A note on iterations-based derivations of high-order homogenization correctors for multiscale semi-linear elliptic equations

Accepted Manuscript A note on iterations-based derivations of high-order homogenization correctors for multiscale semi-linear elliptic equations Vo Anh Khoa, Adrian Muntean PII: DOI: Reference:

S0893-9659(16)30053-2 http://dx.doi.org/10.1016/j.aml.2016.02.009 AML 4950

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Applied Mathematics Letters

Received date: 8 January 2016 Revised date: 15 February 2016 Accepted date: 16 February 2016 Please cite this article as: V.A. Khoa, A. Muntean, A note on iterations-based derivations of high-order homogenization correctors for multiscale semi-linear elliptic equations, Appl. Math. Lett. (2016), http://dx.doi.org/10.1016/j.aml.2016.02.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A Note on Iterations-based Derivations of High-order Homogenization Correctors for Multiscale Semi-linear Elliptic Equations Vo Anh Khoaa,∗, Adrian Munteanb a Mathematics

and Computer Science Division, Gran Sasso Science Institute, L’Aquila, Italy of Mathematics and Computer Science, Karlstad University, Sweden

b Department

Abstract This Note aims at presenting a simple and efficient procedure to derive the structure of high-order corrector estimates for the homogenization limit applied to a semi-linear elliptic equation posed in perforated domains. Our working technique relies on monotone iterations combined with formal two-scale homogenization asymptotics. It can be adapted to handle more complex scenarios including for instance nonlinearities posed at the boundary of perforations and the vectorial case, when the model equations are coupled only through the nonlinear production terms. Keywords: Corrector estimates, Homogenization, Elliptic systems, Perforated domains 2010 MSC: 35B27, 35C20, 76M30, 35B09

1. Background Modern approaches to modeling focus on multiple scales. Given a multiscale physical problem, one of the leading questions is to derive upscaled model equations and the corresponding structure of effective model coefficients (e.g. [1, 2]). This Note aims at exploring the quality of the upscaling/homogenization procedure by deriving whenever possible 5

corrector (error) estimates for the involved unknown functions, fields, etc. and their gradients (i.e. of the transport fluxes). Ultimately, these estimates contribute essentially to the control of the approximation error of numerical methods to multiscale PDE problems. Our starting point is a microscopic PDE model describing the motion of populations of colloidal particles in soils and porous tissues with direct applications in drug-delivery design and control of the spread of radioactive pollutants

10

([3, 4, 5]). We have previously analyzed a reduced version of this system in [6]. Here, we point out a short alternative proof based on monotone iterations ([7]) of the corrector estimates derived in [6] and extend them to higher asymptotic orders. 2. Problem setting We are concerned with the study of the semi-linear elliptic boundary value problem of the form    Aε uε = R (uε ) ,    uε = 0,     ε  ∇u · n = 0,

x ∈ Ωε , (2.1)

x ∈ Γext , x ∈ Γε ,

∗ Corresponding

author Email addresses: [email protected], [email protected] (Vo Anh Khoa), [email protected] (Adrian Muntean)

February 15, 2016

where the operator Aε uε := ∇ · (−dε ∇uε ) involves dε termed as the molecular diffusion while R represents the volume 15

reaction rate. We take into account the following assumptions:
(A1 ) the diffusion coefficient dε ∈ L∞ Rd for d ∈ N is Y -periodic and symmetric, and it guarantees the ellipticity of

Aε as follows:

2

for any ξ ∈ Rd ;

dε ξi ξj ≥ α |ξ|

(A2 ) the reaction coefficient R ∈ L∞ (Ωε × R) is globally L−Lipschitzian, i.e. there exists L > 0 independent of ε

such that

kR (u) − R (v)k ≤ L ku − vk

for u, 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 Y0

Y1

Y

Ω

Ωε Figure 2.1: An admissible perforated domain (left) and basic geometry of the microstructure (right).

Remark 1. In this paper, we denote the space V ε by  V ε := v ∈ H 1 (Ωε )|v = 0 on Γext

endowed with the norm

kvkV ε =

Z

2

Ωε

|∇v| dx

1/2

.

This norm is equivalent (uniformly in the homogenization parameter ε) to the usual H 1 norm by the Poincar´e inequality.

20

3. Derivation of corrector estimates We introduce the following M th-order expansion (M ≥ 2): uε (x) =

M X

m=0


εm um (x, y) + O εM +1 , 2

x ∈ Ωε ,

(3.1)

where um (x, ·) is Y -periodic for 0 ≤ m ≤ M .

Following standard homogenization procedures, we deduce the so called auxiliary problems (see e.g. [9]). To do so, we

consider the functional Φ (x, y) depending on two variables: the macroscopic x and y = x/ε the microscopic presentation, and denote by Φε (x) = Φ (x, y). The simple chain rule allows us to derive the fact that  x  x ∇Φε (x) = ∇x Φ x, + ε−1 ∇y Φ x, . ε ε

(3.2)

The quantities ∇uε and Aε uε must be expended correspondingly. In fact, it follows from (3.2) and (3.1) that ! M
X
ε −1 m M +1 ∇u = ∇x + ε ∇y ε um + O ε m=0

= ε−1 ∇y u0 +

M −1 X m=0


εm (∇x um + ∇y um+1 ) + O εM .

(3.3)

Using the structure of the operator Aε , we obtain the following: Aε uε

= ε−2 ∇y · (−d (y) ∇y u0 ) +ε−1 [∇x · (−d (y) ∇y u0 ) + ∇y · (−d (y) (∇x u0 + ∇y u1 ))] +

M −2 X m=0

εm [∇x · (−d (y) (∇x um + ∇y um+1 ))


+∇y · (−d (y) (∇x um+1 + ∇y um+2 ))] + O εM −1 .

(3.4)

Concerning the boundary condition at Γε , we note: d ∇u · n = di (y) ε ε

ε

−1

∇y u0 +

M −1 X m=0

ε (∇x um + ∇y um+1 ) m

!

· n.

To investigate the convergence analysis, we consider the following structural property: ! M M X X
m R ε um = εm−r R (um ) + O εM −r+1 for r ∈ Z, r ≤ 2. m=0

(3.5)

(3.6)

m=0

At this point we see, if r ∈ {1, 2} solving nonlinear auxiliary problems is then needed. To see the impediment, 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:    A0 u0 = R (u0 ) , in Y1 ,    −d (y) ∇y u0 · n = 0, on ∂Y0 ,     u is Y − periodic in y, 0

   A0 u1 = R (u1 ) − A1 u0 , in Y1 ,    −d (y) (∇x u0 + ∇y u1 ) · n = 0, on ∂Y0 ,     u is Y − periodic in y, 1

   A0 um+2 = R (um+2 ) − A1 um+1 − A2 um , in Y1 ,    −d (y) (∇x um+1 + ∇y um+2 ) · n = 0, on ∂Y0 ,     u m+2 is Y − periodic in y, 3

(3.7)

(3.8)

(3.9)

for 0 ≤ m ≤ M − 2. 25

Here, we have denoted by A0

:= ∇y · (−d (y) ∇y ) ,

A1

:= ∇x · (−d (y) ∇y ) + ∇y · (−d (y) ∇x ) ,

A2

:= ∇x · (−d (y) ∇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 (0)

initial point um = 0 for m ∈ {0, ..., M }. As next step, we consider the following systems corresponding to the nonlinear

auxiliary problems:

   0 −1)  A0 u0(n0 ) = R u(n , in Y1 ,  0   (n ) −d (y) ∇y u0 0 · n = 0, on ∂Y0 ,     (n0 )  u0 is Y − periodic in y,

   (n ) (n −1) (n )   A0 u1 1 = R u1 1 − A1 u0 0 , in Y1 ,      (n ) (n ) −d (y) ∇x u0 0 + ∇y u1 1 · n = 0, on ∂Y0 ,     u(n1 ) is Y − periodic in y, 1

   (nm+2 ) (nm+2 −1) (nm+1 ) (n )   A0 um+2 = R um+2 − A1 um+1 − A2 um m , in Y1 ,      (nm+1 ) (nm+2 ) −d (y) ∇ u + ∇ u · n = 0, on ∂Y0 , x y m+1 m+2     u(nm+2 ) is Y − periodic in y, m+2

for 0 ≤ m ≤ M − 2. Note that the quantity nm is independent of ε.

(3.10)

(3.11)

(3.12)

(n )

Since the approximate auxiliary problems became linear, standard procedures are able to find the solutions um m for

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 VY1 defined by

 VY1 := ϕ|ϕ ∈ H 1 (Y1 ) , ϕ is Y − periodic .

If κp := Cp Lα−1 < 1 holds (here Cp is the Poincar´e constant depending only on the dimension of Y1 ), then we n o (n ) easily obtain that for every m, um m is a Cauchy sequence in H 1 (Y1 ). Hereby, it naturally claims the existence and uniqueness of the nonlinear auxiliary problems (3.7)-(3.9). Moreover, the convergence rate of the iteration procedure is given by



(nm )

um − um

H 1 (Y1 )

κnp m

(1) . nm um 1 1 − κp H (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: ϕε := uε −

M X

m=0

4

εm um .

30

Relying on the auxiliary problems (3.7)-(3.9), note that the function ϕε satisfies the following system:  PM −2   Aε ϕε = R (uε ) − m=0 εm−2 R (um )    −εM −1 (A1 uM + A2 uM −1 ) − εM A2 uM , in Ωε ,     −dε ∇ ϕε · n = εM dε ∇ u · n, on Γε . x x M

(3.13)

Now, multiplying the PDE in (3.13) by ϕ ∈ V ε and integrating by parts, we arrive at * + M −2 X ε ε ε m−2 hd ϕ , ϕiV ε = R (u ) − ε R (um ) , ϕ L2 (Ωε )

m=0

−ε

M −1

−εM

Z

hA1 uM + A2 uM −1 + εA2 uM , ϕiL2 (Ωε )

Ωεint

dε ∇x uM · nϕdSε .

(3.14)

From here on, we 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 *

+ M −2 M

X


ε X m m−2 M −1 R (uε ) − u − ε R (um ) , ϕ ε um + O ε

kϕkL2 (Ωε ) , ≤ CL

ε m=0 m=0 V L2 (Ωε )

(3.15)

where we have essentially used the global Lipschitz condition on the reaction term, the assumption (3.6), and the Poincar´e inequality. Next, we get εM −1 hA1 uM + A2 uM −1 + εA2 uM , ϕiL2 (Ωε ) ≤ CεM −1 kϕkL2 (Ωε ) ,

while using the trace inequality (cf. [10, Lemma 2.31]) to deal with the the last integral, it gives Z εM dε ∇x uM · nϕdSε ≤ CεM −1 kϕkL2 (Ωε ) . Ωεint

(3.16)

(3.17)

Combining (3.15)-(3.17), we provide that

α |hϕε , ϕiV ε | ≤ CεM −1 kϕkL2 (Ωε ) , which finally leads to kϕε kV ε ≤ Cε

M −1 2

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 κp := Cp Lα−1 < 1 for the given Poincar´e constant. Let n o (n ) um m be the schemes that approximate the nonlinear auxiliary problems (3.10)-(3.12). Then (3.10)-(3.12) admit nm ∈N

a unique solution um for all m ∈ {0, ..., M } with the speed of convergence:

Cκnp

(nm )

≤ for all nm ∈ N and m ∈ {0, ..., M } ,

um − um 1 1 − κnp H (Y1 ) where C > 0 is a generic ε-independent constant and n := max {n0 , ..., nM }.

Theorem 5. Let uε be the solution of the elliptic system (2.1) with the assumptions (A1 ) − (A2 ) stated above and suppose that (3.6) holds for r ∈ {1, 2}. For m ∈ {0, . . . , M } with M ≥ 2, we consider um the solutions of the auxiliary problems (3.10)-(3.12). Then we obtain the following corrector estimate:

M

ε X m ε um

u −

m=0

Vε

where C > 0 is a generic ε-independent constant.

5

≤ Cε

M −1 2

,

35

Remark 6. If the Lipschitz constant L depends on the homogenization parameter ε for a given order of O (εq ), q ∈ R, then the same result can be obtained. In fact, such a constant only appears in (3.15). Then an increase in the order M of the expansion is necessary to guarantee the convergence when q is negative. Note that, the more the order M is exploited, the more complicated becomes the computation procedure. On the other hand, such M -dependence broadens the applicability

40

of our approach. For instance, a simple example having an ε-dependent L and satisfying (3.6)) is R(u) = ε−1 u. Here one  M  has L = ε−1 and r = 1, and hence, with M ≥ 3 the corrector estimate is of order of O ε 2 −1 .

Remark 7. An improved version of the above iterations can be proposed by adding a stabilization term. For example, if   (n ) (n −1) the quantity Ls u0 0 − u0 0 (with Ls being a free-to-choose positive number) is inserted into the right-hand side of the PDE of the auxiliary problem (3.10), we are led to a new mild restriction (L + Ls ) Cp / min {α, Ls } < 1.

Remark 8. The structural condition (3.6) must be viewed here as a prototype. Modifying it accordingly allows the 45

treatment of many classes of reaction rates, including those mentioned in [12, 13, 14]. Note that although in many multiscale problems the impediment r ∈ {1, 2} is not present, the high-order corrector (as well as the technicalities

coming with its derivation) are still available. It is worth noting that extensions can also include Arrhenius-like laws (i.e. exponential rates of the type R(u) = e|u| ). The control of the oscillations can be then done in terms of the elementary  inequality ea − eb ≤ max ea |a − b| , eb |a − b| , for a, b ≥ 0, provided L∞ −bounds on the solution are available. 50

Acknowledgment

AM thanks NWO MPE ”Theoretical estimates of heat losses in geothermal wells” (grant nr. 657.014.004) for funding. 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. The authors wish to express their sincere thanks to the anonymous referees for then many constructive comments leading to an 55

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