Finite Difference Approximation of Homogenization Problems for Elliptic Equations

MULTISCALE MODEL. SIMUL. c© 2005 Society for Industrial and Applied MathematicsVol. 4, No. 1, pp. 36–87

FINITE DIFFERENCE APPROXIMATION OFHOMOGENIZATION PROBLEMS FOR ELLIPTIC EQUATIONS∗

RAFAEL ORIVE† AND ENRIQUE ZUAZUA‡

Abstract. In this paper, the problem of the approximation by finite differences of solutions toelliptic problems with rapidly oscillating coefficients and periodic boundary conditions is considered.The mesh size is denoted by h, while ε denotes the period of the rapidly oscillating coefficient. UsingBloch wave decompositions, we analyze the case where the ratio h/ε is rational. We show that ifh/ε is kept fixed, being a rational number, even when h, ε → 0, the limit of the numerical solutiondoes not coincide with the homogenized one obtained when passing to the limit as ε → 0 in thecontinuous problem. Explicit error estimates are given showing that, as the ratio h/ε approximatesan irrational number, solutions of the finite difference approximation converge to the solutions ofthe homogenized elliptic equation. We consider both the one-dimensional and the multidimensionalcase. Our analysis yields a quantitative version of previous results on numerical homogenization byAvellaneda, Hou, and Papanicolaou [RAIRO Model. Math. Anal. Numer., 25 (1991), pp. 693–710].

Key words. finite differences, homogenization, multiscale, Bloch waves

AMS subject classifications. 35B27, 65N06, 65N12

DOI. 10.1137/040606314

1. Introduction. Homogenization of elliptic equations with rapidly oscillatingperiodic coefficients is by now a well-understood problem. Roughly speaking, thelimit equation turns out to be elliptic, with constant coefficients, and the effectivecoefficients may be computed by solving an auxiliary problem on the unit periodiccell. The interested reader may find a fairly complete study of this problem in thebook by Bensoussan, Lions, and Papanicolaou [2].

One of the main applications of homogenization theory is related to thenumerical resolution of elliptic problems with rapidly oscillating coefficients. Moreprecisely, in agreement with the homogenization result mentioned above, instead ofapproximating the problem with rapidly oscillating coefficients one can solvenumerically the homogenized one. The latter is much easier to handle since itscoefficients are constants. For a long time this has been the only feasible numericalapproach to problems with rapidly oscillating coefficients since the directapplication of classical finite difference or finite element methods required meshesof a size h asymptotically smaller than the period of the rapidly oscillating coefficient.This made computations unfeasible in practice.

A new approach to the problem was proposed by Engquist in [15], who intro-duced the notion of convergence essentially independent of the wave of the oscillationin the approximation of oscillatory solutions of hyperbolic problems. The proof of thistype of convergence relies on fundamental results in ergodic theory and convergence

∗Received by the editors April 5, 2004; accepted for publication (in revised form) November 30,2004; published electronically June 3, 2005. This work was supported by grant BFM2002-03345 ofthe MCYT (Spain) and the Networks “Homogenization and Multiple Scales (RTN1-1999-00040)”and “New Materials, Adaptive Systems and Their Nonlinearities: Modelling, Control and NumericalSimulation (HPRN-CT-2002-00284)” of the EU.

http://www.siam.org/journals/mms/4-1/60631.html†Instituto de Matematicas y Fısica Fundamental, Consejo Superior de Investigaciones Cientıficas,

Serrano 123, 28006 Madrid, Spain ([email protected]).‡Departamento de Matematicas, Universidad Autonoma de Madrid, 28049 Madrid, Spain (enrique

[email protected]).

36

APPROXIMATION OF HOMOGENIZATION PROBLEMS 37

of random numbers in Monte Carlo methods (see Niederreiter [28]). This approachhas been successfully applied, in particular, in the context of numerical methods ofhyperbolic problems (Engquist and Hou [16] and Engquist and Liu [17]), finite dif-ference approximations of elliptic equations (Avellaneda, Hou, and Papanicolaou [1]),and multigrid methods of elliptic equations (Engquist and Luo [18]).

In recent years important progress has been made also in the context of finiteelements. The multiscale finite element method (MsFEM) was introduced by Houand Wu [20] and Hou, Wu, and Cai [21] for solving elliptic problems with oscillatorycoefficients. The idea of the MsFEM is to capture small-scale information throughthe base functions (which, in general, are of oscillatory nature) constructed in theelements whose sizes are much larger than the small scales of the problem. For adetailed analysis of the MsFEM we refer the reader to [12], [13], [14], and [21].

Another contribution is due to Matache, Babuska, and Schwab [25] and Matacheand Schwab [26], who, using Fourier–Bochner representation of solutions of the oscil-latory problem, proved that the numerical approximation problem may be attackeddirectly, without using the homogenization theory, by constructing special Galerkinbases adapted to the coefficients of the equation. This approach has also been success-fully applied to equations in rapidly oscillating perforated domains and, in principle,does not require periodicity assumptions.

In this work we present a complete study of the homogenization of the finitedifference schemes approximating a family of elliptic equations with rapidly oscillatingcoefficients.

This problem was previously investigated by Kozlov in [23]. He approximatedthe homogenized coefficients of second-order elliptic equations using linear polyno-mial interpolates of the coefficients of the finite difference system. Later, Piatnitskiand Remy in [31] studied the asymptotic behavior as ε → 0 of the effective coef-ficients for a family of random finite difference schemes. They proved the discreteanalogue of the compensated compactness lemma, adapted to difference operators,originally introduced by Murat in [27] in the continuous case. With it, they defined theH-convergence of difference operators.

In this work we study how the homogenization of the solutions of finite differenceapproximations of elliptic equations with rapidly oscillating coefficients is related tothe homogenized solutions of the corresponding differential operators. A first resultby Avellaneda, Hou, and Papanicolaou in [1] shows that if the mesh size h is of theorder of ε, the period of the oscillating coefficients, and the ratio h/ε is irrationalso that the numerical mesh correctly samples the spatial domain (with respect tothe coefficients of the equation under consideration), then solutions of the numericalapproximation do converge to the solution of the continuous homogenized problem.Moreover, in the multidimensional case, the finite difference approximation schemedoes not provide the right homogenized coefficients unless the components of the ratioh/ε are close enough to an integer number.

In this paper, we further pursue the approach in [1]. We analyze the case wherethe ratio h/ε is rational, using the Bloch wave decomposition at the discrete level. Weobtain explicit error estimates that depend, in particular, on the denominator of therational number h/ε. It is shown that the error tends to zero as the denominator tendsto infinity, and therefore, roughly speaking, the error tends to zero as h/ε approachesan irrational number. Thus, our results are in agreement with those of [1] mentionedabove and provide a constructive way of recovering them by means of classical resultson diophantine approximation and explicit error bounds.

To do this we use the Bloch wave decomposition at both the continuous and

38 RAFAEL ORIVE AND ENRIQUE ZUAZUA

the discrete level. We refer to the following works and the references therein foran introduction to the theory of Bloch waves: Bloch [3], Bensoussan, Lions, andPapanicolaou [2], and Conca, Planchard, and Vanninathan [9].

Our work is inspired by Conca and Vanninathan [10], where a new proof of theconvergence of solutions of elliptic problems with rapidly oscillating periodic coeffi-cients towards the homogenized solution was obtained using Bloch wave decompo-sition. In fact, in [10], it was shown that the problem of homogenization may bereduced to the analysis of the first Bloch mode. It was then established that theBloch waves representing the periodic medium approach Fourier waves representingthe homogenized one, and this fact may be easily interpreted as a homogenizationresult in the Fourier space.

We follow the same approach to analyze the behavior of uεh (this stands for the

numerical approximation with numerical mesh size h of the elliptic problem withrapidly oscillating coefficients of period ε) as ε → 0 and h → 0. In particular, ourapproach allows analyzing whether uε

h converges to the solution u∗ of the homogenizedequation. As we shall see, this can indeed be proved, provided h/ε approximates anirrational number.

But there is a difference in the way h/ε has to be chosen between the one-dimensional (1-d) and the multidimensional case. There are two reasons. First, weneed to guarantee that the discrete operator is positive semidefinite and also to esti-mate the difference between the homogenized coefficients of the continuous problemand the homogenized coefficients associated with the discrete system. According tothis, in the multidimensional case, further restrictions on h/ε are needed, other thanrequiring that h/ε approaches an irrational number.

We restrict ourselves to the simplest case of periodic boundary conditions withε tending to zero along a sequence such that the space domain contains exactly aninteger number of periodic cells of the rapidly oscillating coefficients. This simplifiessignificantly the Bloch representation of the solution at both the continuous andthe discrete level. Important further developments should occur to handle generaldomains. We refer the reader to Conca, Orive, and Vanninathan [7], [8] for theanalysis of the problem of homogenization of continuous elliptic problems in generalbounded domains with Dirichlet boundary conditions. Finally, we note that numericalexperiments using Bloch waves were performed by Conca and Natesan in [6] providinga better approximation to the exact homogenized solution than the classical first-ordercorrector in the case of smooth coefficients.

Organization of the paper. In section 2 we present our main results anddiscuss their significance. In section 3 we define the Bloch waves for discrete spacesand present several properties of Bloch eigenvalues and eigenvectors. The proof ofthese properties is given in Appendix C. Section 4 is devoted to the study of thenumerical problem using Bloch waves. We prove the error estimates in numericalhomogenization in several space dimensions. In section 5, we recover the resultsof [1] in the 1-d case when h/ε is irrational using our analysis and classical resultsin diophantine number theory. In Appendix A we discuss the continuous problemusing the Bloch wave decomposition, and in Appendix B we give a more explicit andsimpler proof of the numerical homogenization result in the 1-d case based on theexplicit representation of solutions.

Notation. Throughout this article the following notation will be used:

For any p = (p1, . . . , pd) ∈ Rd, pi �= 0, we denote1

p=

(1

p1, . . . ,

1

pd

).


For x, z ∈ Rd, we write xz = (x1z1, . . . , xdzd) ∈ Rd.

For x, z ∈ Rd, we denote x · z = x1z1 + · · · + xdzd.

For x ∈ Rd, its euclidean norm is denoted by |x| = (x21 + · · · + x2

d)12 .

For any p ∈ Nd, we write p = p1 · p2 · · · · · pd, the product of all the components pi.

The space of Y -periodic functions in Hsloc(R

d) will be denoted by Hs#(Y ). The norm

in the space Hs(Y ) will be denoted as ‖ · ‖s. In particular, the norm in L2(Y ) (whichcoincides with L2

#(Y )) is ‖ · ‖0.

2. Presentation of the problems and main results.

2.1. Homogenization problem with periodic boundary conditions. Letus introduce the continuous problem to be discretized later on. We consider theelliptic operator

A = − ∂

∂yi

(aij(y)

∂

∂yj

), y ∈ Rd,(2.1)

where the coefficients (aij) satisfy⎧⎪⎪⎨⎪⎪⎩aij ∈ L∞

# (Y ), where Y = [0, 2π[d, i.e., each coefficient aij is a

Y -periodic, bounded measurable function defined in Rd, such that∃α > 0 : aij(y)ηiηj ≥ α|η|2 ∀η ∈ Rd,aij = aji ∀i, j = 1, . . . , d.

(2.2)

For each ε > 0, we consider the operator Aε:

Aε = − ∂

∂xi

(aεij(x)

∂

∂xj

)with aεij(x) = aij

(xε

), x ∈ Rd.

Associated with Aε, we consider the following periodic boundary value problem:⎧⎨⎩Aεuε = f in Y ,

uε ∈ H1#(Y ), m(uε) =

1

|Y |

∫Y

uεdx = 0.(2.3)

If f is in L2#(Y ) with m(f) = 0, then (2.3) is well posed: it has a unique solution. In

this paper, we consider the case where the space domain contains exactly an integernumber of periodic cells of the coefficients {aεij}, i.e.,

1

ε= s ∈ N.(2.4)

The limit of the solutions of (2.3) as ε → 0 solves an elliptic equation related tothe following constant coefficient homogenized operator A∗:

A∗ = −a∗ij∂2

∂xi∂xj.(2.5)

The homogenized coefficients a∗ij are defined as follows:

2a∗ij =1

|Y |

∫Y

(2aij −

∂aj�∂y�

χi − ∂ai�∂y�

χj

)dy,(2.6)


where, for any k = 1, . . . , d, χk is the unique solution of the cell problem⎧⎨⎩Aχk =∂ak�∂y�

in Y ,

χk ∈ H1#(Y ), m(χk) = 0.

(2.7)

The classical theory of homogenization provides the following result (see [2]).Theorem 2.1. Let the coefficients ak� satisfy assumptions (2.2) and uε and u∗

be the solutions of (2.3) and (2.8), respectively. Then, if f belongs to L2#(Y ) with

m(f) = 0, the sequence of solutions uε of (2.3) converges weakly in H1(Y ), as ε → 0,to the so-called homogenized solution u∗ characterized by{

A∗u∗ = f in Y ,

u∗ ∈ H1#(Y ), m(u∗) = 0.

(2.8)

Furthermore, we have

‖uε − u∗‖0 ≤ cε‖f‖0.(2.9)

In Appendix A we give a new proof of (2.9) using Bloch waves as in [10], wherethe Dirichlet problem was studied in both bounded and unbounded domains. Here,we adapt the Bloch waves approach to the case of periodic boundary conditions.

2.2. The finite difference scheme for elliptic PDEs. We now introduce afinite difference approximation of (2.3).

Let h = (h1, . . . , hd) be a vector with positive components,

hi =2π

niwith ni ∈ N,(2.10)

and denote by Γh the following subgroup of Rd:

Γh = {y ∈ Y | y = (z1h1, . . . , zdhd), zj ∈ Z, 1 ≤ j ≤ d}.(2.11)

‖ · ‖h denotes the discrete L2-norm in a mesh Γh,

‖f‖2h = h1 · · ·hd

∑x∈Γh

|f(x)|2,

and (·, ·)h denotes its discrete inner product,

(f, g)hdef=

∑x∈Γh

h1h2 · · ·hdf(x)g(x).

Let ej , j = 1, . . . , d, be the unit vectors in the coordinate directions. Define

∇±hi g(x) =

1

±hi[g(x± hiei) − g(x)]

with i = 1, . . . , d. Note that for any x ∈ Γh, ∇±hi g(x) is an approximation (of order 2)

of ∂ig at (x ± hi ei/2). For numerical purposes it is therefore natural to replace the


pts. where a�� is evaluated

pts. of Γh where u is approximated

pts. where ak�, k �= �, is evaluated

i + 1, j

i, j + 1

i, j

Fig. 2.1. Points involved in the finite difference approximation of (Au) at x = (ih1, jh2).

boundary value problem (2.3) by the discrete one,

d∑i,j=1

−∇−hi

[aεij(x(i, j))∇+h

j uεh(x)

]= f(x), x ∈ Γh,(2.12)

uεh(x + 2πm) = uε

h(x) ∀m ∈ Zd (2π-periodicity),(2.13) ∑x∈Γh

uεh(x) = 0,(2.14)

where

x(i, j) = x +1

2hiei + (1 − δij)

1

2hjej (Kronecker δij).(2.15)

This discrete method is convergent of order 1.When f is continuous, f(x) are the values of f for x ∈ Γh. When f is not

continuous, the value f(x) may be replaced by the average of f on a cell around thepoint x of the mesh. We assume that∑

x∈Γh

f(x) = 0,(2.16)

to guarantee that (2.12)–(2.14) is well posed. This is consistent with the assumptionm(f) = 0 made in the continuous problem.

Note that the numerical scheme provides approximations uεh of uε at the points

of the mesh Γh (points denoted by ◦ in Figure 2.1). However, the coefficients aεij ofthe discrete system (2.12)–(2.14) are evaluated at the points x(i, j) defined in (2.15).Observe that these points do not belong to the mesh Γh. In fact, for i = j, thesepoints are located symmetrically between x and x + hiei (points denoted by + inFigure 2.1). Finally, when i �= j, x(i, j) is located in the diagonal.

From (2.12), for u, v satisfying (2.13), the following identity holds:

∑x∈Γh

d∑i,j=1

−∇−hi


j u(x)]v(x) =

∑x∈Γh

d∑i,j=1

aεij(x(i, j))∇+hj u(x)∇+h

i v(x).

(2.17)

Therefore, as the coefficients {aεij(x(i, j))} are coercive (this will be shown in sec-tion 3.2), the bilinear form (2.17) associated with the linear system (2.12)–(2.13) is


nonnegative. On the other hand, the constant discrete functions, uεh(x) = c for any

x ∈ Γh, solve (2.12)–(2.13) and are its unique solutions when f ≡ 0. Thus, when thecompatibility condition (2.14) holds, there exists a unique uε

h satisfying (2.12)–(2.14).One expects the solution uε

h of (2.12)–(2.14) to be an approximation to theexact solution uε. This is indeed true, but the error estimates depend on the rapidlyoscillating character of the coefficients aεij .

Let us mention the following classical result on the convergence of finite differenceapproximations for elliptic problems (see [4] and [29]).

Theorem 2.2. Let u be the solution of the periodic boundary problem{Au = f in Y ,

u ∈ H1#(Y ), m(u) = 0.

Let uh be the solution of the finite difference approximation. Then, if the order ofaccuracy of the scheme is ν and f belongs to C0,λ, we have

supx∈Γh

|u(x) − uh(x)| ≤ c(hmin(ν,λ) + εh(a))‖f‖Cλ ,

where εh(a) = max[ak�(x) − ak�(x′)], with |x− x′| ≤ h and for k, = 1, . . . , d.

Applying this theorem to (2.3) and (2.12)–(2.14) with ε > 0 fixed, it can be shownthat, when the coefficients ak� are continuous, the solution uε

h(x) of (2.12)–(2.14)converges to uε(x) when h → 0. In particular, we have the following estimate, whenf is Lipschitz:

supx∈Γh

|uεh(x) − uε(x)| ≤ c

(h +

h

ε

),(2.18)

where c > 0 denotes a positive constant which depends on the Lipschitz constant ofthe coefficients and the ellipticity constant, but it is independent of h and ε.

2.3. Main results. In this section, we analyze how well uεh approximates u∗.

From the estimate (2.9) provided by the theory of homogenization and interpolationinequalities, we get for h sufficiently small (see Remark 4.3)

‖uε − u∗‖h ≤ ch + c′ε.(2.19)

On the other hand, by the error estimate (2.18), we obtain

‖uεh − u∗‖h ≤ ‖uε

h − uε‖h + ‖uε − u∗‖h ≤ c1h

ε+ ch + c′ε.(2.20)

Therefore, if h ε, uεh converges to u∗. In this case, the rapid oscillations of the

coefficients aεij are captured by the numerical approximation since the numerical meshis much finer. But this requires h to be asymptotically smaller than ε and makescomputations infeasible in practice.

Existing results in one dimension. Estimate (2.20) does not provide completeinformation on the behavior of numerical solutions since, as we mentioned in theintroduction, the convergence of uε

h to u∗ may hold with h and ε of the same order,and, more precisely, when h/ε → 2πr �= 0 as h, ε → 0, provided r is a suitableirrational number (see [1]).


To see this, we consider the following 1-d problem:⎧⎪⎨⎪⎩− ∂

∂x

(aε(x)

∂uε

∂x(x)

)= f(x), 0 < x < 2π,

uε(0) = b, uε(2π) = c.

(2.21)

The homogenized solution in the 1-d case satisfies⎧⎨⎩−a∗∂2u∗

∂x2= f(x), 0 < x < 2π,

u∗(0) = b, u∗(2π) = c,

(2.22)

where a∗ is defined by

a∗ =

(1

2π

∫ 2π

0

dy

a(y)

)−1

.(2.23)

Theorems 1 and 1a of [1] show that the finite difference approximations associatedwith (2.21) satisfy the following theorems.

Theorem A. For any continuous and bounded function f(x), we have

limε,h→0

supx∈Γh

|uεh(x) − u∗(x)| → 0,(2.24)

provided h/ε = 2πr with r any irrational number with Γh defined in (2.11).Theorem B. Under the assumptions of Theorem A, given τ > 0 there exist

h0 > 0 and a set S(ε, h0) ⊂ [0, h0] defined by

S(ε, h0) =

{0 < h ≤ h0 |

∣∣∣∣khε − 2πj

∣∣∣∣ ≥ τ

|k|3/2 , j = 1, . . . ,

[kh0

ε

]+ 1

for 0 �= k ∈ Z, 0 < ε ≤ 1

},

with Lebesgue measure |S(ε, h0)| ≥ h0(1 − 3τ), such that for 0 < ε ≤ 1

supx∈Γh

|uεh(x) − u∗(x)| ≤ τ.(2.25)

These results guarantee that the numerical homogenization, i.e., the convergenceof the finite difference solution towards the solution of the continuous homogenizedproblem, occurs even when h and ε are of the same order, provided the ratio h/ε iskept irrational.

Main results in one dimension. The following theorem concerns the morenatural case where h/2πε is a rational number.

Theorem 2.3. Assume uεh is the finite difference approximation of the solution

of (2.21), u∗ satisfies (2.22), and f is a continuous function. Assume that h and εare such that

h

ε= 2π

q

pwith H.C.F.(p, q) = 1, q, p ∈ N.(2.26)

Then there exist c1, c2 > 0, independent of h, ε, and f , such that

supx∈Γh

|uεh(x) − u∗(x)| ≤ c1ph +

c2p.(2.27)


Here and in what follows H.C.F. stands for the highest common factor. Thisresult will be proved in Appendix B by means of the explicit formulas of solutions.

Theorem 2.3 concerns the Dirichlet problem (2.21). We have chosen to state ourmain 1-d result for the Dirichlet problem to facilitate the comparison with TheoremsA and B of [1] stated above.

The error estimate (2.27) contains two different terms. The first one tends tozero as h → 0 when h/ε is kept fixed. However, for the second one to tend to zero weneed to take p → ∞, which, in practice, requires h/2πε to approximate an irrationalnumber.

Using discrete Bloch waves we get the following result in one dimension.Theorem 2.4. Let uε

h be the solution of (2.12)–(2.14) in dimension d = 1. Letε be as in (2.4), and consider |h| < h0 satisfying (2.26). Then, for any continuousfunction f with m(f) = 0, there exists a discrete function u∗

q/p such that

‖uεh − u∗

q/p‖h ≤ c |ph| ‖f‖h,(2.28)

with c independent of h, ε. Moreover, u∗q/p is a discrete Fourier approximation with

mesh size h of the solution of⎧⎨⎩−a∗p∂2v

∂x2(x) = f(x), 0 < x < 2π,

v is 2π-periodic, m(v) = 0,

(2.29)

where a∗p is defined by

a∗p =

⎛⎝1

p

p∑j=1

1

a(2π(j + 1/2)/p)

⎞⎠−1

.(2.30)

Furthermore,

‖u∗q/p − u∗‖h ≤ c

1

p,(2.31)

where u∗ is the homogenized solution (2.22) and c > 0 is independent of h, ε.It is important to note that the discrete homogenized problem (2.29) differs from

the continuous one. Consequently, u∗q/p does not coincide with the continuous ho-

mogenized solution u∗. In fact, in (2.31) we give an explicit estimate of the distancebetween u∗

q/p and u∗. We see that it is of order O(1/p). This is in agreement withthe fact that convergence towards the continuous homogenized solution requires theratio h/2πε to be irrational or, in other words, the denominator p of the rational ratioh/2πε = q/p to tend to infinity.

Remark 2.5. Here we compare Theorems 2.3 and 2.4. Both hold under thesame hypotheses. In Theorem 2.3 we get the error estimate in the ∞-norm (2.27).Estimates (2.28) and (2.31) in Theorem 2.4 yield the same error estimate as in (2.27)but in the norm ‖ · ‖h, i.e.,

‖uεh − u∗‖h ≤ c1ph + c2

1

p.(2.32)

We state separately the numerical homogenization result (2.28) (with rational ratio)and the necessity of the ratio to be irrational in (2.31) to converge towards the ho-mogenized solution.


As a consequence of these results with rational ratio, we recover the statementsof Theorems A and B using classical results in the approximation of irrational num-bers by rational ones in section 5. Theorems A and B concern the case of Dirichletboundary conditions, but similar results apply also with periodic ones.

Main results in several space dimensions. In several space dimensions weget a weaker version of Theorems 2.3 and 2.4. According to (2.4) and (2.10), we have

hi

ε= 2π

s

ni, i = 1, . . . , d.(2.33)

In particular, taking into account that s ∈ N, we have{∃ pi, qi ∈ N such that ni = pi ri, s = qi ri,

where ri = H.C.F.(s, ni) with i = 1, . . . , d.(2.34)

Thus, h and ε satisfy

hi

ε= 2π

qipi

with H.C.F.(pi, qi) = 1, i = 1, . . . , d.(2.35)

Let us now state how uεh approximates u∗ in several space dimensions.

Theorem 2.6. Assume that d ≥ 2 and h0 is sufficiently small. Let ε and h beas in (2.4) and (2.35) with |h| < h0. Furthermore, assume that

q

p−

[q

p

]=

ρ

pwith

∣∣∣∣ρp∣∣∣∣ < ca,(2.36)

where ca is sufficiently small and depends only on the lower and upper bounds ofthe coefficients. Let uε

h be the solution of (2.12)–(2.14). Then, for any continuousfunction f with m(f) = 0, there exists a discrete function u∗

q/p such that

‖uεh − u∗

q/p‖h ≤ c |ph| ‖f‖h(2.37)

for all h, ε > 0 as above with c > 0 independent of h, ε, and f . The function u∗q/p is

the discrete Fourier approximation with mesh size h of the solution of⎧⎪⎨⎪⎩−a∗,q/pij

∂2v

∂xi∂xj= f in Y ,

v ∈ H1#(Y ), m(v) = 0.

(2.38)

In general, this solution does not coincide with the homogenized solution (2.8). Wehave the following explicit error estimate:

‖u∗q/p − u∗‖h ≤ c δ ‖f‖h,(2.39)

where δ > 0 is given by

δ = max

(∣∣∣∣ρp∣∣∣∣, 1 − σm

σM,σM

σm− 1

)(2.40)

with σM = max(σi) and σm = min(σi), where σ = q/ρ.


Remark 2.7. As we stated above, the discrete function u∗q/p is the discrete

Fourier approximation of the solution of (2.38), where the coefficients {a∗,q/pij } areconstants and depend on q and p. In general, these coefficients do not coincide withthe homogenized coefficients, and, consequently, the solution of (2.38) does not coin-

cide with the homogenized solution either. We will give an estimate of |a∗ij − a∗,q/pij |

depending on suitable conditions of p, q (see Proposition 3.11). As in the contextof homogenization of continuous problems (see [10]), the discrete homogenized coeffi-cients can be derived from the Hessian of the first discrete Bloch eigenvalue in (3.21).An explicit formula of this Hessian is given in Lemma 3.7.

Remark 2.8. In the 1-d case, the numerical homogenized solution u∗q/p exists

whatever the choice of q and p is. In several space dimensions, we need the extrahypothesis (2.36) to identify the numerical homogenized solution. Condition (2.36)guarantees the ellipticity of the discrete system (2.12) (see section 3.2). This condi-tion is needed since the ellipticity of {aεij(x(i, j))} is not automatic in several spacedimensions.

Remark 2.9. According to Theorem 2.4, uεh converges to u∗

q/p with q, p fixed.

Moreover, u∗q/p differs in general from u∗ with error estimates (2.31) or (2.39) in one

and several space dimensions, respectively, and these error estimates are independentof the sequence h, ε, provided (2.35) is fulfilled. On the other hand, as we shall see insection 5, there exist sequences q, p such that u∗

q/p converges to u∗. In particular, in

one space dimension that happens when q/p tends to an irrational number.In several space dimensions, according to (2.39), we get the same result, provided

δ tends to zero. For this to be true, in view of (2.40) we need two conditions. Thefirst one requires that the residue ρ/p → 0. This condition is similar to that inTheorem 3 in [1], where the residue converges to zero through a sequence of irrationalnumbers. This is significantly more restrictive than in the 1-d case. In one spacedimension, the homogenized coefficient is explicit and is given by (2.23). But, inseveral space dimensions, the homogenized coefficients (2.6) depend on χk, the solutionof (2.7), which, in general, may not be computed explicitly and need to be approximatednumerically. The function χk is approximated with the mesh size ρ/p, which needs tobe sufficiently small (see Appendix C.3).

The other condition which is needed for δ → 0 is that σM/σm → 1, which in-dicates that the ratio between all components of q/ρ tends to one. This conditionis automatically satisfied when hi/ε is independent of i. When the mesh size is dif-ferent in several space directions, the finite difference approximation scheme for χk,the solution of (2.7), with the mesh size ρ/p does not coincide with the discrete ana-logue obtained when applying the discrete Bloch approximation. The latter is givenin (3.24). In Appendix C.3 we give an estimate on the difference of these two quan-tities in terms of |σm/σM − σM/σm|. This explains why this quantity enters in thedefinition of δ in (2.40) appearing in the estimate (2.39).

2.4. Numerical examples. First, we consider an example of numericalhomogenization in one space dimension. We consider the 2π-periodic coefficient

a(y) =1

1 + 0.5 |sin(y/2)| , y ∈ (0, 2π).

The corresponding homogenized coefficient is a∗ = π/(1 + π). The numericalhomogenized coefficient a∗p defined in (2.30) with q = 5, p = 19, as predicted bythe theory, differs with a∗ by an error of the order of 10−3. Considering the constant


Table 2.1

One dimension. Errors of the solutions with q = 5, p = 19.

h ε ‖uε − uεh‖h ‖uε − uε

h‖∞ ‖u∗ − uεh‖h ‖u∗ − uε

h‖∞ ‖u∗q/p − uε

h‖h2π38

110

0.038 0.0217 0.078 0.069 0.0778

2π380

1100

0.0039 0.0027 0.0086 0.0085 0.0079

2π1900

1500

0.0033 0.0018 0.0036 0.0022 0.0016

2π19000

15000

0.0033 0.0018 0.0033 0.0018 1.5 · 10−4

2π190000

150000

0.0033 0.0018 0.0033 0.0018 1.58 · 10−5

Table 2.2

One dimension. Errors of the solutions with different values of q and p.

h ε q p ‖uε − uεh‖h ‖u∗ − uε

h‖h ‖u∗q/p − uε

h‖h |a∗ − a∗p|2π

190001

50005 19 0.0033 0.0033 1.5 · 10−4 2.08 · 10−4

2π19100

15100

51 191 5.37 · 10−5 1.61 · 10−4 1.57 · 10−4 2.06 · 10−6

2π19100

15110

511 1910 3.72 · 10−5 1.56 · 10−4 1.55 · 10−4 2.06 · 10−8

Table 2.3

Numerical homogenized coefficients with different values of p and q.

q1, q2, p1, p2 1, 1, 71, 71 72, 71, 71, 70 72, 72, 71, 71 31, 103, 70, 72 1031, 121, 70, 72

a∗,q/p11 1.3728 1.3727 1.3727 1,3684 1.3656

a∗,q/p22 1.3728 1.3727 1.3727 1.3679 1.3672

a∗,q/p12 0.5010 0.5009 0.5010 0.4939 0.4896

function f = 1, we calculate uεh using the formula (B.5), and we compare it with the

continuous functions uε, u∗, and u∗q/p solutions of (2.21), (2.22), and (2.29), respec-

tively, with homogeneous Dirichlet boundary conditions. As seen in Table 2.1, as hand ε go to zero with h/ε = 2πq/p and q = 5, p = 19, the approximation uε

h does notconverge to u∗ but rather to u∗

q/p.In Table 2.2 we exhibit the results for various choices of p and q. More precisely,

we consider the pairs (p, q) = (19, 5), (191, 51), and (1910, 511). The theory predicts(see (2.32) for the error estimate) that the accuracy of the approximation is improvedwhen q/p approximates an irrational number as p → ∞. This is clearly seen in thecolumn of Table 2.2 devoted to the estimates of ‖u∗ − uε

h‖h, which shows that, bykeeping h and ε essentially fixed but increasing p from 19 to 1910, this error decreasesfrom 3.3 · 10−3 to 1.56 · 10−4.

In two space dimensions we consider the 2π-periodic coefficients

a11(y1, y2) = a22(y1, y2) = 1 + |sin(y1/2) sin(y2/2)|, a12(y1, y2) = 1/2.(2.41)

Using the formula (C.26) we obtain the numerical homogenized coefficients of Table 2.3with different values of p and q. The column corresponding to (q1, q2, p1, p2) =(1, 1, 71, 71) presents the values of the approximation in finite differences of the ho-mogenized coefficients with mesh sizes 2π/71 in both space directions. We give the

explicit value of a∗,q/pkj for k, j = 1, 2. According to the data in Table 2.3 we see


that the results are significantly different in the first three cases corresponding to(q1, q2, p1, p2) = (1, 1, 71, 71), (72, 71, 71, 70), and (72, 72, 71, 71) than in the last two,in which (q1, q2, p1, p2) = (31, 103, 70, 72) and (1031, 121, 71, 71). In the first threecases the values of the numerical homogenized coefficients are rather similar but notin the last two. This is in agreement with the error estimate in Proposition 3.11 withδ defined in (2.40). Indeed, while in the first three cases ρ = (1, 1), σ = (q1, q2), and,accordingly, δ is small, that is not the case in the last two cases. We recall that theerror estimate on the homogenized coefficients (3.31) affects automatically the errorestimate on the homogenized solutions (see (2.39)).

3. Discrete Bloch waves.

3.1. Bloch decomposition. The solutions of the elliptic continuous problemassociated with the operator Aε may be decomposed in Bloch waves, and this allows usto describe the homogenization process as ε → 0. This will be done in Appendix A.1.We need to perform a similar Bloch decomposition for the numerical problem.

In the finite difference system (2.12) the coefficients aεij are hp-periodic in themesh Γh defined in (2.11). Indeed, taking into account that h and ε satisfy (2.35), forany x ∈ Γh and z ∈ Γhp ⊂ Γh we obtain that

aεij (x + z) = aij

(x + z

ε

)= aij

(xε

)= aεij(x).(3.1)

Here, we have used the fact that z/ε ∈ 2πZd. Indeed, z ∈ Γhp, and therefore z = nhpwith n ∈ Zd. Thus, according to (2.35), z/ε = 2πnq ∈ 2πZd. We conclude thatthe coefficients {aεij} are hp-periodic (or qεY -periodic in the mesh Γh by the relation(2.35)).

Now we define the discrete Bloch waves associated with the linear system (2.12)from the following family of spectral problems: To find μ = μ(ξ) ∈ R and ϕε

h =ϕεh(x; ξ) (nonidentically zero) such that

d∑i,j=1

−∇−hi

[aεij (x(i, j))∇+h

j (eix·ξϕεh(x; ξ))

]= μ(ξ)eix·ξϕε

h(x; ξ), x ∈ Γph,

ϕεh(x; ξ) is ph-periodic in x, i.e., ϕε

h(x + pkhkek; ξ) = ϕεh(x; ξ),

(3.2)

where, using that the coefficients are hp-periodic in Γh, we use the mesh Γph defined

by

Γph = {x = (n1h1, . . . , ndhd) | 0 ≤ ni < pi, ni ∈ Z ∀i = 1, . . . , d}.

The discrete Bloch waves are then given by ψεh(x; ξ) = eix·ξϕε

h(x; ξ). By (2.35) we get

ψεh(x + mhp; ξ) = eimhp·ξψε

h(x; ξ), m ∈ Zd.

It is clear that this property remains the same if ξ is replaced by ξ+n(qε)−1, n ∈ Zd,since (qε)−1 is a multiple of 2π. So, there is no loss of generality in confining ξ to thecell [0, (qε)−1[ or in the translated one [−(2qε)−1, (2qε)−1[ .

We consider the following bilinear form associated with (3.2):

aεh(ξ)(u, v) =∑x∈Γp

h

d∑i,j=1

aεij(x(i, j))∇+hj (eix·ξu(x))∇+h

i (eix·ξv(x))(3.3)


for ph-periodic functions u, v. We note that the bilinear form aεh(ξ) is Hermitian, i.e.,

aεh(ξ)(u, v) = aεh(ξ)(v, u).

Indeed, by (3.3), we get

aεh(ξ)(v, u) =∑x∈Γp

h

d∑i,j=1

aεij(x(i, j))∇+hi (eix·ξu(x))∇+h

j (eix·ξv(x)).

The coefficients (aij) and the mesh points (2.15) being symmetric, we deduce thataεh(ξ) is Hermitian. Therefore, the eigenvalues of (3.2), which we shall refer to as dis-

crete Bloch eigenvalues, are real (see [34, p. 25]). We denote them by {μεh,m(ξ)}pm=1,

where p = p1 · · · pd = #(Γph), i.e., the number of points in the mesh Γp

h, and

{ϕεh,m(x; ξ)}pm=1, their corresponding orthonormal eigenvectors.

Given a discrete function f with values f(x) for x ∈ Γh, we define the mth Blochcoefficient of f at the ε scale as follows:

fεh,m(k) = (2π)−

d2 h(p)

12

∑x∈Γh

f(x)e−ik·xϕεh,m(x; k) ∀m ≥ 1, k ∈ Λqε,(3.4)

when h and ε satisfy (2.35), and with

Λqε =

{k = (k1, . . . , kd) ∈ Zd |

[−1

2qiε

]+ 1 ≤ ki ≤

[1

2qiε

]}.(3.5)

The following holds.

Theorem 3.1. Let ε, h > 0 satisfy (2.35) and Γh be defined as in (2.11). Forany Y -periodic discrete function f the following representation formula holds:

f(x) = (2π)−d2 (p)

12

∑k∈Λqε

p∑m=1

fεh,m(k)eik·xϕε

h,m(x; k).

Further, we have Parseval’s identity,

‖f‖2h =

p∑m=1

∑k∈Λqε

|fεm(k)|2,

and Plancherel’s identity,

(f, g)h =

p∑m=1

∑k∈Λqε

fεh,m(k)gεh,m(k).

Proof. We use the same idea as in the case f ∈ L2(Rd), whose detailed proof canbe found in [2, p. 616] and also in [30]. First, we consider the function

f(x; k) = (2π)−d2 h(p)

12

∑z∈Γhp

f(x + z)e−ik·(x+z), k ∈ Λqε.(3.6)


Γhp is a mesh in Y defined as Γh but with hp mesh size. Note that f(x; k) is hp-periodicin x. Indeed,

(2π)d2

h(p)12

f(x + hjpjej ; k) =∑

z∈Γhp

zj �=0

f(x + z)e−ik·(x+z) +∑

z∈Γhp

zj=2π

f(x + z)e−ik·(x+z).

Thanks to the fact that f is Y -periodic and k ∈ Zd, we see that∑z∈Γhp

z1=2π

f(x + z)e−ik·(x+z) =∑

z∈Γhp

z1=0

f(x + z)e−ik·(x+z).

The function f(x; k) being hp-periodic in x, we have {f(x; k) | x ∈ Γph} ∈ Cp. Since

{ϕεh,m(x; k) | x ∈ Γp

h}pm=1 is an orthonormal basis in Cp, we get for x ∈ Γp

h

f(x; k) =

p∑m=1

ϕεh,m(x; k)

∑x′∈Γp

h

f(x′; k)ϕεh,m(x′; k)

=

p∑m=1

ϕεh,m(x; k) (2π)−

d2 h(p)

12

∑x′∈Γp

h

∑z∈Γhp

f(x′ + z)e−ik·(x′+z)ϕεh,m(x′ + z; k).

Note that Γph⊕Γhp = Γh; i.e., for any x ∈ Γh there exist a unique x′ ∈ Γp

h and z ∈ Γhp

such that x = x′ + z, and ϕεh,m defined in (3.2) is hp-periodic in x. Then, by (3.4),

we get

f(x; k) =

p∑m=1

ϕεh,m(x; k) fε

h,m(k).(3.7)

Moreover, using that {ψεh,m(x; k) | x ∈ Γp

h}pm=1 are orthonormal, we get for any

k ∈ Λqε

∑x∈Γp

h

|f (x; k)|2 =

p∑m=1

|fεh,m(k)|2.(3.8)

Now we proceed to show the inverse formula for f . For z, z′ ∈ Γph, it follows that

∑k∈Λqε

eik·(z−z′) =

{0 if z �= z′,

#(Λqε) if z = z′,(3.9)

and, for k ∈ Λqε,

∑z∈Γph

eik·z =

{0 if k �= 0,

#(Γph) if k = 0.(3.10)

The proof of these formulas is immediate. (Note that #(Λqε) = #(Γph).) We get by(3.10)

f(x) = (2π)−d2 (p)

12

∑k∈Λqε

f (x; k) eik·x, x ∈ Γh,


and, by (3.7), we obtain the representation formula of the theorem. On the otherhand, using the definition of f(x; k) and the formula (3.9), we obtain

∑k∈Λqε

∑x∈Γp

h

|f (x; k)|2 =(h)2p

(2π)d

∑x∈Γp

h

∑z∈Γph

|f(x + z)|2#(Λqε),

and, by (2.35) and Γph ⊕ Γhp = Γh, we get∑k∈Λqε

∑x∈Γp

h

|f (x; k)|2 =∑x∈Γh

h1 · · ·hd|f(x)|2.

Parseval’s identity is a consequence of (3.8). Analogously, Plancherel’s identity fol-lows. This concludes the proof of Theorem 3.1.

Using Theorem 3.1, we have the following result.Theorem 3.2. Let {uε

h(x) | x ∈ Γh} be the unique solution of (2.12)–(2.14).

Let uεh,m(k) and fε

h,m(k) be the mth Bloch coefficients of uεh and f , respectively, with

m = 1, . . . , p and k ∈ Λqε. Then we have

μεh,m(k)uε

h,m(k) = fεh,m(k), m = 1, . . . , p ∀k ∈ Λqε.(3.11)

Proof. According to Theorem 3.1

uεh(x) = (2π)−

d2 h(p)

12

p∑m=1

∑k∈Λqε

uεh,m(k)eik·xϕε

h,m (x; k) , x ∈ Γh.(3.12)

Using (3.12) in the discrete problem (2.12), we have, for any x ∈ Γh,

f(x) =h(p)

12

(2π)d2

p∑m=1

∑k∈Λqε

uεh,m(k)

⎛⎝−d∑

i,j=1

∇−hi


j (eik·xϕεh,m(x; k))

]⎞⎠ .

Since {ϕεh,m(x; k) | x ∈ [0, ph] ∩ Γh} satisfies (3.2) we obtain

f(x) = (2π)−d2 h(p)

12

p∑m=1

∑k∈Λqε

uεh,m(k)με

h,m(k)eik·xϕεh,m(x; k), x ∈ Γh.

Then, using the representation formula for f given in Theorem 3.1, (2.12) can bewritten as

p∑m=1

∑k∈Λqε

(μεh,m(k)uε

h,m(k) − fεh,m(k)

)eik·xϕε

h,m(x; k) = 0, x ∈ Γh.

Using the completeness of the Bloch eigenvectors in Cp and their ph-periodicity, weconclude the proof.

3.2. Properties of Bloch waves. In this section we present some properties ofthe discrete Bloch eigenvalues and eigenvectors that we will use to prove the numericalhomogenization results. The proof of these properties will be given in Appendix C.


We have defined the discrete Bloch waves for x ∈ Γph in (3.2). Now, making a

change of variables, we use the discrete Bloch waves in the following mesh:

Γ 2πp

=

{y ∈ Y | y =

(2π

p1n1, . . . ,

2π

pdnd

), 0 ≤ ni < pi, ni ∈ Z, i = 1, . . . , d

}.(3.13)

Note that #(Γ 2πp

) = #(Γph) = p and that y ∈ Γ 2π

pwhenever qεy ∈ Γp

h by (2.35). Now,

for x ∈ Γph, we introduce the following relation:

ϕεh(x; ξ) = ϕp(y; η),(3.14)

where (x, ξ) and (y, η) are related by y = x/(qε), η = qεξ. Recall that ξ belongs

to [−1/(2qε), 1/(2qε)[ . Hence, η ∈ [−1/2, 1/2[d

= Y ′. By the relations (3.14) and(2.35),

∇+hj (eix·ξϕε

h(x; ξ)) =1

qjε∇

2πp

j (eiy·ηϕp(y; η)).

Therefore, we get

∇−hi

[aεij (x(i, j))∇+h

j (eix·ξϕεh(x; ξ))

]= ∇− 2π

p

i

[aij (qy(i, j))

ε2qiqj∇+ 2π

p

j (eiy·ηϕp(y; η))

],

where, for i, j = 1, . . . d,

y(i, j) = y +π

piei + (1 − δij)

π

pjej .(3.15)

Thus, we consider the following family of spectral problems: To find μ = μ(η) ∈ R

and ϕ = ϕp(y; η) (nonidentically zero) such that

−d∑

i,j=1

∇− 2πp

i

[1

qiqjaij (qy(i, j))∇+ 2π

p

j (eiy·ηϕp(y; η))

]= μ(η)eiy·ηϕp(y; η),

ϕp(y, η) is Y -periodic in y.

(3.16)

Its eigenvalues {μm(η)}pm=1 and eigenvectors {ϕp,m(y; η) | y ∈ Γ 2πp} verify

μεh,m(ξ) = ε−2μm(qεξ), ξ ∈

[− 1

2qε,

1

2qε

[,

ϕεh,m(x; ξ) = ϕp,m

(x

qε; qεξ

), x ∈ [0, ph[ ∩ Γh.

(3.17)

We consider the following Hermitian bilinear form associated with (3.16):

a(η)(u, v) =∑

y∈Γ 2πp

d∑i,j=1

1

qiqjaij (qy(i, j))∇

2πp

j (eiy·ηu(y))∇2πp

i (eiy·ηv(y))(3.18)

for η ∈ Y ′ and Y -periodic discrete functions u, v. Note that the coefficients {aij} de-fined in (2.2) are bounded and coercive. However, the values {aij(qy(i, j))} consideredin (3.16) depend on {ij} and are taken in different points of the mesh (see Figure 2.1).


The main consequence of this fact is the loss of the ellipticity in (3.18) in several spacedimensions. In the following lemma, we get the ellipticity of {aij(qy(i, j))} under thekey assumption (2.36).

Lemma 3.3. Assume that the coefficients {aij} defined in (2.2) are Lipschitzcontinuous, α and β are, respectively, their boundedness and ellipticity constants, andq, p ∈ Nd satisfy (2.36). Then, for |ρ/p| < α/(2cdπ), where c is the Lipschitz constantof the coefficients, we have

d∑i,j=1

aij(qy(i, j))ξiηj ≤ 2β|ξ||η| (boundedness),

d∑i,j=1

aij(qy(i, j))ξiξj ≥α

2|ξ|2 (coercivity).

Its proof will be given in Appendix C. As an immediate consequence we get thatthe Hermitian bilinear form a(η) in (3.18) is nonnegative.

Lemma 3.4. Let a(η) be the Hermitian bilinear form in (3.18). Under the hy-potheses of Lemma 3.3, a(η) is positive semidefinite.

Proof. Using formula (3.18) and the coercivity property of Lemma 3.3, we con-clude that

a(η)(v, v) ≥ α

2

∑y∈Γ 2π

p

d∑i=1

∣∣∣∣∇ 2πp

i (eiy·ηv(y))

∣∣∣∣2 ∀η ∈ Y ′.(3.19)

The eigenvalues μm of a(η) are nonnegative and can be ordered in an increasingway:

0 ≤ μ1(η) ≤ · · · ≤ μp(η).

Moreover, thanks to (3.19), we have that

μ1 is simple, μ1(0) = 0, ϕp,1(y; 0) =1

√p1 · · · pd

∀y ∈ Γ 2πp.(3.20)

Lemma 3.5. Assume that the hypotheses of Lemma 3.3 hold. There exists apositive constant c > 0, independent of η, p, and q, such that for any m ≥ 2

μm(η) ≥ c mini=1,...,d

1

q2i

∀η ∈ Y ′.

Proposition 3.6. There exists δ > 0 sufficiently small such that the first eigen-value μ1(η) and the corresponding eigenvector ϕp,1(y; η) are analytic with respect to ηin Bδ.

The proof of the previous results will be given in Appendix C.As in the context of homogenization of continuous elliptic problems, the homoge-

nized coefficients for the discrete problem are given by the Hessian of the first discreteBloch eigenvalue μ1 at η = 0. In particular, we define the discrete homogenizedcoefficients as

a∗,q/pk� =

qkq�2

∂2μ1

∂ηkη�(0).(3.21)


The following proposition gives the values of the derivatives of μ1 at η = 0.Lemma 3.7. Let μ1 and ϕp,1 be the first eigenvalue and eigenvector of the spectral

problem (3.16). Then the origin η = 0 is a critical point of the first Bloch eigenvalue:

∂μ1

∂ηk(0) = 0 ∀k = 1, . . . , d.(3.22)

Moreover, the second derivatives at η = 0 are

∂2μ1

∂ηkη�(0) =

1

p

∑y∈Γ 2π

p

2ak�(qy(k, ))

qkq�

(3.23)

− 1

p

∑y∈Γ 2π

p

d∑j=1

[Θk

q (y)

q�qj∇

2πp

j aj�(qy(, j)) +Θ�

q(y)

qkqj∇

2πp

j ajk(qy(k, j))

],

where {Θkq (y) | y ∈ Γ 2π

p} is the unique solution of⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−d∑

i,j=1

∇− 2πp

i

[aij (qy(i, j))

qiqj∇+ 2π

p

j (Θkq (y))

]=

d∑j=1

1

qkqj∇− 2π

p

j ajk(qy(k, j)),

Θkq (y) Y -periodic,

∑y∈Γ 2π

p

Θkq (y) = 0.

(3.24)

The derivatives of the first Bloch eigenvector are as follows:

∂ϕp,1

∂ηk(y; 0) =

i

(p)12

Θkq (y), k = 1, . . . , d.(3.25)

As a consequence of the analyticity of μ1 and Taylor’s formula we get the followingbounds on μ1 and ϕp,1.

Lemma 3.8. Under the hypotheses of Lemma 3.3, the map η ∈ Y ′ → μ1(η) ∈ R

has a strict global minimum at η = 0, where μ1(0) = 0. Furthermore, there existc1, c2, c > 0, independent of p and q, such that

c1

∣∣∣∣ηq∣∣∣∣2 ≤ μ1(η) ≤ c2

∣∣∣∣ηq∣∣∣∣2 ∀η ∈ Y ′,(3.26)

∑y∈Γ 2π

p

d∑i=1

1

q2i

∣∣∣∣∇ 2πp

i (eiy·ηϕp,1(y; η))

∣∣∣∣2 ≤ c

∣∣∣∣ηq∣∣∣∣2 .(3.27)

Lemma 3.9. Under the hypotheses of Lemma 3.3, there exists c > 0, independentof p, q, such that for any η ∈ Bδ∣∣∣∣μ1(η) −

1

2

∂2μ1

∂ηiηj(0)ηiηj

∣∣∣∣ ≤ c

∣∣∣∣ηq∣∣∣∣2 |η|.(3.28)

Finally, we estimate the difference of the discrete homogenized coefficients (3.21)with the homogenized coefficients a∗k� distinguishing the 1-d and the multidimensionalcase.


Proposition 3.10 (one space dimension). We assume that the coefficient a isLipschitz continuous. a∗ and a∗p are defined by (2.23) and (2.30), respectively. Thenwe get ∣∣a∗ − a∗p

∣∣ ≤ πβ2

2α2

c

p(3.29)

for all p ≥ 1, where β, c, and α are the upper bound, the Lipschitz constant, and thecoercivity constant of the coefficient a, respectively. Moreover, the second derivativeof μ1 at η = 0 satisfies

∂2μ1

∂η2(0) = 2

a∗pq2

.(3.30)

Proposition 3.11 (several space dimensions). We assume that the coefficients{ak�}, their first derivatives, and the functions {χk} defined by (2.7) are Lipschitz

continuous. Let a∗k� and a∗,q/pk� be defined in (2.6) and (3.21), respectively. Then we

get ∣∣∣a∗k� − a∗,q/pk�

∣∣∣ ≤ c δ,(3.31)

where δ is defined in (2.40) and c is independent of δ, q, and p.Remark 3.12. The hypotheses in the multidimensional case are stronger than in

the 1-d case. In the 1-d case, the homogenized coefficient is explicitly given by (2.23),while in several dimensions the homogenized coefficients (2.6) depend on χk, the so-lution of (2.7). In general, the auxiliary functions χk may not be computed explicitlyand need to be approximated numerically. This is the reason for the difference betweenthe 1-d and the several space dimension cases.

4. Homogenization via discrete Bloch waves. In Theorem 3.2 we have seenthat problem (2.12)–(2.14), under the assumption (2.35), is equivalent to

μεh,m(k)uε

h,m(k) = fεh,m(k) ∀m = 1, . . . , p, k ∈ Λqε,(4.1)

with Λqε defined in (3.5). Our goal in this section is to pass to the limit in (4.1) asε → 0. In section 4.1 we show that uε

m(k) with (m ≥ 2) are negligible for any k ∈ Λqε.As a consequence of this result, these Bloch components of uε

h do not play any role inthe limit as ε → 0. It is then sufficient to analyze the limit behavior of the equation

μεh,1(k)uε

h,1(k) = fεh,1(k), k ∈ Λqε.(4.2)

The limit will be given, thanks to the fact that the discrete first Bloch transformrepresenting the periodic medium tends to the discrete Fourier transform representingthe homogeneous one. It is a consequence of the fact that the first Bloch coefficientdefined in Theorem 3.1 tends to the usual discrete Fourier coefficient (see section 4.2).Consequently, the limit of (4.2) is the Fourier transform of the homogenized equation(see section 4.3).

4.1. Estimates on the higher-order Bloch modes. Here we are going toprove that uε

h,m(k) with m ≥ 2 are negligible as ε → 0. We set

vεh(x) = (2π)−d2 (p)

12

∑k∈Λqε

p∑m=2

uεh,m(k)eik·xϕε

h,m(x; k), x ∈ Γh,(4.3)


the projection of the solution uεh of (2.12)–(2.14) over the orthogonal subspace to the

first Bloch component.Proposition 4.1. Let vεh be defined in (4.3). Then ‖vεh‖h ≤ c|qε|2 ‖f‖h.Proof. By Parseval’s identity and (4.1), we deduce that

‖vεh‖2h =

∑k∈Λqε

p∑m=2

|uεh,m(k)|2 =

∑k∈Λqε

p∑m=2

|fεh,m(k)|2

(μεh,m(k))2

.

Recalling that μεh,m(k) = ε−2μm(k/qε) by (3.17), we arrive at

‖vεh‖2h = ε4

∑k∈Λqε

p∑m=2

|fεh,m(k)|2

(μm( kqε ))

2.

At this point, we use the lower bound of the eigenvalues μm with m ≥ 2 proved inLemma 3.5, and we obtain that

‖vεh‖2h ≤ cε4 max

i=1,...,d(qi)

4∑

k∈Λqε

p∑m=2

|fεh,m(k)|2 ≤ c|qε|4 ‖f‖2

h.

As a consequence of this proposition, the problem of passing the limit as ε → 0in (4.1) is reduced to analyzing (4.2) characterizing the first Bloch coefficient.

4.2. First Bloch mode and the discrete Fourier decomposition. Here weprove that the first Bloch transform defined in (3.4) tends to the usual discrete Fouriertransform.

First, we present a brief introduction of discrete Fourier waves (see [11, p. 59]).Suppose that f ∈ L2

loc(Rd) is Y -periodic. Its development in a Fourier series is given

by

f(x) = (2π)−d2

∑k∈Z

fkeik·x,(4.4)

where fk is the kth Fourier coefficient of f defined by

fk = (2π)−d2

∫Y

f(x)e−ik·xdx ∀k ∈ Zd.(4.5)

Identity (4.5) follows from the orthonormality of the plane waves eik·y in L2(Y ). Thenorm in Hs

#(Y ) is defined by

‖f‖s =

⎛⎝∑k∈Zd

(1 + |k|2)s|fk|2⎞⎠ 1

2

.

For n ∈ Nd we consider the finite-dimensional space Sn generated by the functions{eik·x} with k ∈ Δn, a finite subset of Zd defined by

Δn =

{(k1, . . . , kd) ∈ Zd | −

[ni + 1

2

]+ 1 ≤ ki ≤ ni −

[ni + 1

2

], i = 1, . . . , d

}.


Note that Sn is of dimension n since #(Δn) = n. The best approximation of f in Sn

(in the L2-sense) is the function fn ∈ Sn, obtained by truncating the development(4.4) for k ∈ Δn:

fn(x) = (2π)−d2

∑k∈Δn

fkeik·x.(4.6)

Consider also the function f (n) ∈ Sn interpolating f at the points x ∈ Γh,h = 2π/n, and Γh defined in (2.11), such that

f (n)(x) = f(x), x ∈ Γh,

and its Fourier decomposition

f (n)(x) =∑�∈Δn

f�ei�·x,(4.7)

where

f� =1

n1 · · ·nd

∑x∈Γh

f(x)e−i�·x.(4.8)

We observe that #(Γh) = #(Λn) = n. We define the discrete Fourier transform.Definition 4.2. The mapping Cn −→ Cn such that

{f(x)}x∈Γh−→ {f�}�∈Δn

defined according to the formula (4.8) is the discrete Fourier transform.From [5] we have that for 0 ≤ m ≤ s the following estimates hold:

‖f − fn‖m ≤ c1|h|s−m‖f‖s, ‖f − f (n)‖m ≤ c2|h|s−m‖f‖s,(4.9)

where c1, c2 are positive constants independent of n and f .Remark 4.3. The interpolating functions allow us to estimate the discrete norm

‖ · ‖h of continuous functions. Indeed, we have for h = 2π/n that ‖f‖h = ‖f (n)‖0.Now using this property we prove (2.19). Since uε and u∗ are continuous,

‖uε − u∗‖h = ‖uε(n) − u∗(n)‖0 ≤ ‖uε(n) − uε‖0 + ‖u∗ − u∗(n)‖0 + ‖uε − u∗‖0,

and applying (4.9) (note that uε ∈ H1(Y ) and u∗ ∈ H2(Y )) and (2.9), we obtain(2.19).

We prove the convergence of the first Bloch coefficient to the discrete Fouriertransform.

Proposition 4.4. Let f be a continuous function. Let fεh,1(k) be the first Bloch

coefficient of f defined by (3.4). Let h, ε > 0 satisfy (2.35). Let Bδ be the analyticityregion in Proposition 3.6. Then, for (kqε) ∈ Bδ, we get

|fεh,1(k) − (2π)

d2 fk| ≤ c |k| |qε| ‖f‖h,(4.10)

where fk is the discrete Fourier coefficient defined in (4.8). Furthermore, if f is aLipschitz function, there exists c depending on the Lipschitz constant of f such that∑

(kqε)∈Bδ

|fεh,1(k) − (2π)

d2 fk|2 ≤ c |qε|2.(4.11)


Proof. By the definitions of the discrete Fourier coefficients fk and the first Bloch

coefficients fεh,1(k) we can write for k ∈ Λεq

fεh,1(k) − (2π)

d2 fk = (2π)

−d2 h(p)

12

∑x∈Γh

f(x)e−ik·x[ϕεh,1(x; k) − ϕε

h,1(x; 0)]

since ϕεh,1(x; 0) = p−1/2 for any x ∈ Γh (see (3.20)). Since Γh = Γph ⊕ Γp

h and ϕεh,1 is

ph-periodic (see (3.2)), we get

fεh,1(k) − (2π)

d2 fk =

∑x∈Γp

h

f(x; k)e−ix·k[ϕεh,1(x; k) − ϕε

h,1(x; 0)]

with f(x; k) defined in (3.6). Applying the Cauchy–Schwarz inequality, we obtain

|fεh,1(k) − (2π)

d2 fk|2 ≤

∑x∈Γp

h

|f(x; k)|2∑x∈Γp

h

|ϕεh,1(x; k) − ϕε

h,1(x; 0)|2.(4.12)

Note that by (3.17) it follows that∑x∈Γp

h

|ϕεh,1(x; k) − ϕε

h,1(x; 0)|2 =∑

y∈Γ 2πp

|ϕ1(y; kqε) − ϕ1(y; 0)|2.

In Proposition 3.6, we established the analyticity of the map η → ϕ1(·; η). Here weneed only its Lipschitz property to get∑

y∈Γ 2πp

|ϕ1(y; kqε) − ϕ1(y; 0)|2 ≤ c |qε|2 |k|2.(4.13)

In view of (3.8), by Parseval’s identity we get

∑x∈Γp

h

|f (x; k)|2 =

p∑m=1

|fεh,m(k)|2 ≤ ‖f‖2

h.

Then, applying (4.13) in (4.12), we prove (4.10). On the other hand, by (3.8), (4.12),and (4.13), we obtain

∑(kqε)∈Bδ

|fεh,1(k) − (2π)

d2 fk|2 ≤ c |qε|2

∑(kqε)∈Bδ

p∑m=1

|k|2|fεh,m(k)|2.

This concludes the proof of (4.11).

4.3. Numerical homogenization. It is convenient first to pass heuristicallyto the limit in (4.2) to motivate the discrete homogenized limit this produces. Therigorous proofs of the main results are in section 4.4.

First, we study the homogenized problem (2.8) using the discrete Fourier trans-form of section 4.2. We consider the interpolation f (n) of f in Sn as in (4.7). Note

that f0 = 0 since m(f) = 0. We now define

u(n)(x) =∑�∈Λn

u�ei�·x, x ∈ Y ,


with

uk =fk

a∗ijkikjand u0 = 0.(4.14)

It is immediate to see that{A∗u(n) = f (n) in Y ,

u(n) ∈ H1#(Y ), m(u(n)) = 0.

Furthermore, if f ∈ Hs#(Y ), s ≥ 0, and u∗ is the solution of (2.8), we have from (4.9)

that

‖u∗ − u(n)‖2 ≤ ‖f − f (n)‖0 ≤ c|h|s‖f‖s,

and then

‖u∗ − u(n)‖0 ≤ c|h|s+2‖f‖s.(4.15)

We consider (4.2), multiply both sides by (2π)−d/2, and using (3.17), we get

ε−2μ1(qεk)(2π)−d2 uε

h,1(k) = (2π)−d2 fε

h,1(k), k ∈ Λεq.

Expanding μ1 by Taylor’s formula (see Lemma 3.7) and applying (4.10) (concerningthe convergence of the first Bloch coefficient fε

1 (k)), we have

1

2∂2ijμ1(0)qikiqjkj(2π)−

d2 uε

h,1(k) = fk + |k|O(|qε|) + |qk|3O(ε)uεh,1(k).

Here and in what follows we use the notation ∂i = ∂∂ηi

. We define u∗q/p(k) by

a∗,q/pij kikj u

∗q/p(k) = fk and u∗

q/p(0) = 0.(4.16)

We note that u∗q/p(k) are the discrete Fourier coefficients of the solution of (2.38).

Further, since the discrete homogenized coefficients satisfy (3.21), then we can write

(2π)−d/2uεh,1(k) = u∗

q/p(k) + εk,

where εk is the difference of (2π)−d/2uεh,1(k) with u∗

q/p(k). In the following section wegive estimates on the error εk.

Now we calculate the difference between the coefficients u∗q/p(k) and the discrete

Fourier coefficients of the homogenized solution uk defined in (4.14).Proposition 4.5 (several space dimensions). Let u∗ be the solution of the ho-

mogenized problem (2.8). We define the discrete function u∗q/p as

u∗q/p(x) =

∑k∈Λn

u∗q/p(k)eix·k, x ∈ Γh,(4.17)

where u∗q/p(k) is defined by (4.16). Under the hypotheses of Proposition 3.11, we have

‖u∗ − u∗q/p‖h ≤ c (|h| + δ).


Proposition 4.6 (one space dimension). Let u∗ be the homogenized solution.We define the discrete function u∗

q/p as in (4.17). Under the hypotheses of Proposi-tion 3.10, we have

‖u∗ − u∗q/p‖h ≤ c (h + p−1).(4.18)

Proof of Proposition 4.5. First, thanks to Remark 4.3, we get

‖u∗ − u∗q/p‖h = ‖u∗(n) − u∗

q/p‖0,

where u∗(n) is defined interpolating u∗ at the points of Γh. On the other hand, thanksto the definition of u∗

q/p(k) and uk defined in (4.16) and (4.14), we get

u∗q/p(k) − uk = fk

(1

a∗,q/pij kikj

− 1

a∗ijkikj

), k ∈ Λn, k �= 0.

Now, using the coercivity of a∗,q/pij (see Lemma C.7) and of the coefficients {a∗ij}, by

Proposition 3.11 we obtain

|u∗q/p(k) − uk| ≤ cδ

|uk|α2

.

Thus, by Parseval’s identity,

‖u∗q/p − u(n)‖0 ≤ cδ

‖u∗‖0

α2.

Then, by (4.9) and (4.15), we get

‖u∗q/p − u∗(n)‖0 ≤ ‖u(n) − u∗(n)‖0 + ‖u∗

q/p − u(n)‖20 ≤ c|h| + ‖u∗

q/p − u(n)‖0,

and this concludes the proof.We prove Proposition 4.6 following the same steps as in this last proof but using

the one-dimensional estimate of Proposition 3.10.

4.4. Convergence estimates. Applying the techniques and results above, wecan now prove the main estimate in several space dimensions stated in Theorem 2.6.The proof consists of several propositions which correspond to different estimates onthe discrete Bloch space and the discrete Fourier space. To do it, we decompose uε

h

as follows:

uεh(x) = uε

1(x) + vεh(x), x ∈ Γh,

where vεh is defined in Proposition 4.1 and uε1 is defined by

uε1(x) = (2π)−

d2 (p)

12

∑k∈Λqε

uεh,1(k)eix·kϕε

h,1(x; k).(4.19)

Thanks to Proposition 4.1, we can neglect the higher Bloch modes of uεh. In particular,

‖uεh − uε

1‖h = ‖vεh‖ ≤ c |qε|2‖f‖h.(4.20)


Now we decompose uε1 using the following discrete functions:

u∗,ε1 (x) = (p)

12

∑k∈Λqε

u∗q/p(k)eix·kϕε

h,1(x; k),(4.21)

u∗,ε2 (x) =

∑k∈Λqε

u∗q/p(k)eix·k,(4.22)

where u∗q/p(k) is defined in (4.16). We are going to prove the following results.

Proposition 4.7. Let uε1 and u∗,ε

1 be the discrete functions defined in (4.19) and(4.21), respectively. Under the assumptions of Proposition 4.4, we have

‖uε1 − u∗,ε

1 ‖h ≤ c |qε| ‖f‖h.

Proposition 4.8. Let u∗,ε1 and u∗,ε

2 be the discrete functions defined in (4.21)and (4.22), respectively. Then we have

‖u∗,ε1 − u∗,ε

2 ‖h ≤ c |qε| ‖f‖h.

To prove these propositions we need only the analyticity of μ1 and ϕ1 and theconvergence of the first discrete Bloch mode to the discrete Fourier transform. Wenote that uε

1(0) = u∗q/p(0) = 0. Therefore, the contribution of the index k = 0 in the

estimates is negligible, and we do not need to take it into account in the followinganalysis.

Proof of Proposition 4.7. First, we are going to neglect the nonanalytic compo-nents. We use Parseval’s identity and obtain

‖uε1 − u∗,ε

1 ‖2h =

∑k∈Λqε

|uεh,1(k) − (2π)

d2 u∗

q/p(k)|2.

Taking into account that uεh,1(k) and u∗

q/p(k) satisfy (4.2) and (4.16), respectively,and thanks to the existence of c1, c2 > 0 such that

μεh,1(k) ≥ c1|k|2 and a

∗,q/pij kikj ≥ c2|k|2,(4.23)

we can reduce our analysis to the points k ∈ Bδ/qε. In fact, for k ∈ Λqε we get

∑εqk/∈Bδ

|uεh,1(k)|2 ≤

∑εqk/∈Bδ

|fεh,1(k)|2

(c1|k|2)2≤ cδ |hp|4 ‖f‖2

h,

∑εqk/∈Bδ

|u∗q/p(k)|2 ≤

∑εqk/∈Bδ

|fk|2(c2|k|2)2

≤ cδ |hp|4 ‖f‖2h.(4.24)

Thus, we have

‖uε1 − u∗,ε

1 ‖2h ≤

∑k∈Λεq∩Bδ/qε

|uεh,1(k) − (2π)

d2 u∗

q/p(k)|2 + cδ |hp|4 ‖f‖2h.(4.25)

Using the definitions of uεh,1(k) and u∗

q/p(k) we get for k �= 0

uεh,1(k) − (2π)

d2 u∗

q/p(k) =fεh,1(k) − (2π)

d2 f(k)

μεh,1(k)

+ f(k)

[1

μεh,1(k)

− 2

∂ijμ1(0)qikiqjkj

],


where μεh,1(k) is defined by (3.17). For εqk ∈ Bδ we apply Taylor’s expansion of μ1

and obtain

μεh,1(k) = ε−2μ1 (εqk) =

1

2∂2ijμ1(0)qikiqjkj + c |qε| |k|3.(4.26)

Therefore, we get the following estimate:

|uεh,1(k) − (2π)

d2 u∗

q/p(k)| ≤|fε

h,1(k) − (2π)d2 f(k)|

μεh,1(k)

+c|f(k)||qk|3ε

μεh,1(k)∂2

ijμ1(0)qikiqjkj,

and using (4.23), we obtain

|uεh,1(k) − (2π)

d2 u∗

h/ε(k)| ≤ c

|k|2 |fεh,1(k) − (2π)

d2 f(k)| + c|f(k)| |q|ε|k| .

Finally, using the results of Proposition 4.4, we obtain

∑k∈Λεq∩εqBδ

c

|k|4 |fεh,1(k) − (2π)

d2 f(k)|2 ≤ c |qε|2

∑(kqε)∈Bδ

p∑m=1

1

|k|2 |fεh,m(k)|2.

Therefore, we get

∑k∈Λεq∩εqBδ

|uεh,1(k) − (2π)

d2 u∗

q/p(k)|2 ≤ c∑

(kqε)∈Bδ

p∑m=1

|fεh,m(k)|2

|k|2 + c′|qε|2∑

(kqε)∈Bδ

|f(k)|2|k|2 .

Then, since |k| ≥ 1, we conclude the proof, thanks to (4.25).Proof of Proposition 4.8. We observe that for any x ∈ Γh, ϕε

h,1(x; 0) = 1/√p.

Therefore,

u∗,ε1 (x) − u∗,ε

2 (x) =√

p∑

k∈Λqε

u∗q/p(k)eix·k[ϕε

h,1(x; k) − ϕεh,1(x; 0)].

By (3.17), an easy computation shows that

‖u∗,ε1 − u∗,ε

2 ‖2h = (2π)d

∑k∈Λqε

|u∗q/p(k)|2

∑y∈Γ 2π

p

|ϕ1(y; qεk) − ϕ1(y; 0)|2.

Now by (4.24) we reduce our analysis to the points qεk ∈ Bδ as in the previous proof:

‖u∗,ε1 − u∗,ε

2 ‖2h ≤ (2π)d

∑(qεk)∈Λqε∩Bδ

|u∗q/p(k)|2

∑y∈Γ 2π

p

|ϕ1(y; qεk) − ϕ(y; 0)|2 + cδ|qε|4‖f‖2h.

By the analyticity of ϕ1(·; η) in the variable η (see Proposition 3.6), we obtain

‖u∗,ε1 − u∗,ε

2 ‖2h ≤ c

∑(qεk)∈Λqε∩Bδ

|u∗q/p(k)|2|εqk|2 + cδ|qε|4‖f‖2

h

≤ c|qε|2‖f‖2h + cδ|qε|4‖f‖2

h.

This concludes the proof of Proposition 4.8.Now, combining (4.20) and Propositions 4.5, 4.7, and 4.8, we prove Theorem 2.6.


Proof of Theorem 2.6. We consider uεh, the finite difference approximation of

(2.3), and u∗, the homogenized solution of (2.8). For any x ∈ Γh we introduce thedecomposition

uεh(x) − u∗

q/p(x) = (uεh(x) − uε

1(x)) + (uε1(x) − u∗,ε

1 (x)) + (u∗,ε1 (x) − u∗,ε

2 (x))

+ (u∗,ε2 (x) − u∗

q/p(x)),(4.27)

where uε1, u

∗,ε1 , u∗,ε

2 , and u∗q/p are defined in (4.19), (4.21), (4.22), and (4.16), respec-

tively. Since higher Bloch modes can be neglected (see section 4.1), we get (4.20).Then, using Propositions 4.7 and 4.8, we get

‖uεh − u∗

q/p‖h ≤ c(|qε| + |qε|2)‖f‖h + ‖u∗,ε2 − u∗

q/p‖h.(4.28)

Now by (4.17) and (4.22) we observe that

u∗q/p(x) − u∗,ε

2 (x) =∑

k∈Λn/Λqε

u∗q/p(k)eix·k.

Using the same analysis as in (4.24), we have

‖u∗,ε2 − u∗

q/p‖2h ≤

∑k∈Λn/Λqε

|u∗q/p(k)|2 ≤ c|qε|4‖f‖2

h,(4.29)

and we prove (2.37). Using this estimate and Proposition 4.5, we conclude theproof.

We conclude this section with the proof of Theorem 2.4, which is specific to the1-d case.

Proof of Theorem 2.4. We consider the decomposition (4.27). First, we apply theestimates of Propositions 4.7 and 4.8. The assumptions of these propositions are notneeded because (4.23) and (4.26) are satisfied in one dimension with p, q defined as in(2.20). Then we have (4.28).

On the other hand, (4.29) is satisfied in one dimension, and, under the hypothesesof Proposition 3.10, we obtain (4.18). Then, applying (4.29) and (4.18) in (4.28), weconclude the proof.

5. Diophantine approximations. In this section, using our results, we recovera quantitative version of previous results on numerical homogenization by Avellaneda,Hou, and Papanicolaou (see Theorem A and Theorem B in section 2.3).

Proof of Theorem A. Given an irrational number r we are going to prove (2.24)using Theorem 2.3 and classical results of the approximation of irrational numbers byrational ones. We know (see [19, pp. 189–190]) that there exists a sequence of naturalnumbers (pn, qn) such that∣∣∣∣r − qn

pn

∣∣∣∣ ≤ 1√5p2

n

→ 0 when n → ∞.

Now we consider another sequence {an} ⊂ N such that an → ∞. Then, takingε = 1/(anqn) and h = 2π/(anpn), we get by Theorem 2.3 that

supx∈Γh

|uεh(x) − u∗(x)| ≤ c

(1

an+

1

pn

),


and, in particular, when n → ∞ the error converges to 0, while h/2πε goes to r.Thus, we conclude the proof of Theorem A.

Now we prove Theorem B: Given ε ∈ (0, 1), if h ∈ S(ε, h0), we have to prove that(2.25) holds using the estimate of Theorem 2.3.

Proof of Theorem B. Given h ∈ S(ε, h0), when h/2πε ∈ Q we immediately obtain(2.25) by the estimate of Theorem 2.3. Now we study the case h/2πε /∈ Q. We slightlyperturb ε by ε �= ε such that

h

ε= 2π

q

p∈ Q with H.C.F.(q, p) = 1.

First, we calculate the difference between uεh and uε

h, the approximations of thediscrete problems with scale ε and ε, respectively. Using that uε

h and uεh are solutions

of (2.12)–(2.14) with coefficient aεh and aεh, respectively, it is immediate that∑x∈Γh

aεh(x)|∇(uεh(x) − uε

h(x))|2 =∑x∈Γh

[aεh(x) − aεh(x)]∇uεh(x)∇(uε

h(x) − uεh(x)).

Since a is Lipschitz, we get

|aεh(x) − aεh(x)| ≤ c1

h

∣∣∣∣hε − q

p

∣∣∣∣ ∀x ∈ Γh.

Therefore, by the coercivity of a, we get( ∑x∈Γh

|∇(uεh(x) − uε

h(x))|2) 1

2

≤ ch−1

∣∣∣∣hε − q

p

∣∣∣∣ ,(5.1)

where c depends on a. On the other hand, applying Theorem 2.3, we obtain that

supx∈Γh

|uεh(x) − u∗(x)| ≤ c hp + c′

1

p,(5.2)

where c and c′ depend only on α, β, a, and ‖f‖∞.Now we check that there exist q and p, which provide the result we are looking

for. In fact, using (5.1) and (5.2) we get (2.25) when p, q ∈ N satisfy∣∣∣∣hε − 2πq

p

∣∣∣∣ ≤ τh and τ−1 ≤ p ≤ τh−1.(5.3)

By Dirichlet’s theorem (a classical result of the approximation of irrational numbers;see [33, p. 34]), there exist p, q ∈ N such that p ≤ τh−1 and∣∣∣∣hε − 2π

q

p

∣∣∣∣ < h

pτ.(5.4)

To obtain (5.3) it is also necessary that h/pτ ≤ τh. Therefore, p satisfying (5.4) hasto be larger than τ−2; i.e., p must belong to the interval [τ−2, τh−1]. In fact, usingthat h ∈ S(ε, h0), we have that ∣∣∣∣hε − 2π

q

p

∣∣∣∣ ≥ τ

p52

.

From this inequality, we get τp−5/2 ≤ h/pτ , i.e., p ≥ τ4/3h−2/3.


In short, by Dirichlet’s result we know that there exist p, q with p ≤ τh−1 satis-fying (5.4), and, since h ∈ S(ε, h0), then p ∈ [τ4/3h−2/3, τh−1].

Thus, we conclude the proof if τ4/3h−2/3 ≥ τ−2, i.e., if h ≤ τ5. Later, if h0 = τ5,for any h ∈ S(ε, h0), there exist p, q satisfying (5.3), and, therefore, (2.25) is provedusing the estimates of Theorem 2.3.

Appendix A. Homogenization of the continuous problem.

A.1. Bloch wave decomposition. In this section we recall some basic resultson Bloch wave decomposition. We refer the reader to [7], [9], and [10] for more details.

Let us consider the following spectral problem family by the parameter η ∈ Rd:To find λ = λ(η) ∈ R and ψ = ψ(x; η) (nonidentically zero) such that⎧⎪⎨⎪⎩

Aψ(·; η) = λ(η)ψ(·; η) in Rd,

ψ(·; η) is (η, Y )-periodic, i.e.,

ψ(y + 2πm; η) = e2πim·ηψ(y; η) ∀m ∈ Zd, y ∈ Rd,

(A.1)

where A is the elliptic operator in divergence form defined in (2.1). We can writeψ(y; η) = eiy·ηφ(y; η), φ being Y -periodic in the variable y. It is clear from (A.1) that

η can be confined to the dual cell η ∈ Y ′ = [−1/2, 1/2[d. Under these conditions,

it is known (see [10]) that the spectral problem above admits a discrete sequence ofeigenvalues with the following properties:{

0 ≤ λ1(η) ≤ · · · ≤ λn(η) ≤ · · · → ∞,

λm(η) is a Lipschitz function of η ∈ Y ′ ∀m ≥ 1.(A.2)

Besides, the corresponding eigenfunctions denoted by ψm(·; η) and φm(·; η) form or-thonormal bases in the subspaces of L2

loc(Rd) of (η, Y )-periodic and Y -periodic func-

tions, respectively. Moreover, as a consequence of the min-max principle, it followsthat (see [10])

λ2(η) ≥ λ(N)2 > 0 ∀η ∈ Y ′,(A.3)

where λ(N)2 > 0 is the second eigenvalue of A in the cell Y with Neumann boundary

conditions.To express the equation Aεuε = f in an equivalent way in the Bloch space,

we introduce the Bloch eigenvalues {λεm(ξ)}m≥1 and eigenvectors {φε

m(x; ξ)} in theε-scale:

λεm(ξ) = ε−2λm(εξ), φε

m(x; ξ) = φm

(xε; εξ

).(A.4)

Then, given f ∈ L2#(Y ), the mth Bloch coefficient of f at the ε scale is defined as

follows:

fεm(k) = ε−

d2

∫Y

f(x)e−ik·xφεm(x; k)dx ∀m ≥ 1, k ∈ Λε,(A.5)

where

Λε = {k = (k1, . . . , kd) ∈ Zd | [−1/2ε] + 1 ≤ ki ≤ [1/2ε]}.(A.6)


Theorem A.1. Let ε > 0 be defined by (2.4). For any f ∈ L2(Y ) the followingrepresentation formula holds:

f(x) = εd2

∑k∈Λε

∑m≥1

fεm(k)eik·xφε

m(x; k).

Further, we have the following Parseval’s identity:∫Y

|f(x)|2dx = εd∑k∈Λε

∑m≥1

|fεm(k)|2.

More generally, the following Plancherel identity is also valid:∫Y

f(x)g(x)dy = εd∑k∈Λε

∑m≥1

fεm(k)gεm(k) ∀f, g ∈ L2

#(Y ).

The proof of this theorem follows the same steps of the proof of Theorem 3.1.We also have the following result on the dependence of λ1 and φ1 with respect to

the parameter η (see [7] and [10]).Proposition A.2. Assume that the coefficients ak� satisfy (2.2). Then there

exists δ > 0 such that the first eigenvalue λ1 is an analytic function onBδ = {η : |η| < δ} and satisfies

c1|η|2 ≤ λ1(η) ≤ c2|η|2 ∀η ∈ Y ′(A.7)

and

λ1(0) = ∂kλ1(0) = 0, k = 1, . . . , N,

∂2k�λ1(0) = 2a∗k�, k, = 1, . . . , N,(A.8)

∂αλ1(0) = 0 ∀α such that |α| is odd,

where a∗k� are the homogenized coefficients defined in (2.6). Furthermore, there is achoice of the first eigenfunction φ1(y; η) satisfying{

η ∈ Bδ → φ1(y; η) ∈ L∞ ∩ L2#(Y ) is analytic,

φ1(y; 0) = (2π)−d2 .

A.2. Homogenization results. As in [10], we deduce some classical homoge-nization results with periodic boundary conditions as a consequence of the propertiesof the first Bloch eigenvalue and eigenvector.

First, we observe that (2.3) can be easily transformed to a set of algebraic equa-tions for the Bloch coefficients. We show next that the energy of the solution uε of(2.3) contained in all Bloch modes m ≥ 2 goes to zero (Proposition A.3). Thus, itis sufficient to analyze the first Bloch mode corresponding to m = 1 since all highermodes can be neglected in the homogenization process. When passing to the limit inthe first Bloch component (m = 1) we obtain the Fourier series corresponding to thelimit homogeneous medium.

Let us now develop these ideas. Thanks to the relation

Aε(eik·xφεm(x; k)) = λε

m(ξ)eik·xφεm(x; k),


which is satisfied for k ∈ Λε, and, according to Theorem A.1, (2.3) is equivalent to

λεm(k)uε

m(k) = fεm(k) ∀m ≥ 1, k ∈ Λε.(A.9)

Our goal is to pass to the limit in these equations as ε → 0. First, we claim that onecan neglect all the terms corresponding to m ≥ 2.

Proposition A.3. Let

vε(x) = εd2

∑k∈Λε

∞∑m=2

uεm(k)eik·xφε

m(x; k),

where uεm(k) are the Bloch coefficients of the solution of (2.3) with f ∈ L2

#(Y ). Then

‖vε‖0 ≤ cε2‖f‖0, ‖∇vε‖0 ≤ cε‖f‖0.

The proof can be carried out as in Proposition 3.5 of [10], using the Parseval andPlancherel identities in Theorem A.1 and the characterization of the eigenvalues of(A.1) by the min-max principle, together with the inequalities (A.3).

Let us now recall the classical Fourier series decomposition that will arise naturallywhen analyzing the limit behavior of the first Bloch component as ε → 0. Supposethat f ∈ L2

#(Y ) and k ∈ Zd. The kth Fourier coefficient of f is defined by (4.5), andthe inverse formula is given by (4.4). Furthermore, Plancherel’s identity is also valid:∫

Y

f(x)g(x)dx =∑k∈Z

fk gk ∀f, g ∈ L2#(Y ).

In the following proposition we give some convergence results of the first Bloch com-ponent towards the Fourier series.

Proposition A.4. Under the assumptions of Proposition A.2, there exist c, c′ >0 such that for all f ∈ L2

#(Y ) they follow

|ε d2 fε

1 (k) − fk| ≤ cε|k|‖f‖0 ∀k ∈ ε ∈ Bδ ∩ Λε,(A.10) ∑k∈Λε∩B δ

ε

2

λε1(k)

|ε d2 fε

1 (k) − fk|2 ≤ c′ε2‖f‖20(A.11)

with c, c′ > 0 independent of ε, k, and f .The proof is a consequence of the analyticity of the first Bloch eigenvalue φ1(y; η)

with respect to η in Bδ. The proof can be carried out as in Proposition 3.6 of [10].Our next aim is to pass to the limit in (A.9) corresponding to the first Bloch mode.

Proposition A.5. Under the assumptions of Proposition A.2, for kε ∈ Bδ itfollows that

εd2 uε

1(k) −→ u∗k as ε → 0,

where u∗k is the kth Fourier coefficient of the homogenized solution u∗. In particular,

the following estimate is satisfied:

|ε d2 uε

1(k) − u∗k| ≤ cε‖f‖0.(A.12)

Proof. We get

|ε d2 uε

1(k) − u∗k| ≤

2

(λε1(k))2

|ε d2 fε

1 (k) − fk|2 +2|fk|2

λε1(k)a∗ijkikj

|λε1(k) − a∗ijkikj |2, k �= 0.


Using (A.10) and the Taylor expansion of the first Bloch eigenvalue up to second order(see Proposition A.2), we obtain

|ε d2 uε

1(k) − u∗k| ≤ ε2

(|k|2‖f‖2

0

(λε1(k))2

+ c|k|4|fk|2

λε1(k)a∗ijkikj

)for any kε ∈ Bδ, and by the inequalities (A.7), (A.12) is proved. On the other hand,since m(uε) = 0 by (2.3), we have uε

1(0) = 0, which obviously also converges tou∗

0 = 0.We now proceed with the proof of the estimate (2.9) in Theorem 2.1.Proof of Theorem 2.1. We use the same steps as in the proof of Theorem 1.8

in [7]. The Bloch decomposition of uε and the Fourier decomposition of u∗ allow usto write

uε(x) − u∗(x)

= vε(x) + εd2

∑k∈Λε∩B δ

ε

uε1(k)eikx

[φ1

(xε; εk

)− φ1

(xε; 0

)]− (2π)−

d2

∑k∈Uε

u∗ke

ikx

+ εd2

∑k∈Λε∩Uε

uε1(k)eikxφ1

(xε; εk

)+ (2π)−

d2

∑k∈Λε∩B δ

ε

[εd2 uε

1(k) − u∗k]e

ikx

= vε(x) + vε1(x) + vε2(x) + vε3(x) + vε4(x),

where Uε = ZN − (Λε ∩ ε−1Bδ) and vε(x) is defined in Proposition A.3. According tothe second inequality in Proposition A.3 it is sufficient to estimate vεj , j = 1, . . . , 4.

Second, we can neglect the term vε2, thanks to the coercivity of the homogenizedcoefficients and the term vε3 by (A.7). Now, using the analyticity of φ1, we get

‖vε1‖20 ≤ cε2εd

∑k∈Λε∩B δ

ε

|k|2|uε1(k)|2 ≤ cε2‖f‖2

0.

Finally, vε4 satisfies

‖vε3‖20 ≤

∑k∈Λε∩B δ

ε

2

(λε1(k))2

|ε d2 fε

1 (k) − fk|2 +2|fk|2

λε1(k)a∗ijkikj

|λε1(k) − a∗ijkikj |2.

Thus, by (A.7), the coercivity of the homogenized coefficients, and the analyticityof λ1, we conclude the proof of (2.9).

Appendix B. The 1-d case. In this appendix we study the approximation infinite differences for the solutions of the 1-d problem (2.21). In particular, we proveTheorem 2.3.

Taking h = 2π/n, we denote, for i ∈ N,

fi = f(hi), aεi = a

(h

ε

(i +

1

2

)).

We introduce the following system of linear equations with unknown {uεi}n−1

i=1 :{−aεiu

εi+1 + (aεi + aεi−1)u

εi − aεi−1u

εi−1 = h2fi, 1 ≤ i ≤ n− 1,

uε0 = b, uε

n = c.(B.1)


Solutions of (2.21) are bounded in H10 (0, 2π) independently of ε and converge weakly

in H10 (0, 2π) as ε → 0 to u∗ ∈ H1

0 (0, 2π), the solution of (2.22). Associated with(2.22), we have the following finite difference system for h = 2π/n:{

a∗(−u∗i−1 + 2u∗

i − u∗i+1) = h2fi, 1 ≤ i ≤ n− 1,

u∗0 = b, u∗

n = c.(B.2)

Thanks to Theorem 2.2, to prove (2.27) we need only to estimate {uεi}ni=1 and {u∗

i }ni=1.To do it, we write explicitly these vectors. We define

Uε,hi =

1

haεi (u

εi+1 − uε

i ) with 0 ≤ i ≤ n− 1.(B.3)

Then (B.1) can be written as

−(Uε,hi − Uε,h

i−1) = hfi, 1 ≤ i ≤ n− 1,

and, consequently,

Uε,hi = Uε,h

0 −i∑

j=1

h fj , 1 ≤ i ≤ n− 1.(B.4)

Now, by the definition (B.3) of Uε,hi , we get

uεi+1 = b + Uε,h

0

i∑j=0

h

aεj−

i∑j=1

h

aεj

j∑k=1

hfk, 1 ≤ i ≤ n− 1,(B.5)

with Uε,h0 a constant that can be determined by the boundary conditions and f . In

particular, we have

Uε,h0 =

aε,∗h

2π(c− b) +

aε,∗h

2π

n−1∑j=1

(1

aεj

j∑k=1

h2fk

)(B.6)

with

aε,∗h =

⎛⎝ 1

2π

n−1∑j=0

h

aεj

⎞⎠−1

.(B.7)

We observe that (B.5) is an approximation of the explicit formula of the solution of(2.21):

uε(x) = b + Uε0

∫ x

0

ds

aε(s)−

∫ x

0

∫ s

0

f(t)

aε(s)dtds,

where Uε0 is given by

Uε0 =

(c− b +

∫ 2π

0

∫ s

0

f(t)

aε(s)dtds

)(∫ 2π

0

ds

aε(s)

)−1

.


Analogously, the vector {u∗i }ni=1 solution of (B.2) satisfies

u∗i+1 = b +

(i + 1)h

a∗U∗,h

0 −i∑

j=1

j∑k=1

h2

a∗fk,(B.8)

where U∗,h0 is defined by

U∗,h0 = a∗(c− b) +

n−1∑j=0

j∑k=1

h2fk.(B.9)

Now, using (B.5) and (B.8), we write

uεi+1 − u∗

i+1 = Uε,h0

i∑j=0

h

aεj− U∗,h

0

(i + 1)h

a∗+

i∑j=1

j∑k=1

(1

a∗− 1

aεj

)h2fk(B.10)

= Uε,h0

i∑j=0

(h

aεj− h

a∗

)+ (Uε,h

0 − U∗,h0 )

(i + 1)h

a∗+

i∑j=0

(h

a∗− h

aεj

)j∑

k=1

hfk.

Now we establish the following connections between aε,∗h and a∗, necessary toestimate (B.10).

Lemma B.1. Let a be a 2π-periodic Lipschitz function such that 0 < α ≤ a(x) ≤β for any x ∈ (0, 2π). Let h = 2π/n and ε > 0 satisfy (2.26). Then∣∣∣∣∣∣

∫ 2π

0

dy

a(y)− 2π

p

p−1∑j=0

1

aεj

∣∣∣∣∣∣ ≤ c

2α2

1

p.(B.11)

Moreover, if aε,∗h is defined as in (B.7), we have

(i)∣∣a∗ − aε,∗h

∣∣ ≤ cβ2

2α2

1

pif

n

p∈ N,

(ii)∣∣a∗ − aε,∗h

∣∣ ≤ β2

2α2

c

p+

β2

αhp if

n

p/∈ N.

(B.12)

Proof. First, by the relation (2.26), we consider only p-values of the coefficient a,since

aεp+i = a

(2πq + 2π

q

p

(i +

1

2

))= a

(2π

q

p

(i +

1

2

))= aεi .(B.13)

Thus, the only distinct values {aεi} are {aε0, . . . , aεp−1}. We denote

ai = a

(2π

p

(i +

1

2

)).

Since h and ε satisfy (2.26), then {aε0, . . . , aεp−1} ≡ {a0, . . . , ap−1}. As a consequence,we get

p−1∑j=0

1

aεj=

p−1∑j=0

1

aj.(B.14)


Therefore, we obtain (B.11), thanks to the fact that a is Lipschitz continuous. In fact,∣∣∣∣∣∣∫ 2π

0

dy

a(y)− 2π

p

p−1∑j=0

1

aεj

∣∣∣∣∣∣ ≤p−1∑j=0

∫ 2π j+1p

2πjp

∣∣∣∣ 1

a(y)− 1

aj

∣∣∣∣ dy ≤ c

2α2

1

p.

Now we prove (B.12). By the definition of a∗ and aε,h, we get

|a∗ − aε,∗h | ≤ β2

∣∣∣∣∣∣∫ 2π

0

dy

a(y)−

n−1∑j=0

h

aεj

∣∣∣∣∣∣ .(B.15)

Recall that h = 2π/n, and write n = bnp+cn with cn ∈ {0, 1, . . . , p−1} and bn = [n/p].Therefore, when cn = 0

n−1∑j=0

h

aεj=

2π

p

p−1∑j=0

1

aεj.

Coming back to (B.15) and using (B.11), we obtain (i). On the other hand, whencn ∈ {1, . . . , p− 1}, thanks to (B.14),

n−1∑j=0

h

aεj=

p−1∑j=0

hbnaεj

+

cn−1∑j=0

h

aεj=

p−1∑j=0

hbnaj

+

cn−1∑j=0

h

aεj,

and then

∣∣a∗ − aε,∗h

∣∣ ≤ β2

∣∣∣∣∣∣∫ 2π

0

dy

a(y)− 2π

p

p−1∑j=0

1

aj

∣∣∣∣∣∣ + β2

∣∣∣∣2πp − hbn

∣∣∣∣ p−1∑j=0

1

aj+ β2

cn−1∑j=0

h

aεj

≤ β2

2α2

c

p+ 2πβ2

∣∣∣∣1p − bnn

∣∣∣∣ pα + β2 cnh

α,(B.16)

thanks to (B.11). Now, since n = bn + cn, we get

1

p− bn

n=

n− pbnnp

=cnnp

.

Returning to (B.16), since cn ≤ p, we prove (ii).As a consequence of Lemma B.1, we have the following lemma.Lemma B.2. For any 1 ≤ i ≤ n, we get∣∣∣∣∣∣

i∑j=0

h

(1

aεj− 1

a∗

)∣∣∣∣∣∣ ≤ c

2α2bih +

β − α

α2hci,(B.17)

where bi is defined by bi = [i/p] and ci satisfies that ci = i− pbi, ci ∈ {0, . . . , p− 1}.Proof. This proof is analogous to the case n/p /∈ N in (B.12). First, by (B.13)

and since a∗ is constant,

i∑j=0

h

(1

aεj− 1

a∗

)=

p−1∑j=0

hbi

(1

aεj− 1

a∗

)+

ci−1∑j=0

h

(1

aεj− 1

a∗

).(B.18)


Using that α ≤ aε(x), a∗ ≤ β, we have∣∣∣∣∣∣ci−1∑j=0

h

(1

aεj− 1

a∗

)∣∣∣∣∣∣ ≤ β − α

α2hci.

On the other hand, by the definition of a∗,

p−1∑j=0

hbi

(1

aεj− 1

a∗

)= hbip

p−1∑j=0

1

p

(1

aεj− 1

a∗

)= hbip

⎛⎝p−1∑j=0

1

p

1

aεj− 1

2π

∫ 2π

0

dy

a(y)

⎞⎠ .

Coming back to (B.18) and by (B.11), we prove (B.17):∣∣∣∣∣∣i∑

j=0

h

(1

aεj− 1

a∗

)∣∣∣∣∣∣ ≤ hbip

α2

c

p+

β − α

α2hci ≤ c

2α2hbi +

β − α

α2hci.

Proof of Theorem 2.3. We need only to estimate (B.10). We define

E1(i) = Uε,h0

i∑j=0

(h

aεj− h

a∗

), E2(i) = (Uε,h

0 − U∗,h0 )

h(i + 1)

a∗,

E3(i) =

i∑j=0

(h

a∗− h

aεj

)Fj , Fj =

j∑k=1

hfk.

(B.19)

Then uεi+1 − u∗

i+1 = E1(i) +E2(i) +E3(i). Thus, we need to estimate Ej , j = 1, 2, 3,

to prove Theorem 2.3. Uε,h0 , defined in (B.6), is bounded by

|Uε,h0 | ≤ β|c− b| + β

α‖f‖∞.

Then, thanks to (B.17), we get

|E1(i)| ≤(β|c− b| +

β

α‖f‖∞

)(c

2α2bih +

β − α

α2hci

).(B.20)

Now we study E3(i). To do it, we use Abel’s formula:

m∑j=1

vjuj = um+1Vm −m∑j=1

Vj(uj+1 − uj) with Vj =

j∑�=1

v�.(B.21)

Applying it to E3(i), we obtain

E3(i) = Fi+1

i∑j=0

(h

aεj− h

a∗

)−

i∑j=0

hfj+1

j∑k=0

(h

aεk− h

a∗

).

By (B.19), |Fj | ≤ ‖f‖∞. Then, thanks to (B.17), we have

|E3(i)| ≤ 2‖f‖∞(

c

α2

1

p+

β − α

α2hp

).(B.22)


Finally, we estimate E2(i). We observe that

|E2(i)| ≤1

α|Uε,h

0 − U∗,h0 |.

By (B.6) and (B.9), we get

Uε,h0 − U∗,h

0 = (c− b)(aε,∗h − a∗) + (aε,∗h − a∗)n−1∑j=0

Fjh

aεj+ a∗

n−1∑j=0

(h

aεj− h

a∗

)Fj ,

where Fj is defined by (B.19). Using (B.21) in the last term, we obtain

Uε,h0 − U∗,h

0 = (c− b)(aεh − a∗) + (aεh − a∗)n−1∑j=0

Fjh

aεj

+ a∗Fn

n−1∑j=0

(h

aεj− h

a∗

)− a∗

n−1∑j=0

hfj+1

j∑k=0

(h

aεk− h

a∗

).

Then, applying (B.12) and (B.17),

|Uε,h0 − U∗,h

0 | ≤(|c− b| + 1

α‖f‖∞

)(β2

2α2

c

p+

β2

αhp

)(B.23)

+ a∗‖f‖∞(

c

2α2

1

p+

β − α

α2hp

).

By the estimates (B.20), (B.22), and (B.23), we conclude the proof.

Appendix C. Properties of Bloch waves.

C.1. Properties of Bloch eigenvalues. In this section, we prove some resultspresented in section 3.2. First, we see that the matrix {aij(qy(i, j))} is elliptic, thanksto the regularity of the coefficients aij .

Lemma C.1. We assume that the coefficients {aij} defined in (2.2) are Lipschitzand that q, p satisfy (2.35) and

q

p−

[q

p

]=

ρ

pwith ρi ∈ N.(C.1)

Then, for any y ∈ Γ 2πp

defined in (3.13), we have

d∑i,j=1

aij(qy(i, j))ξiηj ≤ β|ξ||η| + cπ

∣∣∣∣ρp∣∣∣∣ |ξ| |η|,

d∑i,j=1

aij(qy(i, j))ξiξj ≥ α|ξ|2 − cπ d

∣∣∣∣ρp∣∣∣∣ |ξ|2.

(C.2)

Proof. We note that by the properties of the coefficients {aij} we have that

d∑i,j=1

aij(qy(i, j))ξiηj = β|ξ||η| + Δ1,

d∑i,j=1

aij(qy(i, j))ξiξj = α|ξ|2 + Δ2


with

Δ1 =

d∑i,j=1

[aij(qy(i, j)) − aij(qy)]ξiηj and Δ2 =

d∑i,j=1

[aij(qy(i, j)) − aij(qy)]ξiξj .

Using (C.1) and the periodicity of the coefficients, we obtain, for any y ∈ Γ 2πp

,

aij(qy) = aij(ρy) and aij(qy(i, j)) = aij

(ρy +

πρipi

ei + (1 − δij)πρjpj

ej

).

Moreover, since the coefficients {aij} are Lipschitz continuous with constant c > 0,then

|aij(qy(i, j)) − aij(qy)| ≤ cπ

(ρ2i

p2i

+ρ2j

p2j

(1 − δij)

) 12

.

Thus, we get

Δ1 ≤ cπ

d∑i,j=1

(ρ2i

p2i

+ρ2j

p2j

(1 − δij)

) 12

|ξi||ηj | ≤ cπ

∣∣∣∣ρp∣∣∣∣ |ξ| |η|,

Δ2 ≤ cπ

d∑i,j=1

(ρ2i

p2i

+ρ2j

p2j

(1 − δij)

) 12

|ξi||ξj | ≤ cπ d

∣∣∣∣ρp∣∣∣∣ |ξ|2.

As a consequence of this lemma, we prove the ellipticity of {aij(qy(i, j))} underassumptions (2.35).

Proof of Lemma 3.3. Thanks to Lemma C.1, the proof is immediate since{aij(qy(i, j))} are bounded and coercive for sufficiently small |ρ/p|. In particular,|ρ/p| ≤ α/(2cdπ), where c is the Lipschitz constant of the coefficients.

With the coercivity of the coefficients we obtain that the discrete problem ispositive semidefinite (see Lemma 3.4) and the second eigenvalue is bounded below bya positive constant. Let us prove it.

Proof of Lemma 3.5. We are going to use the min-max principle (see [34, p. 99]).We consider VC, the set of complex subsets W ⊂ Cp with dimension 2, and VR, theset of subsets W ⊂ Rp whose dimension is 2. Then we have

μ2(η) = minW∈VC

maxv∈W−(0)

a(η)(v, v)

|v|2 .(C.3)

Using the coercivity of the coefficients (see Lemma 3.3), we have

a(η)(v, v) ≥ α∑

y∈Γ 2πp

d∑k=1

(pk

2πqk

)2 ∣∣∣∣ei 2πpk

ηkv

(y +

2π

pkek

)− v(y)

∣∣∣∣2 .(C.4)

We write v ∈ W as v(y) = vr(y) + ivi(y), y ∈ Γ 2πp

. We define for any y ∈ Γ 2πp

w(y) = cos

(2π

pkηk

)vr(y) + sin

(2π

pkηk

)vi(y)

+ i

[cos

(2π

pkηk

)vi(y) − sin

(2π

pkηk

)vr(y)

].


Using this function and the scalar product in C, we have

∑y∈Γ 2π

p

d∑k=1

(pkqk

)2 ∣∣∣∣ei 2πpk

ηkv

(y +

2π

pkek

)− v(y)

∣∣∣∣2

=∑

y∈Γ 2πp

d∑k=1

(pk)2

(qk)2

[∣∣∣∣v(y +2π

pkek

)∣∣∣∣2 + |v(y)|2 − 2Re

⟨v

(y +

2π

pkek

), w(y)

⟩].

By the Cauchy–Schwarz inequality and since |w(y)| = |v(y)|, we get∣∣∣∣Re

⟨v

(y +

2π

pkek

), w(y)

⟩∣∣∣∣ ≤∣∣∣∣v(y +

2π

pkek

)∣∣∣∣ |v(y)|.Therefore, in view of (C.4), in (C.3) we have

μ2(η) ≥ α minW∈VC

maxv∈W−(0)

1

|v|2∑

y∈Γ 2πp

d∑k=1

(pk

2πqk

)2 (∣∣∣∣v(y +2π

pkek

)∣∣∣∣− |v(y)|)2

≥ α minW∈VR

maxv∈W−(0)

1

|v|2∑

y∈Γ 2πp

d∑k=1

(pk

2πqk

)2 (v

(y +

2π

pkek

)− v(y)

)2

.(C.5)

Note that this lower bound is independent of η. Moreover, the discrete eigenvalueproblem associated with the bilinear form (C.5) is the finite difference approximationon Γ 2π

pof ⎧⎪⎨⎪⎩

1

q2k

∂2u

∂y2k

= μu,

u is Y -periodic.

Now the eigenvalues of the discrete system associated with the bilinear form (C.5)may be computed explicitly (see [22, p. 459]). In particular, we have

minW∈VR

maxv∈W−(0)

1

|v|2∑

y∈Γ 2πp

d∑k=1

(pk

2πqk

)2 (v

(y +

2π

pkek

)− v(y)

)2

≥ min

(c

q2i

),

and we conclude the proof.

C.2. Regularity of the first Bloch eigenvalue and eigenvector. In spiteof the fact that the eigenvalue problem defined in (3.16) depends exponentially in η,it is well known that the eigenvalues μm(η) are not, in general, smooth functionsof η ∈ Y ′ because of the possible change in the multiplicity of eigenvalues (see [32,p. 60]). Now, using the min-max principle, we prove that all the eigenvalues areLipschitz continuous.

Proposition C.2. For all m = 1, . . . , p, μm(η) is a Lipschitz continuous functionof η.

Proof. Recall that the Hermitian bilinear form associated with the eigenvalueproblem (3.16) is defined in (3.18). We notice that it can be decomposed as follows:

a(η)(v, v) = a(ξ)(v, v) + R(v; η, ξ),


where, using the notation⎧⎪⎨⎪⎩yk = y +

2π

pkek (ek being the kth canonical vector),

bk�(y) = ak�(qy(k, ))pkp�

qkq�(2π)2,

(C.6)

we can write

R(v; η, ξ) =∑

y∈Γ 2πp

d∑k,�=1

bk�(y)

[ei2π(

ηkpk

− η�p�

) − ei2π(

ξkpk

− ξ�p�

)

]v(yk)v(y�)

+∑

y∈Γ 2πp

d∑k,�=1

bk�(y)

[ei2π

ξkpk − e

i2πηkpk

]v(yk)v(y)

+∑

y∈Γ 2πp

d∑k,�=1

bk�(y)

[e−i2π

ξ�p� − e

−i2πηkpk

]v(y)v(y�).

Taking into account that the coefficients in (C.6) are bounded (see Lemma 3.3) andthe function eiη·y is Lipschitz on the variable η, we have

|R(v; η, ξ)| ≤ c(d, β) |η − ξ|∑

y∈Γ 2πp

|v(y)|2.

Then, using the min-max characterization of eigenvalues, we deduce that

μm(η) ≤ μm(ξ) + c(d, β)|η − ξ|,

and, interchanging η and ξ, we conclude the proof.

Unfortunately, the Lipschitz character of eigenvalues is not enough to derive ho-mogenization results. We are going to prove that μ1(η) is analytic using the fact thatthe eigenvalue μ1(0) (which is equal to 0) is simple. To prove it we use classical resultsof perturbation theory for linear operators in finite-dimensional spaces (see [24]).

Proof of Proposition 3.6. As already remarked, when η = 0, the first eigen-value μ1(0) = 0 is simple. Furthermore, in Lemma 3.5 we prove that all eigenvalues{μm(η)}m≥2 are bounded below for any η ∈ Y ′ by a positive constant. Hence, usingOstrowski’s theorem on continuity of the eigenvalues (see [34, p. 63]) there exists δ > 0such that μ1(η) is simple in η ∈ Bδ = B(0; δ).

Now, applying a consequence of Rellich’s theorem on finite-dimensional spaces(see [32]), we obtain that μ1(η) is analytic in η ∈ Bδ. In fact, note that our finiteeigenvalue problem (3.16) is perturbed by a d-dimensional variable η ∈ Y ′ ⊂ Rd.Thus, by Theorem 5.16 in [24, p. 116], μ1 is an analytic map in Bδ, since μ1 is asimple eigenvalue in this region.

Since the eigenvalue problem (3.16) depends analytically on η, and the first eigen-value does not change its multiplicity and remains analytic with respect to η in Bδ forδ sufficiently small, by Rellich’s theorem the choice of the first eigenvector can bemade so that it depends analytically on η ∈ Bδ.


C.3. Derivatives of the first Bloch eigenvalue and eigenvector. The aimof this section is to obtain an expression of the homogenized coefficients in the finitedifference analysis. We have divided the proof into a sequence of lemmas. First, wecompute the derivatives of μ1 and ϕp,1.

Proof of Lemma 3.7. Given that η → μ1(η) and η → ϕp,1(y; η) are smooth (seeProposition 3.6), it is straightforward to compute their derivatives at η = 0. To doit, it is enough to differentiate the eigenvalue problem

−d∑

i,j=1

e−iy·η∇− 2πp

i

[1


2πp

j (eiy·ηϕp,1(y; η))

]= μ1(η)ϕp,1(y; η)(C.7)

with respect to η twice and evaluate it at η = 0. Since the computations are classical,we present only the essential steps. Define

A(y, η)v(y) = −d∑

i,j=1

e−iy·η∇− 2πp

i

[1

qiqjaij (qy(i, j))∇+ 2π

p

j eiy·ηv(y)

].

With the notation (C.6), we get

A(y, η)v(y) = −d∑

i,j=1

bij(y(i, j))[ei 2πpj

ηjv(yj) − v(y)

]

+

d∑i,j=1

bij(y−i(i, j))

[ei 2πpj

ηje−i 2π

piηiv(yj−i) − e

−i 2πpi

ηiv(y−i)].

Thus, we compute the first-order derivatives in η = 0:

∂kA(y, 0)v(y) = − i2π

pk

d∑i=1

bik(y(i, k))v(yk) + i2π

pk

d∑i=1

bik(y−i(i, k))v(yk−i)(C.8)

− i2π

pk

d∑i=1

bki(y−k(k, i))v(yi−k) + i

2π

pk

d∑i=1

bki(y−k(k, i))v(y−k).

Also, the second-order derivatives are

∂2k�A(y, 0)v(y) =

(2π)2

pkp�[b�k(y

−�(, k))v(yk−�) + bk�(y−k(k, ))v(y�−k)],(C.9)

if k �= , and

∂2kkA(y, 0)v(y) =

(2π)2

(pk)2

⎡⎣ d∑i=1

bik(y(i, k))v(yk) −d∑

k �=i=1

bik(y−i(i, k))v(yk−i)

⎤⎦(C.10)

+(2π)2

(pk)2

⎡⎣ d∑i=1

bki(y−k(k, i))v(y−k) −

d∑k �=i=1

bki(y−k(k, i))v(yi−k)

⎤⎦ .

Differentiating in (C.7), evaluating at η = 0, and taking the scalar product in Cp, weobtain

∂kμ1(0) =∑

y∈Γ 2πp

ϕp,1(y; 0) · ∂kA(y, 0)ϕp,1(y; 0).


Considering (3.20) and (C.9), we have

∂kμ1(0) = i2π

pk

∑y∈Γ 2π

p

d∑i=1

[−bik(y(i, k)) + bik(y−i(i, k))] = 0,

since

∑y∈Γ 2π

p

d∑i=1

bik(y(i, k)) =∑

y∈Γ 2πp

d∑i=1

bik(y−i(i, k)).

On the other hand, the first derivatives of ϕp,1 satisfy⎧⎪⎪⎨⎪⎪⎩A(y, 0)∂kϕp,1(y; 0) = ∂kA(y, 0)ϕp,1(y; 0), y ∈ Γ 2π

p,∑

y∈Γ 2πp

∂kϕp,1(y; 0)ϕp,1(y; 0) = 0

for k = 1, . . . , d. By (3.20) and (C.9), we write

∂kA(y, 0)ϕp,1(y; 0) =2πi

pk(p)12

d∑j=1

[bkj(y(k, j)) − bkj

(y(k, j) − 2π

pjej

)]

=i

(p)12

d∑j=1

∇− 2πp

j

ajkqkqj

(qy(k, j)).

Then the first-order derivatives of ϕp,1 are written as in (3.25). We differentiate againin (C.7), evaluate in η = 0, and take the scalar product to get

∂2k�μ1(0) =

∑y∈Γ 2π

p

ϕp,1(y; 0)∂2k�A(y, 0)ϕp,1(y; 0)

+∑

y∈Γ 2πp

ϕp,1(y; 0) [∂kA(y, 0)∂�ϕp,1(y; 0) + ∂�A(y, 0)∂kϕp,1(y; 0)] .

Using (C.9) or (C.10) and with the notation (C.6), we have∑y∈Γ 2π

p

ϕp,1(y; 0) · ∂2k�A(y, 0)ϕp,1(y; 0) =

2

p

∑y∈Γ 2π

p

1

qkq�a�k(qy(, k)).

On the other hand, in view of (3.25) and (C.8), we get

∑y∈Γ 2π

p

ϕp,1(y; 0) · ∂kA(y, 0)∂�ϕp,1(y; 0) = −1

p

∑y∈Γ 2π

p

Θkq (y)

d∑j=1

1

q�qj∇− 2π

p

j aj�(qy(, j)).

Then we obtain the second-order derivatives of μ1 and conclude the proof ofLemma 3.7.


The 1-d case. First, we observe that, thanks to (3.23), the second-order deriva-tive of μ1 is

1

2∂2μ1(0) =

1

p

1

q2

∑y∈Γ 2π

p

[a(qy(1, 1)) − Θk

q (y)∇− 2πp a(qy(1, 1))

],(C.11)

where {Θq(y) | y ∈ Γ 2πp} satisfies⎧⎪⎪⎨⎪⎪⎩

−∇− 2πp

[a (qy(1, 1))∇ 2π

p (Θq(y))]

= ∇− 2πp a(qy(1, 1)), y ∈ Γ 2π

p,

Θq(y) 2π-periodic and∑

y∈Γ 2πp

Θq(y) = 0.(C.12)

Then, since y(1, 1) is defined by (3.15), we compute the components of Θq:

Θq(2πj/p) =2π

p√p

{p− 2j + 1

2− a∗p

(b∗p −

j∑k=1

1

a(π(2k + 1)q/p)

)}(C.13)

for j = 1, . . . , p, where a∗p is defined in (2.30) and

b∗p =1

p

p∑k=1

p + 1 − k

a(π(2k + 1)/p).(C.14)

Then, replacing (C.13) and (C.14) in (C.11), the second derivative of μ1 satisfies(3.30). Furthermore, a∗ and a∗p are defined in (2.23) and (2.30), respectively. There-fore, using that the coefficient a is Lipschitz, we conclude the proof of Proposition 3.10.

The case of several space dimensions. Now we are going to prove that∂2k�μ1(0) is an approximation of the homogenized coefficient a∗k� defined in (2.6).

To do it, we need to find a relation between χk, the solution of (2.7), and Θkq ,

which satisfies (3.24). Since q, p ∈ N we write

q

p−

[q

p

]=

ρ

pand qi = ρiσi, where σi ∈ R+, σi ≥ 1 ∀i = 1, . . . , d.(C.15)

Then, thanks to the fact that {aij} are Y -periodic, (3.24) coincides with⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩−

d∑i,j=1

∇− 2πρp

i

[aij (y(i, j))

σiσj∇

2πρp

j (Θkρ(y))

]=

d∑j=1

1

σkσj∇

−2πρp

j ajk(y(k, j)),

y ∈ Γ 2πρp, Θk

ρ(y) ρY -periodic, and∑

y∈Γ 2πρp

Θkρ(y) = 0

(C.16)

with

Θkq (y) =

1

ρkΘk

ρ(ρy) ∀y ∈ Γ 2πp.(C.17)

We note that Θkρ is the approximation in finite differences of the solution of the

periodic boundary problem:⎧⎪⎪⎪⎨⎪⎪⎪⎩− ∂

∂yi

(bij(y)

∂χkρ

∂yj

)=

∂bik(y)

∂yiin ρY ,

χkρ ∈ H1

#(ρY ), mρ(χkρ) =

1

ρY

∫ρY

χkρ(y)dy,

(C.18)


where

bij(y) =aij(y)

σiσj∀i, j = 1, . . . , d.

In particular, using classical error estimates (see Theorem 2.2), we obtain

supy∈Γ 2πρ

p

{|χkρ(y) − Θk

ρ(y)|} ≤ cβ

σm

∣∣∣∣ρp∣∣∣∣ ,(C.19)

where σm = min{σi}. Now we study the relation between χkρ and χk, the solution of

(2.7). We have the following result.Lemma C.3. Let χk and χk

ρ, k = 1, . . . , d, be the test functions solving (2.7) and(C.18), respectively. Let δ > 0 be defined in (2.40). Then there exists c, independentof δ, k, ρ, and q, such that

supy∈Γ 2π

p

{|χkρ(y) − χk(ρy)|} ≤ c δ.(C.20)

Proof. Since ρ ∈ Nd and χk is Y -periodic, then χk is also the unique solution of⎧⎨⎩Aχk =∂ak�∂y�

in ρY ,

χk ∈ H1#(ρY ), mρ(χ

k) = 0.

(C.21)

We need another function to compare χkρ to χk. We consider the solution of⎧⎨⎩Aχk

b = σkν∂bk�∂y�

in ρY ,

χkb ∈ H1

#(ρY ), mρ(χkb ) = 0

(C.22)

with ν ∈ R to be chosen later. Thus, we consider

‖∇(χk − χkρ)‖2 ≤ ‖∇(χk − χk

b )‖2 + ‖∇(χkb − χk

ρ)‖2.

Since χk and χkb satisfy (C.21) and (C.22), respectively, using the variational formu-

lation

‖∇(χk − χkb )‖2 ≤ cβ

αsup

i=1,...,d

{∣∣∣∣(1 − ν

σi

)∣∣∣∣} .

On the other hand, since χkb satisfies (C.22), we have∫

ρY

A(χkb − χk

ρ) (χkb − χk

ρ)dy = −∫ρY

(νσkbjk + aij

∂χkρ

∂yi

)∂(χk

b − χkρ)

∂yj.(C.23)

Now, since χkρ verifies (C.18),

∫ρY

(bjk + bji

∂χkρ

∂yi

)∂ϕ

∂yjdy = 0 ∀ϕ ∈ H1

#(ρY ), k = 1, . . . , d.


Using this result in (C.23) and since bij = aij/σiσj , we get∫ρY

A(χkb − χk

ρ) (χkb − χk

ρ)dy =

∫ρY

aij

(σkν

σiσj− 1

)∂χk

ρ

∂yi

∂(χkb − χk

ρ)

∂yj.

Therefore, we obtain

‖∇(χkb − χk

ρ)‖2 ≤ ‖∇χkρ/σ‖2 sup

i,j=1,...,d

{∥∥∥∥aij(y)(σj −νσk

σi

)∥∥∥∥∞

}.

Since χkρ solves (C.18), we have

‖∇χkρ/σ‖2 =

⎛⎝ n∑j=1

1

σ2j

∥∥∥∥∥∂χkρ

∂yj

∥∥∥∥∥2⎞⎠ 1

2

≤ cβ

σkα.

Since aij ∈ L∞(Y ) and considering ν ∈ [σm, σM ], we prove that

‖∇(χk − χkρ)‖2 ≤ c‖a‖∞ sup

i,j=1,...,d

{∣∣∣∣1 − σj

σi

∣∣∣∣} .

We conclude taking the definition (2.40) of δ into account.Thus, thanks to (C.20) and Lemma C.3, we have the following lemma.Lemma C.4. Let χk, k = 1, . . . , d, be the test functions solution of (2.7). Let

δ > 0 be defined in (2.40). Then we get that

supy∈Γ 2π

p

{|ρkΘkq (y) − χk(ρy)|} ≤ c δ(C.24)

with c independent of k, ρ, and q.On the other hand, since the coefficients a∗k� are symmetric and ak� and χk are

Y -periodic, the homogenized coefficients can be written as

a∗k� =1

|Y |

∫Y

(ak�(ρy) −

χk(ρy)

2ρj

∂aj�∂yj

(ρy) − χ�(ρy)

2ρj

∂akj∂yj

(ρy)

)dy.(C.25)

Furthermore, since a∗,q/pk� is defined by (3.21) and q and p satisfy (2.27), we have,

thanks to (3.23), that

a∗,q/pk� =

1

p

∑y∈Γ 2π

p

ak�(ρy(k, ))

(C.26)

− 1

p

∑y∈Γ 2π

p

d∑j=1

[qkΘ

kq (y)

2qj∇− 2π

p

j aj�(ρy(, j)) +q�Θ

�q(y)

2qj∇− 2π

p

j ajk(ρy(k, j))

].

Now we are going to compare (C.25) with (C.26) in the following lemmas.Lemma C.5. We assume that the coefficients {ak�} are Lipschitz continuous.

Then, for any k, = 1, . . . , d and |ρ/p| ≤ δ, we get∣∣∣∣∣∣∣1

|Y |

∫Y

ak�(ρy)dy −1

p

∑y∈Γ 2π

p

ak�(ρy(k, ))

∣∣∣∣∣∣∣ ≤ c δ,(C.27)


where c is the Lipschitz constant of the coefficients.Proof. It is immediate using that the coefficients are Lipschitz continuous.Lemma C.6. We assume that {ak�} and their derivatives are Lipschitz continuous

and {χk} belong to L∞(Y ). Let δ > 0 be defined in (2.40). Then∣∣∣∣∣∣∣1

|Y |

∫Y

χk(ρy)

ρj

∂aj�∂yj

(ρy)dy − 1

p

∑y∈Γ 2π

p

d∑j=1

qkΘkq (y)

qj∇− 2π

p

j aj�(ρy(, j))

∣∣∣∣∣∣∣ ≤ c δ.(C.28)

Proof. Using (C.20) we get

1

p

∑y∈Γ 2π

p

∣∣∣∣∣∣[qkΘkq (y) − σkχ

k(ρy)]

d∑j=1

1

qj∇− 2π

p

j aj�(ρy(, j))

∣∣∣∣∣∣ ≤ c δ,(C.29)

where c depends of the Lipschitz constants of the coefficients. On the other hand,∣∣∣∣∣∣d∑

j=1

χk(ρy)

(σk

qj− 1

ρj

)∇− 2π

p

j aj�(ρy(, j))

∣∣∣∣∣∣ ≤ c supk,j=1,...,d

{∣∣∣∣σk

σj− 1

∣∣∣∣} ,

where c depends of the W 1,∞-norm of the coefficients and the L∞-norm of the testfunction. Then, by (2.40), we get∣∣∣∣∣∣

d∑j=1

χk(ρy)

(σk

qj− 1

ρj

)∇− 2π

p

j aj�(ρy(, j))

∣∣∣∣∣∣ ≤ c δ.

Thus, by this estimate and (C.29), we have only to prove that∣∣∣∣∣∣∣1

|Y |

∫Y

χk(ρy)

ρj

∂aj�∂yj

(ρy)dy − 1

p

∑y∈Γ 2π

p

d∑j=1

χk(ρy)

ρj∇− 2π

p

j aj�(ρy(, j))

∣∣∣∣∣∣∣ ≤ c δ.

But this property is immediate using that the coefficients are C1,1.Finally, applying Lemmas C.3–C.6, we conclude the proof of Proposition 3.11.

C.4. Further properties of the first Bloch eigenvalue. First, we are goingto see that the Hessian of μ1 is coercive. In fact, we get the following lemma.

Lemma C.7. Under the hypotheses of Lemma 3.3, there exist two constants0 < α ≤ β satisfying

d∑i,j=1

qkq�∂2k�μ1(0)ξiηj ≤ β|ξ||η| (boundedness),(C.30)

d∑k,�=1

qkq�∂2k�μ1(0)ξkξ� ≥ α|ξ|2 (coercivity).(C.31)

Proof. We give the proof of (C.31); the other one is very similar. Note that

d∑k,�=1

qkq�∂2k�μ1(0)ξkξ� ≥

d∑k,�=1

a∗k�ξkξ� +

d∑k,�=1

[qkq�2

∂2k�μ1(0) − a∗k�

]ξkξ�.


Then, since the homogenized coefficients are coercive and the error estimate (3.31)holds, the proof follows the arguments of Lemmas 3.3 and C.1.

As a consequence of these results and the analyticity of μ1, we prove Lemmas 3.8and 3.9.

Proof of Lemma 3.8. First, for any η �= 0, μ1(η) > 0. Using the bilinear form(3.18), we write

μ1(η) =∑

y∈Γ 2πp

d∑i,j=1

1


2πp

j (eiy·ηϕp,1(y; η))∇2πp

i (eiy·ηϕp,1(y; η)).

Thanks to Lemma 3.3 and denoting ψp,1(y; η) = eiy·ηϕp,1(y; η), we obtain

α∑

y∈Γ 2πp

d∑i=1

1

q2i

∣∣∣∣∇ 2πp

i ψp,1(y; η)

∣∣∣∣2 ≤ μ1(η) ≤ β∑

y∈Γ 2πp

d∑i=1

1

q2i

∣∣∣∣∇ 2πp

i ψp,1(y; η)

∣∣∣∣2 .(C.32)

If μ1(η) = 0 for some η �= 0, we conclude that ϕp,1(y; η) = ce−iy·η. But ϕp,1 isY -periodic in the variable y only for η = 0 in Y ′. This is in contradiction with thefact that η �= 0. On the other hand, applying Taylor’s formula, we have

μ1(η) =1

2∂2k�μ1(0)ηkη� + O(|η|3).

Thus, thanks to (C.31),

α

∣∣∣∣ηq∣∣∣∣2 + O(|η|3) ≤ μ1(η) ≤ β

∣∣∣∣ηq∣∣∣∣2 + O(|η|3) ∀η ∈ Bδ.

Therefore, there exist c, c′ > 0 such that

c′∣∣∣∣ηq

∣∣∣∣2 ≤ μ1(η) ≤ c

∣∣∣∣ηq∣∣∣∣2 ∀η ∈ Bδ.

Finally, taking into account that μ1(η) > 0 for all η �= 0, we deduce that (3.26) holds.Furthermore, applying (3.26) in (C.32) we obtain (3.27) and conclude the proof.

Proof of Lemma 3.9. Using Taylor’s formula, we get

μ1(η) −1

2∂2ijμ1(0)ηiηj =

∑ijk

cijk∂3ijkμ1(θ)ηiηjηk, where θ ∈ B|η|.

We need to see that ∣∣∣∣∣∣∑ijk

cijk∂3ijkμ1(θ)ηiηjηk

∣∣∣∣∣∣ ≤ c

∣∣∣∣ηq∣∣∣∣2 |η|,

where c is independent of the constant q. We note that the eigenvalue problemdefined in (3.17) depends of q. First, we denote ψp,1(y; η) = eiy·ηϕp,1(y; η) and obtainby (3.27) that

∑y∈Γ 2π

p

d∑i=1

1

q2i

∣∣∣∣∇ 2πp

i ψp,1(y; η)

∣∣∣∣2 ≤ c

∣∣∣∣ηq∣∣∣∣2 .


Now, using similar computations to those in the proof of Lemma 3.7, we have

∂kμ1(η) =∑

y∈Γ 2πp

d∑i=1

1

qkqiaik (qy(i, k))ψp,1(y

k; η)∇2πp

i (ψp,1(y; η))

+∑

y∈Γ 2πp

d∑i=1

1

qkqiaki (qy(i, k))ψp,1(yk; η)∇

2πp

i (ψp,1(y; η)).

Then, by Lemma 3.3 we get

|qk∂kμ1(η)| ≤ c

∣∣∣∣ηq∣∣∣∣ ∀η ∈ Bδ.

On the other hand, ∂kϕp,1(η) satisfies

eiy·η[A(η) − μ1(η)]∂kϕp,1(η) = Fk(η) and 〈∂kϕp,1(η), ϕp,1(η)〉 = 0,

where

Fk(y; η) =

d∑i=1

1

qkqiaik (qy(i, k))∇− 2π

p

i (ψp,1(yk; η))

+

d∑i=1

1

qkqi∇− 2π

p

i [aik (qy(i, k))]ψp,1(yk−i; η)

+

d∑i=1

1

qkqiaik(qy

−k(i, k))∇2πp

i (ψp,1(y−k; η)) + ∂kμ1(η)ψp,1(y; η).

Then there exists C > 0 such that

∑y∈Γ 2π

p

|qkFk(y; η)|2 ≤ c

[1 +

∣∣∣∣ηq∣∣∣∣2]≤ C,

where C depends on the dimension and the coefficients ak�. Therefore,

∑y∈Γ 2π

p

q2k

d∑i=1

1

q2i

∣∣∣∣∇ 2πp

i (eiy·η∂kϕp,1(y; η))

∣∣∣∣2 ≤ C

and, using Lemma 3.5,

∑y∈Γ 2π

p

q2k |∂kϕp,1(y; η))|2 ≤ C|q|2.


On the other hand, when k �= , ∂2k�μ1(η) can be written as

∂2k�μ1(η) =

∑y∈Γ 2π

p

1

qkq�a�k(qy(, k))[ψp,1(y

k; η)ψp,1(y�; η) + ψp,1(y�; η)ψp,1(yk; η)]

−∑

y∈Γ 2πp

d∑i=1

1

qkqiaik(qy(i, k))∇

2πp

i (eiy·η∂�ϕp,1(y; η))ψp,1(yk; η)

−∑

y∈Γ 2πp

d∑i=1

1

qkqiaik(qy(i, k))eiy

k·η∂�ϕp,1(yk; η)∇

2πp

i (ψp,1(y; η))

−∑

y∈Γ 2πp

d∑i=1

1

q�qiai�(qy(i, ))∇

2πp

i (eiy·η∂kϕp,1(y; η))ψp,1(y�; η)

−∑

y∈Γ 2πp

d∑i=1

1

q�qiai�(qy(i, ))e

iy�·η∂kϕp,1(y�; η)∇

2πp

i (ψp,1(y; η)).

Thus, we get

|qkq�∂2k�μ1(η)| ≤ c

⎡⎢⎢⎣1 +

⎛⎜⎝ ∑y∈Γ 2π

p

d∑i=1

∣∣∣∣q�∇ 2πp

i (eiy·η∂�ϕp,1(y; η))

∣∣∣∣2⎞⎟⎠

12

⎤⎥⎥⎦

+ c

⎛⎜⎝ ∑y∈Γ 2π

p

d∑i=1

∣∣∣∣∇ 2πp

i (ψp,1(y; η))

∣∣∣∣2⎞⎟⎠

12⎛⎜⎝ ∑

y∈Γ 2πp

|qk∂kϕp,1(y�; η)|2

⎞⎟⎠12

.

Thus, applying the previous estimates, we obtain

|qkq�∂2k�μ1(η)| ≤ β

[1 + c(d) + C(d)|q|

∣∣∣∣ηq∣∣∣∣] ≤ c(β, d) ∀η ∈ Bδ.

We have similar results for the case k = .Now we study ∂2

k�ϕp,1(η). Note that

eiy·η[A(η) − μ1(η)]∂2k�ϕp,1(η) = Fk�(η),

〈∂2k�ϕp,1(η), ϕp,1(η)〉 = −〈∂kϕp,1(η), ∂�ϕp,1(η)〉.

Thus, we have that

∑y∈Γ 2π

p

q2kq

2�

d∑i=1

1

q2i

∣∣∣∣∇ 2πp

i (eiy·η∂2k�ϕp,1(y; η))

∣∣∣∣2 ≤ C|qη|2 +∑

y∈Γ 2πp

|Fk�(y; η)|2.

We estimate Fk�(η) arguing as for Fk(η), and we see that∑y∈Γ 2π

p

|Fk�(y; η)|2 ≤ c|qη|2.


Finally, according to the previous estimates we proceed analogously to obtain

|qiqjqk∂3ijkμ1(η)| ≤ c(d, β)|q| ∀i, j, k = 1, . . . , d,

and, by Taylor’s formula, we derive (3.28).

Acknowledgment. The authors would like to thank the referees for their helpin improving the manuscript.

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Finite Difference Approximation of Homogenization Problems for Elliptic Equations

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