Advances in Mathematical
Sciences and Applications
Vol. 25, No. 1 (2016), pp. 117–131

GAKKOTOSHOTOKYO JAPANERROR ESTIMATES FOR ELLIPTIC EQUATIONSWITH NOT-EXACTLY PERIODIC COEFFICIENTS

Sina Reichelt
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin(E-mail: sina.reichelt@wias-berlin.de)

Abstract.
This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not-exactly ε-periodic and the ellipticity
constant may degenerate with order O(ε2γ ). Here, ε > 0 denotes the ratio between the
microscopic and the macroscopic length scale, and the coefficients are only periodic withrespect to the microscopic scale. It is shown that for γ = 0 and γ = 1 the error between
the original solution and the effective solution is of order O(ε1/2 ). Therefore suitable test
functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective
solution with respect to the macroscopic scale.

Communicated by Adrian Muntean; Received May 10, 2016.
This research was supported by Deutsche Forschungsgesellschaft within Collaborative Research Center
910: Control of self-organizing nonlinear systems: Theoretical methods and concepts of application via
the project A5: Pattern formation in systems with multiple scales.
AMS Subject Classification: 35B10, 35B27, 35B40, 35D30, 35J70.
Key words: Homogenization, error estimates, periodic unfolding, gradient folding operator.

117

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1

Introduction

Periodic homogenization is a very powerful tool to derive effective equations for e.g.
biological cells [21, 7], binary mixtures [4], modeling of concrete [24, 5], and other processes
in porous media [13, 25, 18, 19]. All these articles deal with systems of coupled reactiondiffusion equations which contain linear, elliptic equations of the form



(1.1)
− div ε2γ A x, xε ∇uε + B x, xε uε = Fε (x) in Ω.
In this text we study equation (1.1) for γ = 0 and γ = 1 on a bounded domain Ω ⊂ Rd
supplemented with homogeneous Neumann boundary conditions. The coefficients underconsideration are not-exactly periodic in the following sense: in addition to the periodicity
with respect to the microscopic scale y = x/ε for ε > 0, the coefficients are allowed todepend arbitrarily on the macroscopic scale x ∈ Ω. For the sake of quantitative estimates,
the x-dependency is assumed to be more regular, namely, Lipschitz- or H1 (Ω)-regular, cf.
assumption (2.7). Within the framework of reaction-diffusion systems γ = 1 models theslow diffusion of order O(ε) for some species, whereas for γ = 0 other species diffuse withorder O(1).
The aim is to quantify the error between the original solution uε of (1.1) and theeffective solution, which is for γ = 0 a one-scale function u(x) and for γ = 1 a two-scale
function U (x, y) solving the equations (2.8) and (4.2). The weak and strong two-scaleconvergence of solutions uε to (1.1) is studied in [24] and [12] for all γ ∈ [0, ∞). Here, we
derive quantitative estimates in the two-scale space Ω × Y , where Y = [0, 1)d , using theperiodic unfolding operator Tε : L2 (Ω) → L2 (Ω × Y ), cf. [2] or (2.1),
γ=0:
γ=1:

kuε − ukL2 (Ω) + k Tε (∇uε ) − [∇u+∇y U ]kL2 (Ω×Y ) ≤ ε1/2 C,
k Tε uε − U kL2 (Ω;H1 (Y )) ≤ ε1/2 C.

(1.2)
(1.3)

The constant C does not depend on ε, but on the given data as well as on the norms of
the effective solution u and U in the space H2 (Ω) and H1 (Ω; H1 (Y )), respectively. In thecase γ = 0 the corrector U is given by the well-known unit-cell problem (4.3).
For two-scale functions F ∈ H1 (Ω; L2 (Y )) which are Y -periodic and in general notcontinuous, neither with respect to x ∈ Ω nor y ∈ Y , the right-hand side of (1.1) isdefined via the folding or averaging operator, cf. (2.2),
Fε (x) = Fbε (x, x/ε)

where

R
Fbε (x, y) = −ε([x/ε]+Y ) F(z, y) dz.

(1.4)

This becomes relevant for systems of parabolic equations which are coupled via Lipschitzterms Fε (uε , vε ), where uε and vε only belong to the space H1 (Ω). For such systems the
rigorous derivation of effective two-scale equations was proved in [16] using the notion ofstrong two-scale convergence via periodic unfolding. In the proceedings article [26] theconvergence rate of order O(ε1/4 ) is shown. For elliptic equations with exactly periodic
coefficients and γ = 0 we refer to [8, 9, 23, 10] for unfolding based error estimates of orderO(ε1/2 ) and higher. For coupled reaction-diffusion equations with γ = 0 only the gradientestimate is of lower order O(ε1/4 ) in [6]. Estimates of order O(ε1/2 ) based on asymptotic
expansion are derived in [4, 19] containing the cases γ = 0 and γ = 1. Therein continuity

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is assumed for the given data and effective solutions, whereas our approach requires nohigher regularity with respect to the microscopic scale.
Our main result (Theorem 2.1) is novel for γ = 1 and for γ = 0 (Theorem 4.1) it
is a slight generalization of the estimates obtained in [8]. There arise two difficulties
deriving the quantitative estimates (1.2)–(1.3) which we briefly outline for γ = 1. The
periodicity defect of Tε uε ∈ L2 (Ω; H1 (Y )) versus U ∈ L2 (Ω; H1per (Y )) is treated as in [9].
Secondly one has to identify a suitable “folding” or “approximating sequence” for the limit
U (x, y) since the naive composition U (x, x/ε) is only well-defined in H1 (Ω) for functions
U ∈ C1 (Ω; H1per (Y )), see [15] for admissible test functions in two-scale convergence. AlsoUε as in (1.4) is not suitable due to insufficient H1 (Ω)-regularity. Therefore, we employ
the gradient folding operator Gε : H1 (Ω; H1per (Y )) → H1 (Ω) as in [12, 16, 26] which isdefined by solving the degenerating elliptic problem (3.3). In Theorem 3.4 we control thefolding mismatch of Uε and Gε U with respect to the H1 (Ω)-norm as it is indicated in [26].
This text is structured as follows. In Subsection 2.1 we introduce basic definitions andnotations and in Subsection 2.2 our assumptions and Theorem 2.1 for γ = 1 are stated.
The derivation of quantitative estimates is done in Section 3, whereby Subsection 3.1 and3.2 contain estimates for the approximation errors and the folding mismatch, respectively.
The proof of Theorem 2.1 is given in Subsection 3.3. Finally, we show in Section 4 how
our approach applies to the case γ = 0 (Subsection 4.1) as well as to parabolic equations(Subsection 4.3). In Subsection 4.2 we compare our estimates with [8].

2Notations and definitions

2.1

Periodic unfolding

Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary ∂Ω. Following [2], we
denote with Y := [0, 1)d the unit cube and Y := Rd /Zd is the torus which we obtain byidentifying opposite faces of Y . Using the mappings [ · ] : Rd → Zd and { · } : Rd →Y, any point x ∈ Rd admits the additive decomposition x = [x] + {x}. Here, [x] :=bx1 c, . . . , bxd c) is the componentwise Gauss bracket and {x} := x − [x] is the remainder.
With this we define for ε > 0 the cells Cε (x) := ε [x/ε] + Y where the macroscopic
part ε[x/ε] ∈ εZd denotes the node associated to Cε (x) and y ∈ Y respective y ∈ Y is
the microscopic part. Since the domain Ω ⊂ Rd is bounded, not all cells Cε (x) are fully
contained in Ω. To handle cells intersecting the boundary ∂Ω we define the following setsΩ−and Ω+
ε := int {x ∈ Ω | Cε (x) ⊂ Ω}ε := int {x ∈ Ω | Cε (x) ∩ Ω}

so that Ω−Ω+ε ⊂ Ω ⊂ε as depicted in Figure 2.1. The Lipschitz property of Ω guarantees√+−
vol(Ωε \Ωε ) ≤ ε2 d meas(∂Ω) with finite surface measure meas(∂Ω).
The periodic unfolding operator Tε : L2 (Ω) → L2 (Ω × Y) is defined as in [2] via(Tε u)(x, y) := u ε[ xε ] + εy)(2.1)

if x ∈ Ω−ε and (Tε u)(x, y) := 0 otherwise. Accordingly, we define the folding operator(also averaging operator) Fε : L2 (Ω × Y) → L2 (Ω) viaZ(Fε U )(x) := −U z, { xε } dz
(2.2)
Cε (x)

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Ω−
ε
Ω

ε

Ω+
ε
ε

Figure 1: Covering of the domain Ω with microscopic cells.
RR−1
if (x, y) ∈ Ω−u(z) dz
ε × Y and (Fε U )(x) := 0 otherwise, where −A u(z) dz := vol(A)Adenotes the average. Both operators are linear and bounded, i.e.
k Tε ukL2 (Ω×Y) ≤ kukL2 (Ω)

andk Fε U kL2 (Ω) ≤ kU kL2 (Ω×Y)(2.3)

by Jensen’s inequality. We emphasize
R that the periodic
R unfolding operator Tε in (2.1)
does not satisfy the integral identity Ω×Y Tε u dx dy = Ω u dx and we refer to [17] for a
unfolding definition which does do so. However, we only use the duality Fε = Tε ∗ , cf. [3,Sect. 2.2], meaning for all u ∈ L2 (Ω) and V ∈ L2 (Ω × Y)ZZu (Fε V ) .

(Tε u) V dx dy =Ω×YΩFurthermore, the unfolding operator satisfies Tε u ∈ L2 (Ω; H1 (Y )) for all u ∈ H1 (Ω) and
it is Tε (ε∇u) = ∇y (Tε u), cf. [3, Eq. (3.1)]. We point out that the spaces L2 (Ω × Y) andL2 (Ω × Y ) can be identified, whereas L2 (Ω; H1 (Y)) is a closed subspace of L2 (Ω; H1 (Y ))with H1 (Y) = H1per (Y ) $ H1 (Y).
Using periodic unfolding via Tε , weak (respective strong) two-scale convergence ofbounded sequences (uε )ε ⊂ L2 (Ω), as introduced in [22], is equivalent to weak (respective strong) convergence of the unfolded sequence (Tε uε )ε in the two-scale space
L2 (Ω × Y). It was first proved in [1, Prop. 1.14] that sequences (uε )ε ⊂ H1 (Ω) satisfying
supε>0 {kuε kL2 (Ω) + kε∇uε kL2 (Ω) } < ∞ weakly two-scale converge (up to subsequence) toa function U ∈ L2 (Ω; H1 (Y)), i.e. Tε uε * U and Tε (ε∇uε ) * ∇y U in L2 (Ω × Y). We emphasize that the unfolded function Tε uε is in general not Y -periodic, while its two-scale
limit U indeed is, and we call this observation periodicity defect as in [8].

2.2Assumptions and statement of the main result for γ = 1

We study the following elliptic equation, which is degenerating as ε → 0,
− div(ε2 Aε ∇uε ) + Bε uε = Fε

in Ω

(2.4)

121
supplemented with homogeneous Neumann boundary conditions, i.e. (ε2 Aε ∇uε ) · ν = 0
on ∂Ω. Here, ν denotes the unit outer normal vector of Ω. The coefficients depend onx ∈ Ω and ε > 0 via


(2.5)
Aε (x) := A x, xε , Bε (x) := B x, xε , and Fε (x) := (Fε F)(x).
Throughout this text we impose the following assumptions on the given data. Let
A(x, y) ∈ Rd×d
sym be a symmetric matrix and let A and B be positive definite, i.e. there
exist constants 0 < α ≤ β < ∞ such that we have uniformly for all (x, y, ξ) ∈ Ω × Y × Rd
α|ξ|2 ≤ A(x, y)ξ · ξ ≤ β|ξ|2and α ≤ B(x, y) ≤ β.

(2.6)

For the derivation of quantitative estimates we need additional regularity with respect tothe macroscopic scale x ∈ Ω, namely let
A ∈ W1,∞ (Ω; L∞ (Y)),

B ∈ W1,∞ (Ω; L∞ (Y)),

and F ∈ H1 (Ω; L2 (Y)).

(2.7)

Thanks to the continuity of A and B with respect to x ∈ Ω, the definitions of Aε and Bεin (2.5) are indeed well-defined. Passing to the limit ε → 0 in (2.4) gives the two-scaleconvergence of uε to a limit function U ∈ L2 (Ω; H1 (Y)) solving the effective equation (cf.
[24, 12, 16])
− divy (A∇y U ) + BU = F

in Ω × Y(2.8)

with periodic boundary conditions on the torus Y and no boundary conditions on Ω. By
the Lax–Milgram Theorem and the regularity (2.7) of the given data, the solution U of
(2.8) even belongs to the better space H1 (Ω; H1 (Y)). This follows by considering difference
quotients in x ∈ Ω, cf. [27, Prop. 2.3.17]. All solutions are uniformly bounded in the sense
sup kuε kL2 (Ω) + kε∇uε kL2 (Ω) + kU kH1 (Ω;H1 (Y)) ≤ Const.(A, B, F).
(2.9)
ε>0

Our main result is in the case γ = 1 as follows.
Theorem 2.1. Let the given data satisfy the assumptions (2.5)–(2.7) and let uε and U
solve the elliptic equation (2.4) and (2.8), respectively. Then, there exists a constantC > 0 depending on the norm in (2.9) such thatk Tε uε − U kL2 (Ω;H1 (Y )) ≤ ε1/2 C.
The proof (see Subsection 3.3) combines ideas of [26, Thm. 3.2] and [8, Prop. 4.3], and
it is shown that Gε U is an approximate solution of the original equation (2.4). Therefore,
we proceed in three steps:
1. The choice of an admissible test function for the effective equation (2.8) leads to theperiodicity defect.
2. We reformulate by using the duality Fε = Tε ∗ and control the approximation errors.
3. Choosing a suitable test function for the original equation is equivalent to finding
the correct “recovery sequence” (in terms of Γ-convergence) which is Gε U here.

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3

Error estimates

3.1

Estimating the periodicity defect and approximation errors

In this subsection we provide preparatory estimates to control the approximation error
for the folding operator and the periodicity defect. To handle the error for microscopic
cells Cε (x) which are not fully contained in the domain Ω, we exploit for functions U ∈H1 (Ω; L2 (Y)) the boundary estimate
kU kL2 ((Ω\Ω−ε )×Y) ≤ ε1/2 CkU kH1 (Ω;L2 (Y)) ,(3.1)

where the constant C > 0 only depends on the Lipschitz property of Ω. For one-scale
functions, estimate (3.1) can be found in e.g. [8, Eq. (4.6)] or [9, Eq. (2.4)] and for a full
proof we refer to [27, Lem. 2.3.5]. The error between Tε Fε and idL2 (Ω×Y) is given via:
Lemma 3.1. For every U ∈ H1 (Ω; L2 (Y)) it holds
kU − Tε Fε U kL2 (Ω×Y) ≤ (ε + ε1/2 )CkU kH1 (Ω;L2 (Y)) ,where the constant C > 0 only depends on the domains Ω and Y .
d−Proof. Using the paving property Ω−
ε = ∪λi Cε (x), where λi = [x/ε] ∈ Z for x ∈ Ωε , andapplying the Poincaré–Wirtinger inequality yields

kU − Tε Fε U k2L2 (Ω−ε ×Y) =
≤

XZ
λi

X
λi

Z

ε(λi +Y )YZ
U (x, y) − −2U (z, y) dz dx dyCε (x)(diam(Cε (x))2 k∇x U k2L2 (ε(λi +Y ) ≤ ε2 Ck∇x U k2L2 (Ω×Y) .

Here, we used that the Poincaré–Wirtinger constant is bounded by the diameter of theconvex set Cε (x), namely, εdiam(Y ). Since kU −Tε Fε U kL2 (Ω×Y) = kU −Tε Fε U kL2 (Ω−ε ×Y) +kU kL2 ((Ω\Ω−ε )×Y) the boundary estimate (3.1) gives the desired result.
Introducing for ε > 0 and ϕ ∈ H1 (Ω) the normkϕkε := kϕkL2 (Ω) + kε∇ϕkL2 (Ω)(3.2)

we can control the periodicity defect as follows.
Theorem 3.2 ([9, Thm. 2.2]). For every ϕ ∈ H1 (Ω), there exists a Y-periodic functionΦε ∈ L2 (Ω; H1 (Y)) such thatkΦε kH1 (Y ;L2 (Ω)) ≤ Ckϕkε

andk Tε ϕ − Φε kH1 (Y ;H1 (Ω)∗ ) ≤ (ε + ε1/2 )Ckϕkε ,where the constant C > 0 only depends in the domains Ω and Y .

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3.2

Gradient folding operator and folding mismatch

By the Sobolev extension Theorem, there exists a linear and continuous operator,
cf. e.g. [20, Thm. 3.9], E : H1 (Ω; L2 (Y)) → H1 (Rd ; L2 (Y)) such that (EU )|Ω×Y = U andkEU kH1 (Rd ;L2 (Y)) ≤ CkU kH1 (Ω;L2 (Y)) . Moreover, let E be such that its restriction to onescale functions u ∈ H1 (Ω) satisfies (Eu)|Ω = u and kEukH1 (Rd ) ≤ CkukH1 (Ω) . Taking careof cells Cε (x) intersecting the boundary of Ω, we define the extended folding operator
Fε+ : H1 (Ω; L2 (Y)) → L2 (Rd ) viaZ+(Fε U )(x, y) := − (EU ) z, { xε } dz.
Cε (x)The gradient folding operator Gε : H1 (Ω; H1 (Y)) → H1 (Ω+ε ) is defined as in [16] following
11
1[17, 12]: for U ∈ H (Ω; H (Y)) given, Gε U := ubε ∈ H (Ω+ε ) is the unique solution of theelliptic problemZ(buε − Fε+ U )ϕ + (ε∇buε − Fε+ (∇y U )) · ε∇ϕ dx = 0 for all ϕ ∈ H1 (Ω+
(3.3)
ε ).
Ω+
ε

The Lax–Milgram Theorem implies that Gε U is well-defined and bounded in the sensek Gε U kε ≤ kEU kH1 (Rd ;H1 (Y)) . Throughout the following calculations the operator E isomitted in notation, however, all generic constants C will also depend on its norm.
Having defined two different folding operators Fε and Gε , we want to control theirdifference which we call folding mismatch. Therefore we introduce the so-called scalesplitting operator Qε : H1 (Ω) → W1,∞ (Rd ) following [3, Def. 4.1]: for x ∈ Cε (x) and everyκ = (κ1 , . . . , κd ) ∈ {0, 1}d , we set(
xl −ε[x/ε]l
if κl = 1(κ )εx̄l l :=
xl −ε[x/ε]l1−
if κl = 0εand(Qε w)(x) :=Xκ∈{0,1}d (κ )(κ )(Fε+ w) ε[ xε ] + εκ · x̄1 1 · · · x̄d d .

(3.4)

The function Qε w interpolates the values of Fε+ w at the nodes ε[x/ε] via Q1 -Lagrange
elements as customary in the finite elements methods. Indeed, it is Fε+ w ∈ L∞ (Ω+ε ) such2that for z ∈ L (Y) the productsx 7→ Fε+ w (x)z xε
and x 7→ Qε w (x)z xε(3.5)
belong to the space L2 (Ω+ε ), see e.g. [15, Thm.4]. According to [3, Prop. 4.5], there exists
a constant C > 0 only depending on Ω and Y such thatk Qε wkH1 (Ω) ≤ CkwkH1 (Ω)

for all w ∈ H1 (Ω).

(3.6)

The following auxiliary estimate is proved in [27, Lem. 3.6] or [26, Lem. 2.3.9] based on
ideas from [8, Prop. 3.2].

124
Lemma 3.3. For w ∈ H1 (Ω) and z ∈ L2 (Y), it holdsk(Fε+ w − Qε w)z( ε· )kL2 (Ω+ε ) ≤ εCkwkH1 (Ω) kzkL2 (Y) ,where the constant C > 0 only depends on the dimension d.
Finally, we have collected all ingredients to control the folding mismatch.
Theorem 3.4. For every U ∈ H1 (Ω; H1 (Y)), the folding mismatch isk Gε U − Fε U kL2 (Ω) + kε∇(Gε U ) − Fε (∇y U )kL2 (Ω) ≤ ε1/2 CkU kH1 (Ω;H1 (Y)) ,k Gε U − Fε+ U kL2 (Ω+ε ) + kε∇(Gε U ) − Fε+ (∇y U )kL2 (Ω+ε ) ≤ εCkU kH1 (Ω;H1 (Y)) ,

(3.7)
(3.8)

where the constant C > 0 only depends on the domains Ω and Y .
Proof. For brevity we set L2ε := L2 (Ω+ε ) troughout this proof.
Step 1: Treatment of the boundary. The triangle inequality, the identity (Fε U )|Ω−ε =(Fε+ U )|Ω−ε , Jensen’s inequality, and the boundary estimate (3.1) imply
k Gε U − Fε U kL2 (Ω) ≤ k Gε U − Fε+ U kL2 (Ω) + k Fε+ U kL2 (Ω\Ω−ε )≤ k Gε U − Fε+ U kL2ε + (ε + ε1/2 )CkU kH1 (Ω;L2 (Y)) .

An analog estimate holds for the gradient term so that it remains to prove (3.8) and (3.7)
follows immediately.
Step 2: Proof of estimate (3.8) for products U (x, y) = w(x)z(y). Let U satisfy thedecomposition U (x, y) = w(x)z(y) with w ∈ H1 (Ω) and z ∈ H1 (Y). We introduce thefunction ϑε (x) := (Qε w)(x)z(x/ε) which belongs to the space H1 (Ω+ε ) by construction.
The definition of Gε U in (3.3) is equivalent toZ(Gε U − ϑε )ϕ + ε∇(Gε U − ϑε ) · ε∇ϕ dxΩ+εZ=(Fε+ U − ϑε )ϕ + (Fε+ (∇y U ) − ε∇ϑε ) · ε∇ϕ dx for all ϕ ∈ H1 (Ω+
ε ).
Ω+
ε

Choosing ϕ = Gε U − ϑε as well as applying Hölder’s inequality, Lemma 3.3, and theboundedness of Qε in (3.6) gives
k Gε U − ϑε kε ≤ k Fε U − ϑε kL2ε + k Fε (∇y U ) − ε∇ϑε kL2ε= k(Fε+ w − Qε w)z( ε· )kL2ε + k(Fε+ w − Qε w)∇y z( ε· )kL2ε + kε∇x (Qε w)z( ε· )kL2ε≤ εCkwkH1 (Ω) kzkH1 (Y) .
Finally, the triangle inequality yields
k Gε U − Fε U kL2ε + kε∇(Gε U ) − Fε (∇y U )kL2ε≤ k Gε U − ϑε kε + k Fε U − ϑε kL2ε + k Fε (∇y U ) − ε∇ϑε kL2ε
≤ 2εCkwkH1 (Ω) kzkH1 (Y)

(3.9)

125
and the desired estimate (3.8) follows for functions U of product form.
Step 3: Proof of estimate (3.8) for general functions U (x, y). Let {Φi }∞i=1 be anorthonormal basis in H1 (Y) which is also orthogonal in L2 (Y). Then we can express1
U ∈ H1 (Ω+ε ; H (Y)) (extended by E) via the linear combination
U (x, y) =∞XZui (x)Φi (y) where ui (x) :=U (x, y)Φi (y) dy.

(3.10)

Y

i=1

By construction it is ui ∈ H1 (Ω+ε ) and we set Ui (x, y) := ui (x)Φi (y).
The assumptions on Φi imply Φi (·/ε) ⊥ Φj (·/ε) and ∇y Φi (·/ε) ⊥ ∇y Φj (·/ε) in L2ε for
all i 6= j, i.e. {Φi (·/ε)}i is an orthogonal system in H1 (Ω+ε ) for each ε > 0. (Here, u ⊥ v
if and only if (u, v)L2ε = 0.) Indeed, using the paving property Ω+
ε = ∪λi Cε (x), whered+λi = [x/ε] ∈ Z for x ∈ Ωε , a substitution of variables yields
ZXZxxΦi ( ε )Φj ( ε ) dx =Φi ( xε )Φj ( xε ) dx
Ω+
ε

λi

=

Cε (x)

X 1 Z
λi

εd

x
[ ]+Y
ε

Φi (y)Φj (y) dy = 0(3.11)

and analogously for ∇y Φi . Applying the folding operator Fε+ to Ui gives(Fε+ Ui )(x) = (Fε+ ui )(x)Φi ( xε ) and (Fε+ [∇y Ui ])(x) = (Fε+ ui )(x)∇y Φi ( xε ).
R
Since (Fε+ ui )(x) = −Cε (x) ui (z) dz is constant on each cell Cε (x), we have as well Fε+ Ui ⊥Fε+ Uj and Fε+ (∇y Ui ) ⊥ Fε+ (∇y Uj ) in L2ε for all i 6= j. Therefore, it suffices to consider
the basis functions Φi . By the definition of Gε we have for viε := Gε ΦiZviε − Φi ( ε· ) ϕ + ε∇viε − ∇y Φi ( ε· ) · ε∇ϕ dx = 0 for all ϕ ∈ H1 (Ω+(3.12)ε ).
Ω+
εInserting the test function ϕ(x) = Φj (x/ε) in (3.12) yields with (3.11) for all i 6= jZviε Φj ( ε· ) + ε∇viε · ∇y Φj ( ε· ) dx = 0.
Ω+εFurthermore, choosing ϕ(x) = vjε (x) in (3.12) yields with the latter equalityZviε vjε + ε∇viε · ε∇vjε dx = 0.
Ω+εHence, it is viε ⊥ vjε in H1 (Ω+ε ) for all i 6= j. We continue to estimate the folding mismatch+ε
of Fε and Gε and set ui := Gε Ui . Replacing Fε+ Φi and Fε+ (∇y Φi ) with Fε+ Ui andFε+ (∇y Ui ) in (3.12), respectively, and exploiting that Fε+ ui is constant on each cell Cε (x)implies uεi ⊥ uεj in H1 (Ω+ε ) for all i 6= j.
Finally, we apply the result of Step 2 with w = ui and z =PΦi . Since Fε+ and Gεεare linear and continuous operators, we have in particular uε = ∞i=1 ui . Therefore, we126square estimate (3.9) so that the mixed product terms vanish for i 6= j, namely,kuε − Fε+ U k2L2ε + kε∇uε − Fε+ (∇y U )k2L2ε
P
P∞
ε
+
2
ε
+
2
=k ∞
i=1 (ui − Fε Ui )kL2ε + k
i=1 (ε∇ui − Fε (∇y Ui ))kL2εP∞Pε2= i=1 kuεi − Fε+ Ui k2L2ε + ∞i=1 kε∇ui − Fε (∇y Ui )kH
P∞ 2≤ i=1 ε Ckui k2H1 (Ω) kzk2H1 (Y) = ε2 CkU k2H1 (Ω;H1 (Y)) .
P
2The last equality follows by Parseval’s identity kU k2H1 (Ω+ ;H1 (Y)) = ∞i=1 kui kH1 (Ω+ ) .
ε

3.3

ε

Proof of Theorem 2.1

Proof of Theorem 2.1. Step 1: Periodicity defect. The weak formulation of the effective
equation (2.8) readsZA∇y U · ∇y Φ + BU Φ − FΦ dx dy = 0 for all Φ ∈ L2 (Ω; H1 (Y)).
(3.13)Ω×Y

Thanks to Theorem 3.2 we can choose Φε ∈ L2 (Ω; H1 (Y)) to estimate the periodicity
defect of Tε ϕ ∈ L2 (Ω; H1 (Y )). Indeed, exploiting the higher x-regularity (2.7) of thegiven data and the limit U ∈ H1 (Ω; H1 (Y)) yields
ZA∇y U · [∇y (Tε ϕ) − ∇y Φε ] + BU − F [Tε ϕ − Φε ] dx dyΩ×Y≤ kA∇y U kL2 (Y;H1 (Ω)) k∇y (Tε ϕ) − ∇y Φε kL2 (Y;H1 (Ω)∗ )
+ k|BU | + |F|kL2 (Y;H1 (Ω)) k Tε ϕ − Φε kL2 (Y;H1 (Ω)∗ )≤ Const.(A, B, F, U )(ε + ε1/2 )Ckϕkε .

∗Note that the identification of the spaces L2 (Y; H1 (Ω)) and L2 (Y; H1 (Ω)∗ ) as well as
L2 (Y; H1 (Ω)) and H1 (Ω; L2 (Y)) holds due to the underlying tensor product structure of
the two-scale space L2 (Y; H1 (Ω)). Overall testing (3.13) with Φε gives
ZA∇y U · ∇y (Tε ϕ) + BU Tε ϕ − F Tε ϕ dx dy ≤ ε1/2 Ckϕkε .
Ω×YStep 2: Approximation errors. Using Fε = Tε ∗ and ∇y (Tε ϕ) = Tε (ε∇ϕ) givesZ
Fε (A∇y U ) · ε∇ϕ + Fε (BU )ϕ − Fε ϕ dx ≤ ε1/2 Ckϕkε .
(3.14)Ω

Thanks to the Lipschitz continuity of A and B with respect to x ∈ Ω and the boundedness(2.3) of Fε , we obtainkFε (A∇y U ) − Aε Fε (∇y U )k2L2 (Ω)
Z Z
2



=
−
A z, xε − A x, xε ∇y U z, xε dz dxΩ−ε2Cε (x)≤ ε k∇x AkL∞ (Ω×Y) k∇y U kL2 (Ω×Y)

127
and analogously k Fε (BU ) − Bε Fε (U )k2L2 (Ω) ≤ ε2 k∇x BkL∞ (Ω×Y) kU kL2 (Ω×Y) . Insertingthese two estimates into (3.14) yields
Z
Aε Fε (∇y U ) · ε∇ϕ + Bε Fε U ϕ − Fε ϕ dx ≤ ε1/2 Ckϕkε .
ΩStep 3: Folding mismatch. Thanks to Theorem 3.4 we can replace the folded functions
Fε U and Fε (∇y U ) with the gradient folding Gε U and ε∇(Gε U ), respectively, such thatZ
Aε ε∇(Gε U ) · ε∇ϕ + Bε (Gε U )ϕ − Fε ϕ dx ≤ ε1/2 Ckϕkε .
(3.15)Ω

The choice ϕε = uε − Gε U ∈ H1 (Ω) is indeed admissible in (3.15) and ϕε is also a test
function for the original equation (2.4), i.e.
ZAε ε∇uε · ε∇ϕε + Bε uε ϕε − Fε ϕε dx = 0.
(3.16)Ω

Subtracting (3.15) from (3.16) and exploiting the positive definiteness (2.6) gives
αkuε − Gε U k2ε ≤ ε1/2 Ckuε − Gε U kε .

(3.17)

Finally, applying the triangle inequality, the boundedness (2.3) of Tε , estimate (3.7) for
the folding mismatch, and the approximation error yields
k Tε uε − U kL2 (Ω;H1 (Y ))≤ kuε − Gε U kε + k Gε U − Fε U kL2 (Ω) + kε∇(Gε U ) − Fε (∇y U )kL2 (Ω)+ k Tε Fε U − U kL2 (Ω×Y) + k Tε Fε (∇y U ) − ∇y U kL2 (Ω×Y) ≤ ε1/2 C

which finishes the proof.

4

Discussion of further error estimates

4.1

The non-degenerating case for γ = 0

We consider the non-degenerating elliptic equation− div(Aε ∇uε ) + Bε u = Fε

in Ω

(4.1)

with homogeneous Neumann boundary conditions and given data as in (2.5)–(2.7). It is
well-known that uε * u weakly in H1 (Ω) and u solves the effective equation− div(Aeff ∇u) + Beff u = Feff

in Ω

(4.2)

with the same boundary conditions. The effective matrix Aeff is given via the standard
unit-cell problem (for arbitrary vectors ξ ∈ Rd )Z
ξ · Aeff (x)ξ = min(ξ + ∇y Φ) · A(x, y)(ξ + ∇y Φ) dy,(4.3)
1Φ∈Hav (Y)

Y

128
R
where H1av (Y) = {Φ ∈ H1 (Y) | Y Φ dy = 0}. The effective data Beff and Feff are the usual
averages
Z
Z
B(x, y) dy and Feff (x) =F(x, y) dy.
(4.4)
Beff (x) =
Y

Y

d×d
The regularity of A in (2.7) implies Aeff ∈ W1,∞ (Ω; Rsym). If the domain Ω is additionallyconvex or of class C 2 , we obtain higher regularity of u, namely u ∈ H2 (Ω), see e.g. [11, 14].
This higher regularity of the limit solution u is needed to derive quantitative estimatesand it is not nearly as trivial as in the degenerating case γ = 1. The corrector U is theunique minimizer in (4.3) corresponding to ξ = ∇u(x) and it satisfies U ∈ H1 (Ω; H1 (Y))
thanks to the higher regularity of A and u with respect to x ∈ Ω. Hence, we can state
our main result for γ = 0.

Theorem 4.1. Let the assumptions (2.5)–(2.7) hold and let uε and u solves the elliptic
equation (4.1) and (4.2), respectively. If u belongs to H2 (Ω), then there exists a constantC > 0 such thatkuε − ukL2 (Ω) + k Tε (∇uε ) − [∇u + ∇y U ]kL2 (Ω×Y) ≤ ε1/2 C.
Proof. The proof is analog to the one of Theorem 2.1 for γ = 1 and one shows thatu + ε Gε U is an approximate solution of the original equation (4.1). To control the
periodicity defect in Step 1 we apply [9, Thm. 2.3], i.e. for every ϕ ∈ H1 (Ω), there exists afunction Φε ∈ L2 (Ω; H1 (Y)) and a constant C > 0 such that kΦε kH1 (Y ;L2 (Ω)) ≤ CkϕkH1 (Ω)
and k Tε (∇ϕ) − [∇ϕ + ∇y Φε ]kL2 (Y;H1 (Ω)∗ ) ≤ (ε + ε1/2 )CkϕkH1 (Ω) .
Remark 4.2 (Boundary conditions). For different boundary conditions the error estimates are functioning as before, however, one has to modify the approximating sequence
so that the new boundary conditions are satisfied. In the case of homogeneous Dirichletboundary conditions, a suitable candidate is u + ε Gε (ρε U ), where ρε denotes a cut-off
function to guarantee zero conditions at the boundary ∂Ω.

4.2Comparison with [8]The corrector U admits the decomposition (see e.g. [15, Sec. 4])
U (x, y) =

d
X
∂u
i=1

∂xi

(x)zi (x, y),(4.5)

where zi ∈ W1,∞ (Ω; H1av (Y)) is the unique minimizer corresponding to the standard basis
vector ei in Rd . If Aε is exactly periodic as in [8], i.e. Aε (x) = A(x/ε), then zi ∈ H1av (Y)is it as well. In this case we have the following error estimates for solutions uε and u of(4.1) and (4.2), respectively, with B ≡ 0 and either homogeneous Neumann or Dirichletboundary conditions.
Theorem 4.3 ([8, Prop. 4.3]). If u ∈ H2 (Ω), then there exists C > 0 such that ·P∂ukuε − ukL2 (Ω) + ∇uε − [∇u + di=1 Qε ∂xzi ( ε )] L2 (Ω) ≤ ε1/2 C.
i

129
In any case – exactly periodic or not – the question is how to “fold” U (x, y)? We remarkthat x 7→ U (x, x/ε) ∈ L2 (Ω) is in general not satisfied for U ∈ L∞ (Ω; L2 (Ω)), whereas for∞2
products x 7→ w(x)z(x/ε) ∈ L2 (Ω) holds
toPdtrue for∂uw∈ L (Ω) and z ∈ L (Y) according
1[15, Thm.4]. Hence, the construction i=1 Qε ∂xi (x)zi (x/ε) is admissible in H (Ω) for
exactly periodic coefficients. However, in the not-exactly periodic situation, the choice∂u(x)zi (x, x/ε) is not admissible. This technicality is circumvented in our approachQε ∂xiby using the gradient folding operator and choosing the recovery sequence u + ε Gε U
respective u + ε Gε (ρε U ). Nevertheless, for more regular data such as A ∈ C1 (Ω; L∞ (Y))
and hence zi ∈ C 1 (Ω; H1av (Y)), the naive folding zi (x, x/ε) is again well-defined in H1 (Ω)
and the proof of [8, Prop. 4.3] seems to be valid, too.
The approach of [8] applies as well to systems of reaction-diffusion equations, cf. [6].
And our approach is also generalizable to semilinear parabolic equations with globally
Lipschitz continuous right-hand sides, see Subsection 4.3.

4.3

Error estimates for parabolic equations

Our approach can be generalized to semilinear parabolic equations. Following e.g.
[4, 6, 19, 27] one aims to apply Gronwall’s Lemma and therefore estimates
1 dkuε − Fε U k2L2 (Ω) =
2 dt

Z
ZΩ

=
Ω

(u̇ε − Fε U̇ )(uε − Fε U ) dxZ(u̇ε − Fε U̇ )ϕε dx + (u̇ε − Fε U̇ )(Gε U − Fε U ) dx.
Term 1Ω

Term 2Choosing the test function ϕε = uε − Gε U and inserting the reformulated equations,
Term 1 can be treated as in the elliptic case. To control Term 2, in particular to obtainthe convergence rate ε1/2 , we need the uniform boundedness supε>0 ku̇ε kL2 (Ω) < ∞ andestimate (3.8) of order O(ε) for the folding mismatch, i.e. |Term 2| ≤ εC. This can beachieved by assuming either Ω is polyhedral or by extending all solutions, given data, and+ ∗+operators from Ω to Ω+ε such that the duality Fε = (Tε ) is applicable.

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