0) + · · ·+∇ynun(0))

· (∇y1φ1 + · · ·+∇ynφn)dydx.

Once the approximation uL(0) has been computed, we can then compute an approx-imation for ∂u

∂xk(0) in V. Other spatial derivatives of u(0) can be computed similarly.

From this, we can approximate ∇∂2u0

∂t2 and find uL0tt.

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If g0 = 0 as we consider in the next section, the problem becomes much easier asu(0) = 0. Note that if g0 = 0, in order for condition (4.7c) to hold, g0 ∈ H3

0 (D) andfor (4.7e) to hold, A0(x) needs to be in W 2,∞(D), i.e., A(x,y) needs to be in W 2,∞

with respect to x.

Appendix C. We now present the proof of Proposition 6.1.Proof. First, we establish the regularity

(C1) u0 ∈ H2(0, T ;V ) ∩ L∞(0, T ;H2(D)),∂u0∂t

∈ L∞(0, T ;V ∩H2(D)).

As g0 = 0, condition (4.5) holds. From (4.10), we have ∂2u0

∂t2 ∈ L∞(0, T ;V ) and∂3u0

∂t3 ∈ L∞(0, T ;H). From (4.9), we have

−∇ ·(A0∇∂u0

∂t

)=∂f

∂t− ∂3u0

∂t3∈ L∞(0, T ;H).

We, therefore, deduce that ∂u0

∂t ∈ L∞(0, T ;H2(D)).For uε, we have that

(C2)∂2

∂t2

(∂uε

∂t

)−∇ ·

(Aε∇∂uε

∂t

)=∂f

∂t,

with the compatibility condition

∂uε

∂t(0) = g1 ∈ V,

∂

∂t

∂uε

∂t(0) = f(0)−∇ · (Aε(·)∇g0(·)) = f(0) ∈ H.

Thus

(C3)∂uε

∂t∈ L2(0, T ;V ) ∩H1(0, T ;H).

Let

uε1(t, x) = u0(t, x) + εu1

(t, x,

x

ε

).

We first show that

‖∇ · (Aε∇uε1)−∇ · (A0∇u0)‖L∞(0,T ;H−1(D)) ≤ cε.

For elliptic problems, Jikov, Kozlov, and Oleınik [19] shows this for the case whereA(x, y) = A(y) and for smooth u0. However, the result also holds for the case whereA(x, y) depends on x and for the weaker regularity u0 ∈ H2(D). This is done in detailin Hoang and Schwab [16]. We thus only present the main ideas of the proof here.

We note that

(Aε∇uε1)i(t, x) = A0ij(x)

∂u0∂xj

(t, x)

+ gji

(x,x

ε

)∂u0∂xj

(t, x) + εwk

(x,x

ε

)Aij

(x,x

ε

)∂2u0∂xj∂xk

(t, x)

+ εAik

(x,x

ε

)∂wj

∂xk

(x,x

ε

)∂u0∂xj

(t, x),

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where

gji (x, y) = Aij(x, y) +Aik(x, y)∂wj

∂yk(x, y)−A0

ij(x).

From (6.1) and (6.2), for every x ∈ D, we have∫Y

gji (x, y)dy = 0 and∂

∂yigji (x, y) = 0.

Adapting the proof of Jikov, Kozlov, and Oleınik [19] for the case A(x, y) = A(y),Hoang and Schwab [16] show that there are functions αk

ij(x, y) which are Y -periodic

with respect to y such that αkij = −αk

ji and

gki (x, y) =∂

∂yjαkij(x, y),

and that when A(x, y) ∈ (C1(D, C1#(Y )))d×d, αk

ij(x, y) ∈ C1(D, C#(Y )). We there-fore have

(Aε∇uε1(t, x)−A0(x)∇u0(t, x))i = ε∂

∂xj

(αkij

(x,x

ε

)∂u0∂xk

(t, x)

)+ (rε)i(t, x),

where

(rε)i(t, x) = −ε∂αk

ij(x, y)

∂xj

∣∣∣y=x/ε

∂u0(t, x)

∂xk(t, x)− εαk

ij

(x,x

ε

)∂2u0∂xk∂xj

(t, x)

+ εwk

(x,x

ε

)Aij

(x,x

ε

)∂2u0∂xk∂xj

(t, x)

+ εAik

(x,x

ε

)∂wj

∂xk

(x,x

ε

)∂u0∂xj

(t, x).

As αkij(x, y) ∈ C1(D, C(Y )), we deduce from (C1) that ‖(rε)i(·)‖L∞(0,T ;H) ≤ cε.

Using αkij = −αk

ji we get

(C4) ‖∇ · (Aε∇uε1(t, x)) −∇ · (A0∇u0(t, x))‖L∞(0,T ;H−1(D)) ≤ cε.

Let τε ∈ C∞0 (D) be such that τε = 1 outside an ε neighborhood of ∂D and ε|∇τε(x)| ≤

c for all ε > 0. Let

wε1(t, x) = u0(t, x) + ετε(x)wk

(x,x

ε

)∂u0∂xk

(t, x)(C5)

= uε1(t, x)− ε(1− τε(x))wk

(x,x

ε

)∂u0∂xk

(t, x).

We have that

∇(uε1 − wε1)(t, x) = −ε∇τε(x)wk

(x,x

ε

)∂u0∂xk

(t, x)(C6)

+ (1− τε(x))∇ywk

(x,x

ε

)∂u0∂xk

(t, x)

+ ε(1− τε(x))∇wk

(x,x

ε

)∂u0∂xk

(t, x)

+ ε(1− τε(x))wk

(x,x

ε

)∇∂u0∂xk

(t, x).

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Thus

∇ · (Aε(∇(uε1 − wε1))

= ε∇ ·(Aε(x)(1 − τε(x))∇wk

(x,x

ε

)∂u0∂xk

(t, x)

)

+ ε∇·(Aε(x)(1 − τε(x))wk

(x,x

ε

)∇∂u0∂xk

(t, x)

)

+∇·(Aε(x)

(−ε∇τε(x)wk

(x,x

ε

)+ (1− τε(x))∇yw

k

(x,x

ε

))∂u0∂xk

(t, x)

).(C7)

Since ∂2u0

∂t2 ∈ L∞(0, T ;V ), we deduce

(C8)

∥∥∥∥∂2wε1

∂t2− ∂2u0

∂t2

∥∥∥∥L∞(0,T ;H)

≤ cε.

Using

∂2uε

∂t2−∇ · (Aε∇uε) = ∂2u0

∂t2−∇ · (A0∇u0),

from (C4), (C7), and (C8) we deduce that

∂2(uε − wε1)

∂t2−∇ · (Aε∇(uε − wε

1))

=∂2u0∂t2

− ∂2wε1

∂t2+∇ · (Aε∇(wε

1 − uε1)) +∇ · (Aε∇uε1)−∇ · (A0∇u0)

=∂2u0∂t2

− ∂wε1

∂t2+∇ · (Aε∇uε1)−∇ · (A0∇u0)

− ε∇ ·(Aε(1− τε)∇wk

(·, ·ε

)∂u0∂xk

)− ε∇ ·

(Aε(1− τε)wk

(·, ·ε

)∇∂u0∂xk

)

−∇ ·(Aε

(−ε∇τεwk

(·, ·ε

)+ (1− τε)∇yw

k

(·, ·ε

))∂u0∂xk

)

= X −∇ ·(Aε

(−ε∇τεwk

(·, ·ε

)+ (1− τε)∇yw

k

(·, ·ε

))∂u0∂xk

),

where X denotes the sum of the first six terms. From (C1), (C4), and (C8) wededuce that ‖X‖L∞(0,T ;H−1(D)) ≤ cε. Since ∂u0

∂t ∈ L∞(0, T ;V ∩H2(D)), we have that∂wε

1

∂t ∈ L∞(0, T ;V ) and supt∈(0,T ) ‖∂wε

1

∂t (t)‖V < c for c independent of ε. Therefore,∫ t

0

(∂2(uε − wε

1)

∂s2(s, ·), ∂(u

ε − wε1)

∂s(s, ·)

)H

ds

+

∫ t

0

∫D

Aε(x)∇(uε(s, x)− wε1(s, x)) ·

∂

∂s∇(uε(s, x)− wε

1(s, x))dxds

=

∫ t

0

(X(s, ·), ∂(u

ε − wε1)

∂s(s, ·)

)H

ds

+

∫ t

0

∫D

Aε(x)

(−ε∇τε(x)wk

(x,x

ε

)+ (1− τε(x))∇yw

k

(x,x

ε

))

· ∂u0∂xk

(s, x)∂

∂s∇(uε(s, x) − wε

1(s, x))dxds

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≤ c‖X‖L2(0,T ;H−1(D))

(∥∥∥∥∂uε∂s

∥∥∥∥L2(0,T ;V )

+

∥∥∥∥∂wε1

∂s

∥∥∥∥L2(0,T ;V )

)

+

∫ t

0

∫D

Aε(x)

(−ε∇τε(x)wk

(x,x

ε

)+ (1 − τε(x))∇yw

k

(x,x

ε

))

·[∂

∂s

(∂u0∂xk

(s, x)∇(uε(s, x)− wε1(s, x))

)− ∂2u0∂s∂xk

(s, x)∇(uε(s, x)− wε1(s, x))

]dxds.

Let Dε ⊂ D be an ε neighborhood of ∂D outside which τε(x) = 1. As uε(0, x) = 0and wε

1(0, x) = 0, and∥∥∥∂uε∂t

(0)− ∂wε1

∂t(0)∥∥∥H

=∥∥∥∂u0∂t

(0, ·)− ∂wε1

∂t(0)∥∥∥H

≤ cε∥∥∥ ∂∂t

∇u0(0)∥∥∥H,

we deduce that∥∥∥∂(uε − wε1)

∂t(t)∥∥∥2H+ ‖∇ (uε(t)− wε

1(t)) ‖2H ≤ cε

+ c∥∥∥∂u0∂xk

(t)∥∥∥L2(Dε)

‖∇(uε(t)− wε1(t))‖H

+ c

∫ T

0

∥∥∥ ∂2u0∂t∂xk

(s)∥∥∥L2(Dε)

ds sup0≤t≤T

‖∇(uε(t)− wε1(t))‖H .

Following Hoang and Schwab [16], as ∂D is Lipschitz, for all functions φ ∈ C∞(D)we have ‖φ‖2

L2(Dε)≤ cε2‖φ‖2H1(D) + cε‖φ‖2L2(∂D), so

(C9) ‖φ‖L2(Dε) ≤ cε1/2‖φ‖H1(D) ∀φ ∈ H1(D).

We, therefore, have ∥∥∥∂u0∂xk

(t, ·)∥∥∥L2(Dε)

≤ cε1/2∥∥∥∂u0∂xk

(t, ·)∥∥∥H1(D)

and ∥∥∥ ∂2u0∂t∂xk

(t, ·)∥∥∥L2(Dε)

≤ cε1/2∥∥∥ ∂2u0∂t∂xk

(t, ·)∥∥∥H1(D)

.

From (C1), we deduce that for all t ∈ (0, T )∥∥∥∂(uε − wε1)

∂t(t)∥∥∥2H+ ‖∇(uε − wε

1)(t)‖2H ≤ cε+ cε1/2‖∇(uε − wε1)(t)‖H

+ cε1/2 supt∈(0,T )

‖∇(uε − wε1)(t)‖H ,

which implies ∥∥∥∂(uε − wε1)

∂t(t)∥∥∥H+ ‖∇(uε − wε

1)(t)‖H ≤ cε1/2.

From (C1) and (C6) we have that∥∥∥∂(uε1 − wε1)

∂t

∥∥∥L∞(0,T ;H)

≤ cε.

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Further, from (C1) and (C7) by using (C9) we get

‖∇(uε1 − wε1)‖L∞(0,T ;H) ≤ cε1/2.

We then get the conclusion.

Appendix D. We prove Proposition 6.4 in this appendix.We consider the expression

Eε(t) =

∫D

(∂uε∂t

(t)− ∂u0∂t

(t))2dx

+

∫D

∫Y

T εn (A

ε)(T εn (∇uε(t))− (∇u0(t) +∇y1u1(t) + · · ·+∇ynun(t)))

· (T εn (∇uε(t))− (∇u0(t) +∇y1u1(t) + · · ·+∇ynun(t)))dydx.

From (6.3) and the conservation of energy of wave equations (see Lions and Margenes[20], also Brahim-Otsmane, Francfort, and Murat [5]) we have∫

D

(∂uε∂t

)2dx+

∫D

∫Y

T εn (A

ε)T εn (∇uε(t)) · T ε

n (∇uε(t))dydx

=

∫D

(∂uε∂t

)2dx+

∫D

Aε∇uε(t) · ∇uε(t)dx

=

∫D

g21dx+ 2

∫ t

0

∫D

f∂uε

∂tdxdt

→∫D

g21dx+ 2

∫ t

0

∫D

f∂u0∂t

dxdt.

From this and (6.4), we have that

limε→0

Eε(t) =

∫D

g21dx+ 2

∫ t

0

∫D

f∂u0∂t

dxdt

−[∫

D

(∂u0∂t

)2dx+

∫D

∫Y

A(x,y)(∇u0(t) +∇y1u1(t) + · · ·+∇ynun(t))

· (∇u0(t) +∇y1u1(t) + · · ·+∇ynun(t))dydx

].

Since A0 is the homogenized coefficient of the multiscale coefficient A(x,y), we have∫D

∫Y

A(x,y)(∇u0(t) +∇y1u1(t) + · · ·+∇ynun(t))

· (∇u0(t) +∇y1u1(t) + · · ·+∇ynun(t))dydx

=

∫D

A0∇u0(t) · ∇u0(t)dx.

From (4.3), we deduce that∫D

(∂u0∂t

)2dx+

∫D

A0∇u0 · ∇u0dx =

∫D

g21dx+ 2

∫ t

0

∫D

f∂u0∂t

dxdt.

Thus limε→0Eε(t) = 0.

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Next, we show that the convergence is uniform. From (C2) in Appendix C,we find that ∂uε

∂t is uniformly bounded in H1(0, T ;H)⋂L2(0, T ;V ) with respect to

ε and ∂u0

∂t is in H1(0, T ;H)⋂L2(0, T ;V ). Also from (4.9), ∂u0

∂t ∈ L2(0, T ;V ) so∂u∂t ∈ L2(0, T ;V).

We show that Eε(t) is equicontinuous with respect to ε. Considering the firstterm, we have for s < t∣∣∣∣∣

∫D

[(∂uε

∂t(t)− ∂u0

∂t(t)

)2

−(∂uε

∂t(s)− ∂u0

∂t(s)

)2]dx

∣∣∣∣∣=

∣∣∣∣∣∫D

((∂uε

∂t(t)− ∂uε

∂t(s)

)−(∂u0∂t

(t)− ∂u0∂t

(s)

))

·(∂uε

∂t(t) +

∂uε

∂t(s)− ∂u0

∂t(t)− ∂u0

∂t(s)

)dx

∣∣∣∣∣≤(∥∥∥∥∂uε∂t (t)− ∂uε

∂t(s)

∥∥∥∥H

+

∥∥∥∥∂u0∂t (t)− ∂u0∂t

(s)

∥∥∥∥H

)

·(∥∥∥∥∂uε∂t (t)

∥∥∥∥H

+

∥∥∥∥∂uε∂t (s)

∥∥∥∥H

+

∥∥∥∥∂u0∂t (t)

∥∥∥∥H

+

∥∥∥∥∂u0∂t (s)

∥∥∥∥H

).

From the regularity ∂uε

∂t ∈ H1(0, T ;H) in (C3), we have that ‖∂uε

∂t (t)‖H is uniformly

bounded for all t. Similarly, ‖∂u0

∂t (t)‖H is uniformly bounded for all t. Therefore,∣∣∣∣∣∫D

[(∂uε

∂t(t)− ∂u0

∂t(t)

)2

−(∂uε

∂t(s)− ∂u0

∂t(s)

)2]dx

∣∣∣∣∣≤ c

(∥∥∥∥∫ t

s

∂2uε

∂t2(τ)dτ

∥∥∥∥H

+

∥∥∥∥∫ t

s

∂2u0∂t2

(τ)dτ

∥∥∥∥H

)

≤ c

∫ t

s

(∥∥∥∥∂2uε∂t2(τ)

∥∥∥∥H

+

∥∥∥∥∂2u0∂t2(τ)

∥∥∥∥H

)dτ

≤ c(t− s)1/2

(∫ t

s

(∥∥∥∥∂2uε∂t2(τ)

∥∥∥∥2H

+

∥∥∥∥∂2u0∂t2(τ)

∥∥∥∥2H

)dτ

)1/2

≤ c(t− s)1/2

due to ‖∂uε

∂t ‖H1(0,T ;H) is uniformly bounded with respect to ε and ∂u0

∂t ∈ H1(0, T ;H).We can deal with the second term in the expression of Eε(t) in a similar manner,

using the fact that uε is uniformly bounded in H1(0, T ;V ) and u ∈ H1(0, T ;V). Wethen get the conclusion from the Arzela–Ascoli theorem.

REFERENCES

[1] A. Abdulle and M. J. Grote, Finite element heterogeneous multiscale method for the waveequation, Multiscale Model. Simul., 9 (2011), pp. 766–792.

[2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992),pp. 1482–1518.

[3] G. Allaire and M. Briane, Multiscale convergence and reiterated homogenisation, Proc. Roy.Soc. Edinburgh Sect. A, 126 (1996), pp. 297–342.

[4] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Struc-tures, Stud. Math. Appl. 5, North–Holland, Amsterdam, 1978.

[5] S. Brahim-Otsmane, G. A. Francfort, and F. Murat, Correctors for the homogenizationof the wave and heat equations, J. Math. Pures Appl. (9), 71 (1992), pp. 197–231.

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[6] P. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North–Holland, Amsterdam, 1978.

[7] D. Cioranescu, A. Damlamian, and G. Griso, The periodic unfolding method in homoge-nization, SIAM J. Math. Anal., 40 (2008), pp. 1585–1620.

[8] T. Dupont, L2-estimates for Galerkin methods for second order hyperbolic equations, SIAMJ. Numer. Anal., 10 (1973), pp. 880–889.

[9] W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003),pp. 87–132.

[10] B. Engquist, H. Holst, and O. Runborg, Multiscale methods for the wave propagation inheterogeneous media, Commun. Math. Sci., 9 (2011), pp. 33–56.

[11] M. Griebel and P. Oswald, Tensor product type subspace splittings and multilevel iterativemethods for anisotropic problems, Adv. Comput. Math., 4 (1995), pp. 171–206.

[12] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman(Advanced Publishing Program), Boston, 1985.

[13] H. Harbrecht and C. Schwab, Sparse tensor finite elements for elliptic multiple scale prob-lems, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 3100–3110.

[14] V. H. Hoang, Sparse finite element method for periodic multiscale nonlinear monotone prob-lems, Multiscale Model. Simul., 7 (2008), pp. 1042–1072.

[15] V. H. Hoang and Ch. Schwab, High-dimensional finite elements for elliptic problems withmultiple scales, Multiscale Model. Simul., 3 (2005), pp. 168–194.

[16] V. H. Hoang and Ch. Schwab, Analytic regularity and polynomial approximation of stochas-tic, parametric elliptic multiscale PDEs, Anal. Appl. (Singap.), 11 (2013), 1350001.

[17] T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in compositematerials and porous media, J. Comput. Phys., 134 (1997), pp. 169–189.

[18] L. Jiang, Y. Efendiev, and V. Ginting, Analysis of global multiscale finite element methodsfor wave equations with continuum spatial scales, Appl. Numer. Math., 60 (2010), pp. 862–876.

[19] V. V. Jikov, S. M. Kozlov, and O. A. Oleınik, Homogenization of Differential Operatorsand Integral Functionals, Springer-Verlag, Berlin, 1994, translated from the Russian by G.A. Yosifian [G. A. Iosifyan].

[20] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag,New York, 1972; translated from the French by P. Kenneth.

[21] G. Nguetseng, A general convergence result for a functional related to the theory of homoge-nization, SIAM J. Math. Anal., 20 (1989), pp. 608–623.

[22] H. Owhadi and L. Zhang, Numerical homogenization of the acoustic wave equations with acontinuum of scales, Comput. Methods Appl. Mech. Engrg., 198 (2008), pp. 397–406.

[23] F. Santosa and W. Symes, A dispersive effective medium for wave propagation in periodiccomposites, SIAM J. Appl. Math., 51 (1991), pp. 984–1005.

[24] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987;translated from the German by C. B. Thomas and M. J. Thomas.

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high dimensional finite elements for multiscale wave equations

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