Mathematical multi-scale model of water purification

In this work we consider a mathematical model of the water treatment process and determine the effective characteristics of this model. At the microscopic length scale we describe our model in terms of a lattice random walk in a high-contrast periodic medium with absorption. Applying then the upscaling procedure we obtain the macroscopic model for total mass evolution. We discuss both the dynamic and the stationary regimes, and show how the efficiency of the purification process depends on the characteristics of the macroscopic model.


Introduction
The problem of water purification has great practical importance and gives rise to many interesting mathematical questions.Mathematical modelling of water treatment has become increasingly popular in recent years, see e.g.[4,11,10].In the present work we deal with mathematical models for the treatment of wastewater in biofilm reactors and in filters filled with granules which are made of nanoporous super-hydrophilic materials.We use here a combination of a probabilistic approach and a homogenization technique for modelling the purification process.
To clarify the motivation of the model we shortly describe one of the biofilters used in industrial water purification process.A biofilm reactor is a tank of cylindrical shape which is about one meter high and of diameter about 20 cm.It is packed with parallelepipeds consisting of thin pressed polymer fibers.A typical volume of such a parallelepiped is 15-20 cm 3 , and the fibers are small rods whose length is about 1 cm.Each such a rod is covered with a thin biologically active biofilm, these biofilms being filled with bacteria for which impurities within water are a nutrition.Water is supplied to the upper cross section of the device and then trickles down drop by drop along the rods so that the biofilms covering the rods are getting wet.The material of biofilms is designed in such a way that its diffusion coefficient is much smaller then that in the surrounding fluid domain.The polluted water penetrates the biofilms and the impurities are consumed by the bacteria.The intensity of this process depends on the concentration of both the bacteria and the impurities at the biofilms boundary.The averaged speed of water also influences the said intensity.The biofilter is efficient if the said averaged speed is sufficiently small.All together there are several millions such rods in the device, they are called basic elements of the biofilter.We would like to construct an adequate model of the mentioned above consumption process for one rod and then to model the whole process of water purification.Our goal is to evaluate the drop in the water pollution level.Several models of this type have been considered in a number of works, in particular in [8,3].
In [3] the consumption of impurities in one basic element is described by a system of differential equations including a diffusion equation in 3D cylindrical domain and a transport equation at the rod border.This problem does not have an analytic solution.So it is natural to simulate its solution numerically.To this end in [3] the whole cylindrical tank is divided into layers in the vertical direction and calculate numerically the drop of the impurity concentration at each layer.Also, in [3] the asymptotic analysis of the system is performed provided a small thickness of the rods.
We turn to another model of water treatment.One of the most common pollutants of the wastewater are petrol and oils impurities, and the water purification from oil and petrol products refer to the highly important environmental problems.One of the modern methods of water purification is described in [7].It is the filtering method using granules made of innovative nano-porous superhydrophilic materials.For the practical implementation a design of the classical pressure filter for granulated filter bed was chosen.The system is represented by a vertical cylindrical filter with a distribution system below and above.The filter is filled with granulated bed of the grade 0.7 -1.7 mm, preliminary impregnated with water.The filter has a height of 1.5 m and an average pore diameter of 6.5 nm.The diffusion coefficient in the granules is much smaller than that in the surrounding solute.
In the present work we suggest a mathematical model based on a probabilistic interpretation of the water purification process described in [7].It is assumed that the movement of the impurities at the micro scale is described in terms of a Markov process.Namely, the impurities can enter the porous granules with a positive probability and then either be absorbed there or leave.Also, it is assumed that the basic purification elements are located periodically.We then divide each periodicity cell into a finite number of cubes, introduce the lattice formed by the centers of these cubes and perform the corresponding discretisation of the Markov process.For the obtained random walk we define the transition probabilities between the sites of the same cell or neighbouring cells.This yields the description of the model at the microscopic level.
Our goal is to provide the macroscopic description of this process based on upscaling procedure.It will be shown that the coefficients involved in the macroscopic model, i.e. the effective characteristics of the water purification process, can be expressed through the characteristics of the model at the microscopic scale by means of solving a system of linear algebraic equations.The size of the system only depends on the number of points in the period.
The main characteristics of the quality of water purification is the exponent that specifies the rate of decay of impurities concentration depending on the distance to the upper cross section of the filter.In this work we provide some examples of calculating such exponent.
The advantage of the proposed model is that it can be easily adapted to any geometry of absorbing films.The model considered in this work can be used for better understanding complex treatment systems as well as for optimising the parameters of water purification devices in accordance with the restrictions on the device productivity and the purification quality.
Let us formulate the mathematical problem underlying the considered model.In Sect. 2 we introduce the random walk X(n) on Z d , d ≥ 1 in a periodic high-contrast medium that models the process of water purification on a microscopic scale.We consider the random walk as a discretization of the diffusion of impurities within a device with appropriate restrictions on their movement.In the microscopic model considered in this work, in addition to the random walk there is also a partial absorption at the astral sites.To describe the absorption process we modify the random walk model adding the absorbing state {⋆}.Thus, our model at the micro level is the random walk with absorption, we call this process X (t).In Sect. 3 we study the large time behaviour of this process under the upscaling procedure.To do this we assume that the transition probabilities of the random walk depend on a small parameter ε > 0, and study the limit behaviour of the rescaled process X ε (t), as ε → 0. It turns out that there exists a nice and very useful description of the limit process as a two component continuous time Markov process X(t) = (X (t), k(t)).Its first component X (t) evolves in the space R d ∪ {⋆}, while the second component is a jump Markov process k(t) with a finite number of states.
In Sect. 4 we provide an example of the macroscopic (effective) model, both in dynamic and in stationary regimes.In Sect. 5 one can find the derivation of formulae for the effective characteristics of the macroscopic model.In Sect.6 we calculate the effective matrix and the effective drift for one example, and show the connection between models at micro and macro scales.
The mathematical background of the present work has been developed in [9], where we studied a symmetric random walk in high-contrast medium and constructed the limit process on the extended state space.In this work, we supplement the symmetric random walk with an additional drift and absorption, which leads to a significant modification of our previous scheme.A crucial step in our approach is constructing several periodic correctors which are introduced as solutions of auxiliary difference elliptic equations on the period.Earlier the corrector techniques in the discrete framework have been developed in [6] for proving the homogenization results for uniformly elliptic difference schemes.
Various phenomena in media with a high-contrast microstructure have been widely studied by the specialists in applied sciences and then since '90th high-contrast homogenization problems have been attracting the attention of mathematicians.Homogenization problems for partial differential equations describing high-contrast periodic media have been intensively investigated in the existing mathematical literature.In the pioneer work [2] a parabolic equation with high-contrast periodic coefficients has been considered.It was shown that the effective equation contains a non-local in time term which represents the memory effect.In the literature on porous media these models are usually called double porosity models.Later on in [1], with the help of two-scale convergence techniques, it was proved that the solutions of the original parabolic equations two-scale converge to a function which depends both on slow and fast variables.
In this section we provide a detailed description of the random walk.Given a probability space (Ω, F.P), we consider a random walk Denote the transition matrix of the random walk by P = {p(x, y), x, y ∈ Z d }.We assume that the random walk satisfies the following properties: -Periodicity.The functions p(x, x + ξ) are periodic in x with a period Y for all ξ ∈ Z d .In what follows we identify the period Y with the corresponding d-dimensional discrete torus T d .
-Finite range of interactions.There exists c > 0 such that -Irreducibility.The random walk is irreducible in Z d .
In this paper we consider a family of transition probabilities p (ε) (x, y) that satisfy all above properties and depend on a small parameter ε > 0. The transition probabilities p (ε) (x, y) describe the so-called high-contrast periodic structure of the environment.We suppose that the transition matrix P (ε) is a small perturbation of a fixed transition matrix P 0 = {p 0 (x, y)} that corresponds to a symmetric random walk, i.e.
We say that y We will use further the notation Thus the normalization condition can be rewritten as The transition matrix P (ε) has the following form In order to characterize the matrices P 0 , D and V we divide the periodicity cell into two sets and assume that B ⊂ T d is a connected set such that its periodic extension denoted B ♯ is unbounded and connected.Here the connectedness is understood in terms of the transition matrix P 0 .Two points x ′ , x ′′ ∈ Z d are called connected if there exists a path x 1 , . . ., x L in Z d such that x 1 = x ′ , x L = x ′′ and p 0 (x j , x j+1 ) > 0 for all j = 1, . . ., L − 1.As a consequence we get that We also denote by A ♯ the periodic extension of A.
In addition to the general assumptions ( 3) and ( 7) the following conditions on the matrices P 0 , D and V are imposed: the elements of matrices D and V satisfy the relation Remark 1.In particular, the above conditions imply that v(x, y) ≥ 0, if at least one of x or y ∈ A ♯ and x = y.
From the periodicity of D and V it also follows that Summarizing all above conditions, we conclude that the non-zero transition probabilities defined by ( 5) have the following structure: The above choice of the transition probabilities reflects a slow drift (of the order ε) given by matrix D in the fast component, and also a significant slowdown (of the order ε 2 ) of the random walk inside the slow component.
Further, we add to the above random walk an absorption process consistent with the structure of the periodic environment, assuming that the absorption occurs only inside the inclusions A ♯ .For the description of the complete process we will denote by S = Z d ∪ {⋆} the state space of the new process, where {⋆} is the absorption state.Then the transition matrix of the complete process with absorption has the following form where We note that random walks with transition probabilities of the form has have been studied in [9].In the present work we supplement the model with drift and absorption.
Our goal is to derive the effective evolution equation under the diffusive scaling.

Rescaled process
In what follows we study the scaling limit of the random walk on S with transition matrix Q (ε) and use ε as the scaling factor.Denote In what follows the symbols x and y are used for the variables on Z d (fast variables), while the symbols z and w for the variables on εZ d (slow variables).Notice that the state {⋆} does not change under the scaling.
We introduce now the rescaled process.Denote by T ε the transition operator associated with the transition matrix (9): where q ε (z, w) are elements of the matrix Q (ε) , see (9).Namely, q ε (z, w) = p ε (z, w) = p( z ε , w ε ), when z, w ∈ εZ d , where p ε (z, w) are elements of the matrix P (ε) defined in (9); is the difference generator of the rescaled process X ε (t) on εS = εZ d ∪ {⋆} with transition operator T ε .The rescaled process has two components: where is the rescaled random walk on εZ d , and the latter component s(t) lives on {⋆}.
The goal of the paper is to describe the limit behavior of the rescaled process X ε (t), as ε → 0, to construct the limit process, and to find the explicit expressions for all effective characteristics of the limit process.

Extended random walk
Homogenization of non-stationary processes in high contrast environments often results in the effective equations with nonlocal in time terms representing the memory effect.As was shown in [9] the limit process for a random walk in a high contrast environment remains Markov if we equip the original random walk with additional component(s) and consider the obtained random walk in the extended state space.
In this subsection we describe the constructions of an extended random walk introduced in [9].We equip the random walk X ε (t) (the first component in (13)) with an additional component(s) in the same way as it has been done in [9].Assume that the set A defined in (6) contains M ∈ N sites of T d : A = {x 1 , . . ., x M }.For each k = 1, . . ., M we denote by {x k } ♯ the periodic extension of the point We assign to each z ∈ εZ d the index k(z) ∈ {0, 1, . . ., M } depending on the component in decomposition (14) to which z belongs: With this construction in hands we introduce the metric space with a metric that coincides with the metric in εZ d for the first component of (z, k(z)) ∈ E ε .
The index k(⋆) = ⋆ is assigned to the state z = {⋆}.Thus the extended version of the absorption state is {⋆, ⋆}, but for simplicity we will keep the notation {⋆}.Denote by S Eε = E ε ∪ {⋆}, and in what follows instead of X ε (t) we will consider the process We denote the space of bounded functions on S Eε by B(S Eε ) and construct the transition operator T ε of the process X ε (t) on S Eε using the same transition probabilities as in operator ( 11): with Then T ε is the contraction on B(S Eε ): Remark 2. Since the point (z, k(z)) ∈ E ε is uniquely defined by its first coordinate z ∈ εZ d , then we can use z ∈ εZ d as a coordinate in E ε (considering E ε as a graph of the mapping k : εZ d → {0, 1, . . ., M }).In particular, for the transition probabilities of the random walk on E ε we keep the same notations q ε (z, w) as in (11).

Limit process
In this subsection, we construct a limit process, which is a Markov process completely determined by its generator.We denote E = R d × {0, 1, . . ., M }, and C 0 (E) stands for the Banach space of continuous functions vanishing at infinity.Together with E we consider S E = E ∪ {⋆} and denote C 0 (S E ) = C 0 (E) ⊕ C. Then F ∈ C 0 (S E ) can be represented as and the norm in C 0 (S E ) is equal to where Consider the operator where 1 {k=0} is the indicator function, Θ •∇∇f 0 = Tr(Θ∇∇f 0 ), Θ is a positive definite matrix defined below in (63), b is a vector of the effective drift defined also below by (62).Both of these effective characteristics of the limiting process are written in terms of the first corrector of the corresponding problem on the cell.The operator L A is a generator of a Markov jump process with Notice that the parameters α jk , j, k = 0, 1, . . ., M , are non-negative and define intensities of the limit Markov jump process on the period Y .
The operator L is defined on the core and is a dense set in C 0 (E).One can check that the operator L on C 0 (S E ) satisfies the positive maximum principle, i.e. if F ∈ C 0 (S E ) and max E∪{⋆} F (z, k) = F (z 0 , k 0 ), then LF (z 0 , k 0 ) ≤ 0. Since L A is a bounded operator in C 0 (S E ), the operator λ − L is invertible for sufficiently large λ.Then by the Hille-Yosida theorem the closure of L is a generator of a strongly continuous, positive, contraction semigroup T (t) on C 0 (S E ).
Let us describe the limit process X(t) generated by the operator L. It is a two component continuous time Markov process X(t) = {X (t), k(t)}, where the first component X (t) lives in R d ∪ {⋆}, the second component is a continuous time jump Markov process k(t) on the state space K = {0, 1, 2, . . ., M, ⋆}.The process k(t) does not depend on the other components; its transition rates α ij are expressed in terms of the transition probabilities of the original random walk, see (20).The probability of jump between any two states i, j ∈ {0, 1, 2, . . ., M }, i = j, is equal to α ij .The absorbing state {⋆} is reachable only from the "astral" states {1, 2, . . ., M } with the same intencity m.Thus, the matrix corresponding to the generator L A has the following form When k(t) = 0, the first component X (t) evolves along the trajectories of a diffusion process in R d with the corresponding effective characteristics, while when k(t) = 0, the first component X (t) remains still until k(t) takes again the value 0. Thus the trajectories of X (t) coincide with the trajectories of a diffusion process in R d on those time intervals where k(t) = 0.As long as k(t) = 0 the first component X (t) does not move, and only the second component k(t) of the process evolves.Additionally, the process k(t) can jump from the astral states {1, . . ., M } to the absorbing state {⋆} with intensity m, and upon reaching this state, the process never leaves it.

Main result. The convergence of semigroups
In this subsection we formulate the main result of this work on convergence (upscaling) to the limit process constructed in the previous subsection.Let l ∞ 0 (E ε ) be a Banach space of functions on E ε vanishing as |z| → ∞ with the norm and denote l ∞ 0 (S Eε ) = l ∞ 0 (E ε ) ⊕ C. For every F ∈ C 0 (S E ) we define the function π ε F ∈ l ∞ 0 (S Eε ) as follows: and π ε F (⋆) = F (⋆). Then π ε defines a bounded linear transformation π ε : C 0 (S E ) → l ∞ 0 (S Eε ).Theorem 1.Let T (t) be a strongly continuous, positive, contraction semigroup on C 0 (S E ) with generator L defined by (18)-( 19), and T ε be the linear operator on l ∞ 0 (S Eε ) defined by (17).Then for every as ε → 0.
Proof.The proof of (26) relies on the approximation techniques from [5] used for the proof of convergence of semigroups.According to results of [5, Theorem 6.5, Ch.1] the semigroups convergence stated in ( 26) is equivalent to the statement which is the subject of the next lemma.
Lemma 1.For every F ∈ D, where D was defined by (21), there exists Proof.For every we present a function F ε that guarantees the convergence ( 27)-(28).Let us take F ε ∈ l ∞ 0 (S Eε ) in the following form and F ε (⋆) = F (⋆).Here h(y), g(y), q j (y), j = 1, . . ., M, are periodic bounded functions that will be defined below.The boundedness together with (29) immediately imply that Let us turn to the second convergence stated in (28).Since (L ε F ε )(⋆) = (LF )(⋆) = 0, it suffices to show that In the proof of (30) we use the same arguments as in the paper [9].According to (17) and ( 9) the operator L ε can be written as where We consider next separately the cases when z ∈ εB ♯ , and z ∈ εA ♯ .Since the second component in E ε is a function of the first one, in the remaining part of the proof for brevity write F ε (z) instead of 29) can be written as a sum where To estimate we will use two following propositions.
Proposition 1.There exist bounded periodic functions h(y) = {h i (y)} d i=1 and g(y) = {g im (y)} d i,m=1 (correctors) and a positive definite matrix Θ > 0, such that where F P ε is defined in (33).The proof of this proposition is based on the corrector techniques, it is given in the Appendix.Proposition 2. There exist bounded periodic functions q j (x), j = 1, . . ., M , on B ♯ such that where α 0j > 0 are constants defined in (20), and F Q ε is introduced in (34).The proof of Proposition 2 is the same as in [9].We give a proof in the Appendix for complete presentation. Since where then Propositions 1 -2 together with (36) and (39) yield Next we consider the case when z ∈ εA ♯ , and prove that sup 29), (31) and continuity of functions f k it follows that as ε → 0.Here we have used the fact that x, y ∈ Y are variables on the periodicity cell, and v(x k , x j ) are the elements of the matrix V .On the other hand, according ( 19) and (25) π ε LF (z) for z ∈ ε{x k } ♯ has the following form where constants α k0 , α kj are given by (20).Thus, relations (42) and ( 43) imply (41).Finally, (30) is a consequence of (40) and (41), and Lemma 1 is proved.
It remains to recall that (26) is a straightforward consequence of the above approximation theorem.This completes the proof of Theorem 1.

Dynamics of pollution. Stationary regime
In this section we consider an example of the limit dynamics in the case when the astral set A contains one point.We also derive an equation on the first component ρ 0 (x, t) of the astral diffusion describing a visible dynamics of the pollution density.
Denote by ρ(x, t) = ρ 0 (x, t), ρ 1 (x, t), ρ 2 (t) the three-component density of pollution, where ρ 0 (x, t) is the density outside of micro-granules, ρ 1 (x, t) is the density inside of micro-granules, ρ 2 (t) is the density of pollution accumulated (or absorbed) as a result of cleaning by time t.
The corresponding model at microscopic scale is an one-point astral model with absorption.Then for the limit dynamics, we obtain the following evolution equations for ρ(x, t) (44) with initial data ρ(x, 0) = π 0 (x), π 1 (x), π 2 .Here Θ, b are the effective diffusion matrix and the effective drift depending on the geometry of the micro-scale model, λ(0) > 0, λ(1) > 0 are the rates of exchanging between inside and outside regions: λ(0) is the intensity of the water flows into cleaning inclusions, while λ(1) is the intensity of flows from inclusions.All of them are the parameters of the limit model.Below in App.1-2 we will show how the effective parameters Θ and b can be found from the micro-scale model.
The solution of the second equation in (44) has the form After substitution of ρ 1 (x, t) to the first equation in (44) we obtain the following evolution equation on ρ 0 : with a boundary conditions where x d is the direction of the drift.Here ϕ(x) ≥ 0 is the profile of the initial concentration on the upper cross section.
Assuming that the initial profile ϕ is a constant function one can reduce the dimension in problem (46)-( 47) and obtain a one-dimensional stationary problem that reads Thus, the rate of the purification process is equal to R pur = 1 2θ √ b 2 + 4θκ − b , and for sufficiently small θ we get R pur ≈ κ b .Remark 3. If the astral set A contains more that one points, i.e. |A| = M > 1, then the kernel K(t − s) in (45) is a linear combination of exponents e −κ j (t−s) with κ j > 0, j = 1, . . ., M .
It follows from (54) that the left-hand side of (53) takes the form: Thus the periodic vector function h(x) is taken as a solution of the equation where l(x) = x is the linear function.The solvability condition for equation (56) reads Since p ξ (x) = p −ξ (x + ξ), this condition holds true, which implies the existence of the unique, up to an additive constant, periodic solution h(x) of equation ( 56).
We follow the similar reasoning to find an equation for the periodic matrix function g(x), x ∈ B ♯ .We will also obtain below expressions for effective matrix Θ and drift b.
Collecting in (52) all terms of the order O(1), using relation (56) on the function h(x) and relation ( 8) on matrix D we get: Let x ε = y ∈ B, and denote by Φ(h) the following matrix and vector functions In order to ensure the convergence in (37) we should find a constant matrix Θ, a periodic matrix function g(y) and a constant vector b such that The latter equation implies that The solvability condition for (60) reads Thus Θ is uniquely defined as follows: and g(y) is a solution of equation ( 60).This solution is uniquely defined up to a constant matrix.We notice that the matrix Θ defined by (63) is positive definite, i.e. (Θη, η) > 0 ∀η = 0.The proof is given in [9].This complete the proof of Proposition 1.
6 Appendix 2. One example with calculation of effective parameters.
In this section we consider one example of a model at the microscopic scale and calculate for this example the effective parameters Θ and b of the limit model that in particular used in the description of the stationary regime, see equation ( 46).Let the periodicity cell Y be a square 3 × 3 of the two-dimensional lattice Z 2 , A is the one-point subset of Y located at the center of Y , and B = Y \A, t.e.|B| = 8.Let us renumber the elements from B in accordance with their position on the cell Y : We define the symmetric matrix P 0 | B ♯ = {p 0 (x, y), x, y ∈ B ♯ } describing the free moving of the water outside of cleaning elements as follows: p 0 (x, x ± e 1 ) = p 0 (x, x ± e 2 ) = 1  4 , if x ∈ {s 1 , s 3 , s 6 , s 8 }; -p 0 (x, x ± e 1 ) = 1  4 , p 0 (x, x + e 2 ) = 1 2 , if x = {s 2 }; -p 0 (x, x ± e 2 ) = 1 4 , p 0 (x, x − e 1 ) = 1 2 , if x = {s 4 }; -p 0 (x, x ± e 2 ) = 1 4 , p 0 (x, x + e 1 ) = 1 2 , if x = {s 5 }; p 0 (x, x ± e 1 ) = 1 4 , p 0 (x, x − e 2 ) = 1 2 , if x = {s 7 }.Other elements of the matrix P 0 | B ♯ equal to 0.
Next the matrix D is introduced, which determines a nonzero drift (of the order ε) in the microscopic scale model.Taking into account relation (8) we consider the following elements of D = {d(x, y)}, x, y ∈ B ♯ : d(x, x ± e 2 ) = ∓K, if x / ∈ {s 2 , s 7 }, and d(x, y) = 0, otherwise. (64) Thus, the model at the microscopic scale on the component B ♯ complementary to the astral component is completely defined.
As follows from results of the previous section the effective parameters, the matrix Θ and the vector b, are given by ( 63) and (59) respectively.Since p ξ (x) = p 0 (x, x + ξ), d ξ (x) = d(x, x + ξ), x, x + ξ ∈ B ♯ , have been already introduced above in this section, we have to find the vector function h(x) = {h(x), x ∈ B}.This function is called corrector, and it is taken as a solution of the equation (56).Thus, in our example the corrector is the same as a set of eight vectors h B = {h(s 1 ) ∈ Z 2 , h(s 2 ) ∈ Z 2 , . . ., h(s 8 ) ∈ Z 2 }, B = {s 1 , . . ., s 8 }.
We explain now how to find this vector function.It is worth noticing that (56) is a system of uncoupled equations, and we can solve it for each coordinate separately.To find the vector of the first coordinates h B 1 = {h 1 (s 1 ), h 1 (s 2 ), . . ., h 1 (s 8 )} we take in (56) l 1 (x) = (x 1 , 0) and rewrite (56) in the following way: (P 0 − I)h 1 (x) = −(P 0 − I)l 1 (x) =: where g 1 (x) = −(P 0 − I)l 1 (x) = (0, 0, 0, 1/2, −1/2, 0, 0, 0) is the function defined on B and periodic on Z 2 .It can be represented as follows: 0 0 0 1/2 • −1/2 0 0 0 Thus, it follows from (65) that the vector of the first coordinates h B 1 is equal to Similarly, setting l 2 (x) = (0, x 2 ), we obtain the vector of the second coordinates h B 2 of the set h B by the formula where K is the same constant as in (64).Here we used that by the periodicity assumption for any x ∈ B and any ξ ∈ Z 2 (x + ξ) modB = x, with some x ∈ B, so that h(x + ξ) = h(x).
Finally the matrix Θ of order 2 × 2 can be found by the formula (63), where each term in the sum (63): is determined in terms of the corrector h(x) and the matrix elements of P 0 | B ♯ .We used here that be the Banach space of bounded functions on Z d vanishing at infinity with the norm f = sup x∈Z d |f (x)|.Similarly, we consider the Banach space of bounded functions l ∞ 0