A Note on Iterations-based Derivations of High-order Homogenization Correctors for Multiscale Semi-linear Elliptic Equations

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by  Khoa Vo

A Note on Iterations-based Derivations of High-order Homogenization Correctors for
Multiscale Semi-linear Elliptic Equations
Vo Anh Khoaa,∗, Adrian Munteanb
aMathematics and Computer Science Division, Gran Sasso Science Institute, L’Aquila, Italy
bDepartment of Mathematics and Computer Science, Karlstad University, Sweden
Abstract
ThisNoteaimsatpresentingasimpleandefficientproceduretoderivethestructureofhigh-ordercorrectorestimatesfor
6 the homogenization limit applied to a semi-linear elliptic equation posed in perforated domains. Our working technique
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relies on monotone iterations combined with formal two-scale homogenization asymptotics. It can be adapted to handle
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morecomplexscenariosincludingforinstancenonlinearitiesposedattheboundaryofperforationsandthevectorialcase,

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when the model equations are coupled only through the nonlinear production terms.
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Keywords: Corrector estimates, Homogenization, Elliptic systems, Perforated domains
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2010 MSC: 35B27, 35C20, 76M30, 35B09

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1. Background
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a Modern approaches to modeling focus on multiple scales. Given a multiscale physical problem, one of the leading
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questions is to derive upscaled model equations and the corresponding structure of effective model coefficients (e.g.
[

1 [1,2]). ThisNoteaimsatexploringthequalityoftheupscaling/homogenizationprocedurebyderivingwheneverpossible
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corrector (error) estimates for the involved unknown functions, fields, etc. and their gradients (i.e. of the transport
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5 fluxes). Ultimately, these estimates contribute essentially to the control of the approximation error of numerical methods
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1 to multiscale PDE problems.
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. Our starting point is a microscopic PDE model describing the motion of populations of colloidal particles in soils
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0 and porous tissues with direct applications in drug-delivery design and control of the spread of radioactive pollutants
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1 ([3, 4, 5]). We have previously analyzed a reduced version of this system in [6]. Here, we point out a short alternative
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v proof based on monotone iterations ([7]) of the corrector estimates derived in [6] and extend them to higher asymptotic
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orders.
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2. Problem setting
We are concerned with the study of the semi-linear elliptic boundary value problem of the form

Aεuε =R(uε), x∈Ωε,
uε =0, x∈Γext, (2.1)
∇uε·n=0, x∈Γε,
∗Correspondingauthor
Email addresses: [email protected], [email protected](VoAnhKhoa),[email protected](AdrianMuntean)
January 11, 2016
where the operator Aεuε := ∇·(−dε∇uε) involves dε termed as the molecular diffusion while R represents the volume
reaction rate. We take into account the following assumptions:
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(A ) the diffusion coefficient dε ∈L∞(cid:0)Rd(cid:1) for d∈N is Y-periodic and symmetric, and it guarantees the ellipticity of
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Aε as follows:
dεξ ξ ≥α|ξ|2 for anyξ ∈Rd;
i j
(A ) the reaction coefficient R ∈ L∞(Ωε×R) is globally L−Lipschitzian, i.e. there exists L > 0 independent of ε
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such that
(cid:107)R(u)−R(v)(cid:107)≤L(cid:107)u−v(cid:107) foru,v ∈R.
It is worth noting that the domain Ωε ∈Rd considered here approximates a porous medium. The precise description
of Ωε is showed in [6, Section 2] and [8]. In Figure 2.1 (left), we sketch an admissible geometry of our medium, pointing
out the sample microstructure in Figure 2.1 (right). We follow the notation from [6].
Γ = ∂Ω
Y
0
Y Y
1
Ω
Ωε
Figure2.1: Anadmissibleperforateddomain(left)andbasicgeometryofthemicrostructure(right).
Remark 1. In this paper, we denote the space Vε by
Vε :=(cid:8)v ∈H1(Ωε)|v =0onΓext(cid:9)
endowed with the norm
(cid:18)(cid:90) (cid:19)1/2
(cid:107)v(cid:107) = |∇v|2dx .
Vε
Ωε
This norm is equivalent (uniformly in the homogenization parameter ε) to the usual H1 norm by the Poincar´e inequality.
3. Derivation of corrector estimates
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We introduce the following Mth-order expansion (M ≥2):
M
uε(x)= (cid:88) εmu (x,y)+O(cid:0)εM+1(cid:1), x∈Ωε, (3.1)
m
m=0
2
where u (x,·) is Y-periodic for 0≤m≤M.
m
Followingstandardhomogenizationprocedures,wededucethesocalledauxiliaryproblems(seee.g. [9]). Todoso,we
considerthefunctionalΦ(x,y)dependingontwovariables: themacroscopicxandy =x/εthemicroscopicpresentation,
and denote by Φε(x)=Φ(x,y). The simple chain rule allows us to derive the fact that
(cid:16) x(cid:17) (cid:16) x(cid:17)
∇Φε(x)=∇ Φ x, +ε−1∇ Φ x, . (3.2)
x ε y ε
The quantities ∇uε and Aεuε must be expended correspondingly. In fact, it follows from (3.2) and (3.1) that
(cid:32) M (cid:33)
∇uε = (cid:0)∇ +ε−1∇ (cid:1) (cid:88) εmu +O(cid:0)εM+1(cid:1)
x y m
m=0
M−1
= ε−1∇ u + (cid:88) εm(∇ u +∇ u )+O(cid:0)εM(cid:1). (3.3)
y 0 x m y m+1
m=0
Using the structure of the operator Aε, we obtain the following:
Aεuε = ε−2∇ ·(−d(y)∇ u )
y y 0
+ε−1[∇ ·(−d(y)∇ u )+∇ ·(−d(y)(∇ u +∇ u ))]
x y 0 y x 0 y 1
M−2
(cid:88)
+ εm[∇ ·(−d(y)(∇ u +∇ u ))
x x m y m+1
m=0
+∇ ·(−d(y)(∇ u +∇ u ))]+O(cid:0)εM−1(cid:1). (3.4)
y x m+1 y m+2
Concerning the boundary condition at Γε, we note:
(cid:32) M−1 (cid:33)
(cid:88)
dε∇uε·n=d (y) ε−1∇ u + εm(∇ u +∇ u ) ·n. (3.5)
i y 0 x m y m+1
m=0
To investigate the convergence analysis, we consider the following structural property:
(cid:32) M (cid:33) M
R (cid:88) εmu = (cid:88) εm−rR(u )+O(cid:0)εM−r+1(cid:1) forr ∈Z,r ≤2. (3.6)
m m
m=0 m=0
Atthispointwesee,ifr ∈{1,2}itthengeneratespontaneouslynonlinearauxiliaryproblems. Toseetheimpediment,
let us focus on r =2. By collecting the coefficients of the same powers of ε in (3.4) and (3.5), we are led to the following
systems, which we refer to the auxiliary problems:

A0u0 =R(u0), inY1,
−d(y)∇ u ·n=0, on∂Y , (3.7)
y 0 0
u isY −periodic iny,
0

A0u1 =R(u1)−A1u0, inY1,
−d(y)(∇ u +∇ u )·n=0, on∂Y , (3.8)
x 0 y 1 0
u isY −periodic iny,
1

A0um+2 =R(um+2)−A1um+1−A2um, inY1,
−d(y)(∇ u +∇ u )·n=0, on∂Y , (3.9)
x m+1 y m+2 0
u isY −periodic iny,
m+2
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for 0≤m≤M −2.
Here, we have denoted by
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A := ∇ ·(−d(y)∇ ),
0 y y
A := ∇ ·(−d(y)∇ )+∇ ·(−d(y)∇ ),
1 x y y x
A := ∇ ·(−d(y)∇ ).
2 x x
Remark 2. In the case r ≤0, it is trivial to not only prove the well-posedness of these auxiliary problems (3.7)-(3.9), but
also to compute the solutions by many approaches due to its linearity. For details, the reader is referred here to [10].
The idea is now to linearize the auxiliary problems. Inspired by the fact that a fixed-point homogenization argument
seems to be applicable in this framework, and also by the way a priori error estimates are proven for difference schemes,
we suggest an iteration technique to ”linearize” the involved PDE systems. We start the procedure by choosing the
initial point u(0) =0 for m∈{0,...,M}. As next step, we consider the following systems corresponding to the nonlinear
m
auxiliary problems:
 (cid:16) (cid:17)
A0u(0n0) =R u(0n0−1) , inY1,
−d(y)∇ u(n0)·n=0, on∂Y , (3.10)
y 0 0
u(n0) isY −periodic iny,
0
 (cid:16) (cid:17)
A0u(1n1(cid:16)) =R u1(n1−1) −A(cid:17)1u(0n0), inY1,
−d(y) ∇ u(n0)+∇ u(n1) ·n=0, on∂Y , (3.11)
x 0 y 1 0
u(n1) isY −periodic iny,
1
 (cid:16) (cid:17)
A0u(mn+m(cid:16)2+2) =R u(mn+m2+2−1) −A(cid:17)1um(n+m1+1)−A2u(mnm), inY1,
−d(y) ∇ u(nm+1)+∇ u(nm+2) ·n=0, on∂Y , (3.12)
x m+1 y m+2 0
u(nm+2) isY −periodic iny,
m+2
for 0≤m≤M −2. Note that the quantity n is independent of ε.
m
Since the approximate auxiliary problems became linear, standard procedures are able to find the solutions u(nm) for
m
0≤m≤M . Note that these problems admit a unique solution (see, e.g. [10, Lemma 2.2]) on V, i.e. the quotient space
of V defined by
Y1
V :=(cid:8)ϕ|ϕ∈H1(Y ),ϕisY −periodic(cid:9).
Y1 1
On the other side, if κ := C Lα−1 < 1 holds (here C is the Poincar´e constant depending only on the dimension of
p p p
(cid:110) (cid:111)
Y ),weeasilyobtainthatforeverym, u(nm) isaCauchysequenceinH1(Y ). Hereby,itnaturallyclaimstheexistence
1 m 1
and uniqueness of the nonlinear auxiliary problems (3.7)-(3.9). Moreover, the convergence rate of the iteration procedure
is given by
(cid:13) (cid:13) κnm (cid:13) (cid:13)
(cid:13)u(nm)−u (cid:13) ≤ p (cid:13)u(1)(cid:13) .
(cid:13) m m(cid:13)H1(Y1) 1−κnpm (cid:13) m (cid:13)H1(Y1)
Remark 3. For more details in this sense, see [6, Theorem 9] and [11, Theorem 2.2].
To prove the corrector estimate, we suppose that the solutions of the auxiliary problems (3.7)-(3.9) belong to the
space L∞(Ωε;V). Let us introduce the following function:
M
(cid:88)
ϕε :=uε− εmu .
m
m=0
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Relying on the auxiliary problems (3.7)-(3.9), note that the function ϕε satisfies the following system:

Aεϕε =R(uε)−(cid:80)Mm=−02εm−2R(um)
−εM−1(A u +A u )−εMA u , inΩε, (3.13)
1 M 2 M−1 2 M
−dε∇ ϕε·n=εMdε∇ u ·n, onΓε.
x x M
Now, multiplying the PDE in (3.13) by ϕ∈Vε and integrating by parts, we arrive at
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(cid:42) M−2 (cid:43)
(cid:88)
(cid:104)dεϕε,ϕ(cid:105) = R(uε)− εm−2R(u ),ϕ
Vε m
m=0 L2(Ωε)
−εM−1(cid:104)A u +A u +εA u ,ϕ(cid:105)
1 M 2 M−1 2 M L2(Ωε)
(cid:90)
−εM dε∇ u ·nϕdS . (3.14)
x M ε
Ωε
int
From here on, we have to estimate the integrals on the right-hand side of (3.14)., which is a standard procedure; see
[10, 6] for similar calculations. Thus, we claim that
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42)R(uε)−Mm(cid:88)=−02εm−2R(um),ϕ(cid:43)L2(Ωε)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)≤CL(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)uε−m(cid:88)M=0εmum+O(cid:0)εM−1(cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Vε(cid:107)ϕ(cid:107)L2(Ωε), (3.15)
wherewehaveessentiallyusedtheglobalLipschitzconditiononthereactionterm,theassumption(3.6),andthePoincar´e
inequality. Next, we get
(cid:12) (cid:12)
εM−1(cid:12)(cid:104)A u +A u +εA u ,ϕ(cid:105) (cid:12)≤CεM−1(cid:107)ϕ(cid:107) , (3.16)
(cid:12) 1 M 2 M−1 2 M L2(Ωε)(cid:12) L2(Ωε)
while using the trace inequality (cf. [10, Lemma 2.31]) to deal with the the last integral, it gives
(cid:12) (cid:12)
(cid:12)(cid:90) (cid:12)
εM(cid:12) dε∇ u ·nϕdS (cid:12)≤CεM−1(cid:107)ϕ(cid:107) . (3.17)
(cid:12) x M ε(cid:12) L2(Ωε)
(cid:12) Ωε (cid:12)
int
Combining (3.15)-(3.17), we provide that
α|(cid:104)ϕε,ϕ(cid:105) |≤CεM−1(cid:107)ϕ(cid:107) ,
Vε L2(Ωε)
which finally leads to (cid:107)ϕε(cid:107)Vε ≤CεM2−1 by choosing ϕ=ϕε, very much in the spirit of energy estimates.
Summarizing, we state our results in the frame of the following theorems.
Theorem 4. Suppose (3.6) holds for r ∈ {1,2} and assume κ := C Lα−1 < 1 for the given Poincar´e constant. Let
p p
(cid:110) (cid:111)
u(nm) be the schemes that approximate the nonlinear auxiliary problems (3.10)-(3.12). Then (3.10)-(3.12) admit
m
nm∈N
a unique solution u for all m∈{0,...,M} with the speed of convergence:
m
(cid:13) (cid:13) Cκn
(cid:13)u(nm)−u (cid:13) ≤ p for alln ∈Nandm∈{0,...,M},
(cid:13) m m(cid:13)H1(Y1) 1−κnp m
where C >0 is a generic ε-independent constant and n:=max{n ,...,n }.
0 M
Theorem 5. Let uε be the solution of the elliptic system (2.1) with the assumptions (A )−(A ) stated above and suppose
1 2
that (3.6) holds for r ∈ {1,2}. For m ∈ {0,...,M} with M ≥ 2, we consider u the solutions of the auxiliary problems
m
(3.10)-(3.12). Then we obtain the following corrector estimate:
(cid:13) (cid:13)
(cid:13)(cid:13)(cid:13)uε− (cid:88)M εmum(cid:13)(cid:13)(cid:13) ≤CεM2−1,
(cid:13) (cid:13)
m=0 Vε
where C >0 is a generic ε-independent constant.
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Acknowledgment
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AMthanksNWOMPE”Theoreticalestimatesofheatlossesingeothermalwells”(grantnr. 657.014.004)forfunding.
VAK would like to thank Prof. Nguyen Thanh Long, his old supervisor, for the whole-hearted guidance when studying
at Department of Mathematics and Computer Science, Ho Chi Minh City University of Science in Vietnam.
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