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Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

This work is devoted to the development and analysis of a linearization algorithm that can be used for the numerical implementation of microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in L2 norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove convergence at the micro-scale based on the energy method. Meanwhile, we adopt the classical homogenization result in line with corrector estimate to show the convergence of the scheme at the macro-scale. In the numerical section, we use the standard finite element method to assess the efficiency and convergence of our proposed algorithm.


Microscopic problem
Let Ω ε be a Lipschitz perforated domain contained in a polygonal bounded domain Ω ⊂ R d (d = 2, 3). In this sense, Ω ε possesses a uniformly periodic microstructure defined by a length scale ε. This ε is a small parameter (0 < ε 1) since the size of the pores are usually much smaller than the characteristic length of the reservoir. We are herein concerned with the asymptotic behavior in a stationary case of the function u ε : Ω ε → R that describes the spread of concentration of solutes dissolved in a saturated porous tissue shaped by the perforated domain Ω ε with a cubic periodicity cell Y = [0, 1] d . The molecular diffusion coefficient A : Y → R d×d is assumed to vary in the cell Y , while we consider in this scenario the presence of a volume reaction R : R → R subject to an internal source f : Ω → R. We also take into account the no-flux boundary condition at the internal boundaries, denoted by Γ ε , whilst giving the homogeneous Dirichlet boundary condition at the exterior boundary, denoted by Γ ext . Essentially, this context can be understood by the following elliptic problem: (P ε ) : in Ω ε , −A(x/ε)∇u ε · n = 0 across Γ , u ε = 0 across Γ ext , where L ε is a symmetric operator given by (1.2)

Background
In this paper, we follow up on our earlier works [1,2] that focus on the asymptotic analysis of semi-linear elliptic problems posed in perforated domains. Cf. [1], we briefly design an efficient linearization scheme to prove the weak solvability of (P ε ) in a Hilbert setting and derive the macroscopic equation for u ε as α = 0. Furthermore, rates of convergence are obtained to fulfill the asymptotic analysis of (P ε ). The presence of non-negative scalings stems from our mathematical concerns about the asymptotic behaviors of u ε when ε tends to 0 and their corresponding rates. As a result, this analysis unveils the approximate shape of u ε at the macro-scale and thereupon delineates the steps to get its approximation in a less time-consuming way. As is known in the homogenization community, solving (P ε ) directly is computationally expensive since the space discretization is inversely proportional to the scale parameter, ε.
In principle, the microscopic solution to the problem (P ε ) converges to a macroscopic function that solves a certain homogenized problem since the scaling variable α is eventually the main factor that determines the presence of the reaction term R at the macro-scale. We remark that the case α > 0 can be subsequently caught in the context of low-cost control problems on perforated domains, cf. [3]. Meanwhile, the case α = 0 can be viewed as an inception of the periodic homogenization of cross-and thermo-diffusion models, cf. e.g. [4,5]. In general, arguments obtained from studies of the variable scalings can be helpful in the qualitative analysis of eigen-elements for elliptic boundary value problems with rapidly oscillating coefficients in a perforated cube (cf. e.g. [6,7]), while it can be further adapted to complex scenarios (cf. e.g. [8] and references cited therein). Moreover, we note that the consideration of an elliptic model and the linearization scheme obtained in the sequel could be helpful in solving the corresponding parabolic types. In the parabolic context, one often applies the backward Euler method to handle the time discretization that consists in its stability. This procedure essentially leads to a sequence of nonlinear elliptic equations due to the implicit approach and thus motivates us to explore the present problem.

Novelty of our new method
In this work, we show that the linearization scheme we design is essential in proving the well-posedness of (P ε ), but also in deriving the approximate macroscopic solution with certain error estimates. Basically, our theoretical analysis will proceed in accordance with the following diagram: Here, (P k ε ) denotes the approximate problem of (P ε ) by linearization, whilst its macroscopic equation is structured in (P k 0 ). The notion behind this approach is to linearize nonlinearities in the model using a suitable choice of the stabilization parameter. As the nonlinearity is supposed to be degenerate at a single point, we are aided by a regularization approach during the linearization process. In this way, we arrive at a regularized form of the nonlinearity, where we can figure out the error estimate between u ε and u k ε in L 2 -norm. In our proof, the stabilization is ε-dependent only when the scaling factor α is positive. Meanwhile, ε does not contribute to the convergence of the linearization scheme for any α ≥ 0. As ε → 0 in the homogenization process, the stabilization constant becomes ε-independent for any α ≥ 0. Henceforth, several stability estimates for the macroscopic solution are easily obtained.
As another vantage of our proposed scheme, we point out that if the solvability of (P ε ) and the corresponding macroscopic equation are already known, one can also use the scheme directly to get the approximation u k 0 . In this case, we mean Furthermore, in the diagram (1.4) we can prove that the rate of convergence is k-independent (cf. [9]), compared to the result we obtain in Corollary 2 for the diagram (1.3). It is worth mentioning that as we aim to show the unique weak solvability of the microscopic problem by a linearization technique, we follow diagram (1.3).

Outline of the paper
The paper is organized as follows. Section 2 is dedicated to introducing notations and necessary assumptions on the input of the problem. In Section 3, we propose an iterations-based variational scheme to linearize the microscopic problem (P ε ). Accordingly, we obtain the well-posedness of (P ε ) as well as the rate of convergence by the linearization we choose. Settings of the homogenization are involved in Section 4, where we also state the structures of the cell problems and the limit equation at every step of linearization. Additionally, several types of stability analysis of the scheme at the macro-scale are justified. Section 5 is devoted to the numerical test of the scheme and we close this paper with some concluding remarks in Section 6.

Preliminaries
In the sequel, all the constants C are independent of the scaling parameter ε, but their precise values may differ from line to line and may change even within a single chain of estimates. We use either the superscript or subscript ε to indicate its dependence. Depending on the situation, we denote by |·| the absolute value of a function, the finitedimensional Euclidean norm of a vector or the volume of a domain. For brevity, we herein skip the mathematical descriptions of the perforated domain of interest. Instead, we only provide Figure 1 for a graphically schematic representation of the scaling procedure within a natural soil leading to the consideration of periodically perforated domains with its unit cell. In addition, the reader can be referred to [10,11,12] and those collectively mentioned in Section 1 for some concrete results concerning such domains.
e. in R and the following conditions hold true: • There exist δ 0 , r x0 > 0 independent of ε such that where B (x 0 , δ x0 ) denotes a ball centered at x 0 with a radius r x0 .
• For x ∈ B (x 0 , r x0 ), F is non-decreasing and changes its monotonicity at 0.
Next, we introduce the space equipped with the norm Cf. [13, Lemma 2.1], one can show the uniform-in-ε equivalence between this norm and the usual H 1 -norm by the Poincaré-type inequality.
Moreover, we denote by H 1 0 (Ω) the Hilbert space of weakly differentiable functions u : Ω → R that vanishes on the boundary in the sense of trace. We also use below the space H 1 # (Y l ) to indicate functions in H 1 (Y l ) that is Y -periodic and has zero mean value. Accordingly, the definition of the Bochner space L 2 (Ω; H 1 # (Y l )) will easily follow.
This is now the moment to state our working assumptions on data involved in the microscopic problem. Those include (A 1 ) The diffusion A is essentially bounded, Y -periodic, symmetric and globally Lipschitz. It satisfies the uniform ellipticity condition in the sense that we can find ε-independent constants γ, γ > 0 such that The internal source f belongs to L 2 (Ω).

Settings of the iterative variational algorithm
There are several linearization methods investigated in the past and each with its modifications and improvements to serve certain classes of nonlinear partial differential equations. To give a very cursory glance, one may concern the Newton method, cf. [14], whose convergence requires the initial guess to be close to the true solution, albeit its quadratic convergence. The Jäger-Kačur scheme is also renowned for its outstanding performance in the approximations of one-dimensional parabolic problems with a linear convergence; see [15]. In this paper, our method is conventionally in line with the so-called L-scheme extensively studied in many distinctive types of parabolic equations; cf. [16] and references cited therein for a short background concerning this typical scheme.
In this sense, the so-called stabilization term is added to stabilize the entire linearized equation in the standard variational formulation. Thereby, a linear convergence is obtained under a suitable choice of the stabilization constant.
Definition 2. For each ε > 0, a function u ε ∈ H 1 (Ω ε ) is said to be a weak solution to (P ε ) if it satisfies At this stage, we take into account the degeneracy of the reaction term R, especially in the ball where its derivative is zero and non-decreasing (see again in Definition 1). It is worth citing here the Jackson type estimates in the approximation theory of monotone functions by monotone polynomials. In principle, cf. [17], for every monotone non-decreasing functionf ∈ C k [−1, 1], there are non-decreasing polynomials p n , whose degree does not exceed n, such that where C here is independent off and n, and ω indicates the modulus of continuity off (k) . Cf. [18,19] for 1] with certain conditions, iff possesses strict lower and upper bounds of the available derivatives, then for sufficiently large n, the best polynomial approximation tof also satisfies the same property. Besides, the corresponding derivatives of the best approximation approach the derivatives off , respectively. Cf. [20] for a survey of recent developments of the polynomial approximation, the approximation process can also preserve the monotonicity off .
The aforementioned references enable us to assume the existence of a regularization scheme for the degenerate R. In this regard, a function F γ for γ > 0 being as a regularization parameter is said to be a regularization of a degenerate function F if one has • |F − F γ | ≤ Cγ κ for κ > 0 and for any x ∈ R.
Technically, the smallness of the lower bound of F γ is taken to regularize the degeneracy of F at x 0 as it is zero at this degenerate point. Thus, one may tacitly look for F γ such that F γ (x 0 ) = γδ 0 and should attempt to preserve the "shape" of F through the regularization process. Note that this approach does not mean that the regularization scheme must be a linear mapping; in general, the process can still be nonlinear.
After regularization of the reaction term, the linearized problem in the variational setting is given as follows.
where the stabilization constant M ≥ δ 1 and the regularization parameter γ k > 0 are selected later. The initial guess is taken as u 0 ε = 0.
In the following theorem, we prove that this sequence is well-defined and exists uniquely in H 1 (Ω ε ). Proof. It suffices to consider the first-loop problem of (3.4) (i.e. k = 1), which reads as wheref := f − ε α R γ k (0). Hereby, we can introduce the bilinear form B : . Hence, we complete the proof of the theorem by using the standard Lax-Milgram theorem by virtue of the natural ε-independent coerciveness and continuity of B in H 1 (Ω ε ).

Lemma 2.
Let {p k } k∈N * and {q k } k∈N * be sequences of nonnegative real numbers that obey the following recursion where a k and b k are also nonnegative real numbers. Then, it holds Proof. The proof is trivial and it can be found in Appendix A.
Theorem 2. Assume (A 1 )-(A 3 ) hold and let γ k > 0 be non-increasing with κ > 0. Then by choosing M = η + ε α δ 1 for η > 0, the sequence u k ε k∈N * of the variational problem (3.4) possesses the following property γ for k ≥ 2 as a difference function between the kth and (k − 1)th steps of approximation. Then the difference equation is provided by Observe that g γ k ≤ M − ε α γ k δ 0 and in view of the fact that resulting from the regularization factor we apply, we estimate that .
Upon the monotonicity of γ k , we use the Young inequality to get Using again the Young inequality, we also obtain the upper bound of I 2 as follows: Thereby, after some rearrangements, we find that At present, we take M = η + ε α δ 1 for η > 0 (independent of ε and k) in (3.11). Then we apply the Poincaré inequality (cf. Lemma 1) to arrive at and it thus follows from Lemma 2 that where we have denoted by Naturally, b k ∈ (0, 1) and thus the product of b i approaches 0 when k tends to infinity in the sense that This completes the proof of the theorem.
In (3.8), we observe that the stability of the scheme is essentially dependent on the partial sums of the series of a k , particularly of the choice of the regularization parameter γ k . Since in this paper we obtain a strong convergence along with a particular error estimate, harmonic series are not reliable in ensuring that the sequence u k ε k∈N * is Cauchy in H 1 (Ω ε ). This is hindered by the convergence-towards-zero of the series of a k when using the standard triangle inequality. In simpler terms, harmonic series (or even hyperharmonic series) are mostly either divergent or convergent to a non-zero constant. It is worth noting that the product of b k possesses an exponential-like decay as k → ∞. Then the same behavior should be applied to the series of a k by looking for a geometric progression of a k .
Proof. According to Theorem 2, it is straightforward to find an ε-independent upper bound for (3.8) as follows: where we have essentially used the binomial identity. Thereby, for any k, r ∈ N * we obtain using the standard triangle inequality. This way we prove that u k ε k∈N * is Cauchy in H 1 (Ω ε ). Consequently, there exists a unique function u ε ∈ H 1 (Ω ε ) such that u k ε → u ε as k → ∞. Furthermore, it is straightforward to get that Thus, it enables us to confirm the existence and uniqueness of u ε to the microscopic problem (P ε ) in H 1 (Ω ε ). This completes the proof of the theorem.

Corollary 1.
Under the assumptions of Theorem 3, the following rate of convergence holds Proof. The proof of this corollary is obvious by the aid of (3.21) when taking r → ∞.
Remark 1. It is worth mentioning here that if we already know that u ε ∈ H 1 (Ω ε ) is a unique solution of the microscopic problem (P ε ), then the partial sums of the series (3.13)) just needs to approach 0 as k → ∞ with a certain convergence rate, by considering the energy-like estimate of the difference between the linearized problem and (P ε ). Accordingly, the whole error estimate is controlled by such a rate of that partial sums.
This exactly mimics the proof in [9] where a minimal polynomial rate k −ω for ω ∈ (0, 1), which basically leads to the harmonic progression, is sufficiently taken into account.
Additionally, it is straightforward to obtain the stability analysis of the scheme u k ε k∈N * we construct above, which, in principle, provides concretely ε-independent a priori estimates in H 1 (Ω ε ). This somewhat enables us to get the existence of a weak solution to the problem (P ε ) in H 1 (Ω ε ) by the standard compactness argument, if we are able to derive, at least, the weak convergence of the reaction term in L 2 (Ω ε ) after passing to the limit. However, the uniqueness result may not always be achievable by this strategy. In this work, we do not go beyond this matter and will leave it for the future works.

Settings of the homogenization
In the previous section, the rigorous error estimate for the linearization scheme has been obtained in H 1 (Ω ε ); cf. Corollary 1. In this section, we only exploit the L 2 error estimate, although it is well-known from the corrector estimate for the homogenization limit that the microscopic solution of a linear elliptic equation approaches the macroscopic solution of the corresponding homogenized elliptic equation with a rate O ε 1 2 in H 1 (Ω ε ); cf. [13, Corollary 2.29]. From here on, it is very easy to adapt this result since our approximate problem defined in (3) is all linear at every step k.
Our goal here is to introduce the structures of the homogenized problem (P 0 ) for (P ε ) as well as its cell problems for the sake of computations in Section 5. When doing so, we remark that u ε satisfies an a priori estimate by means of u ε H 1 (Ω ε ) ≤ C established in Theorem 1 and by taking into account the usual zero extension on u ε from H 1 (Ω ε ) to H 1 (Ω). Accordingly, we only need to take the test function ϕ = ψ 0 (x) + εψ 1 x, x ε for ψ 0 ∈ H 1 0 (Ω), ψ 1 ∈ L 2 (Ω, H 1 # (Y l )) in (3). Henceforward, the compactness result allows us to extract subsequences from bounded sequences and to obtain the passage to the two-scale limit. For detailed results concerning the two-scale convergence method for the linear elliptic equation, we refer the interested reader to e.g. [21,8] under the theoretical results of the two-scale convergence postulated in [22,23].
After plugging that typical test function we have then by passing to the limit ε → 0, we are led to the following limit equation: • Case α > 0: • Case α = 0: These two cases are almost the same since we are at the linearization stage. Typically, we choose ψ 0 = 0 in both two cases to get where we have used the integration by parts with respect to y. In this way, we use the separation of variables to Here, χ k i is called as the cell function that solves the following cell problem: It is worth noting that the cell problem for every step k remains unchanged, so that our computations will be less expensive in the sense that we do not need to compute the vector-valued χ = χ k = χ k i 1≤i≤d at every k. Furthermore, one can prove that such χ k ∈ H 1 # (Y l ). Due to the non-convexity of Y l , the regularity of the unique function χ stops at H 1+s (Y l ) for s ∈ (−1/2, 1/2), no matter how smooth the involved data are; see [24] for detailed concerns. Now, choosing ψ 1 = 0 and then applying the integration by parts with respect to x, we obtain the equation for u k 0 as follows: Here, this equation is endowed with the Dirichlet boundary condition u k 0 = 0 at ∂Ω and A 0 is known as the homogenized coefficient given by where δ ij stands for the constants of the identity matrix. Additionally, this coefficient also satisfies the uniform ellipticity condition by virtue of the well-known Voigt-Reiss inequality; see e.g. [25]. Thus, proof of the wellposedness of the macroscopic problem in H 1 0 (Ω) is standard.

Theorem 4. Under the assumptions of Theorem 3 and suppose that
where c p > 0 is the standard Poincaré 2 constant. Then, the iterative sequence u k 0 k∈N * is Cauchy in H 1 0 (Ω).
Proof. It is trivial to prove that the functions u k 0 are all in H 1 0 (Ω). Observe that (4.4) is structured by the cases of α, we thus also divide the proof here into two parts. In the first part, we treat the following equation: Recall that the problem under consideration is associated with the zero Dirichlet boundary condition.
We now use the test functionφ ∈ H 1 0 (Ω) to arrive at the following variational formulation: whereā : H 1 0 (Ω) × H 1 0 (Ω) → R is a bilinear mapping given bȳ Then, it is straightforward to compute the difference equation by putting v k . This function essentially satisfies the following equation: (4.10) and then, by takingφ = v k 0 it leads to Due to the standard Poincaré inequality, we have 12) and by using Lemma 2, we estimate that Similar to proof of Theorem 3, we can prove that u k 0 k∈N * is Cauchy in H 1 0 (Ω). Next, we consider the following equation: (4.14) We proceed as above by taking into account the variational formulation of the difference equation. By so doing, (4.11) becomes where we have denoted byḡ γ k (t) = R γ k (t) − (η + δ 1 ) t.
Since ḡ γ k ≤ η + δ 1 − γ k δ 0 and R γ k−1 − R γ k ≤ Cγ κ k−1 , we then apply the Young inequality to obtain and after some rearrangements and applying the standard Poincaré inequality, we arrive at Henceforward, with the aid of Lemma 2 we have the same estimate as (3.8) and by the choice of γ k in Theorem 3, we consequently prove that u k 0 k∈N * is Cauchy in H 1 0 (Ω). This completes the proof of the theorem.
As argued in Remark 1, the geometric progression is required to prove the existence and uniqueness of the microscopic problem (P ε ) in H 1 (Ω ε ), and to avoid the case α > 0 where the error bound is arbitrarily slow. Since we are now dealing with the macroscopic framework, we can get a better "stability analysis". This argument is shown in the following theorem as a stability analysis of the macroscopic scheme in L ∞ (Ω). Additionally, this can be applied to Theorem 4 where the assumption (4.6) is no longer necessary by not using the Poincaré inequality.
Theorem 5. Consider the case α = 0. Suppose that the internal source f and the regularization R γ k are smooth.
Under the assumptions of Theorem 4, if we can choose the iterative sequence u k 0 k∈N * is stable in the sense that Proof. In this proof, we also follow the same vein as proof of Theorem 4 where we consider two typical structures of the limit equation (4.4) and investigate their corresponding difference equation by the standard variational setting.
For brevity, we do not mention again those equations, but the governing difference equations. Due to the linear problem as well as the smoothness of f and R γ k , it is also trivial to prove that the functions u k 0 are in L ∞ (Ω), cf. [26]. In this regard, we recall the difference equation where the test functionφ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) is now taken into account. Now we set and put from Ω to the set W 1 k . Thus, below we will accompany the notation W 1 k to indicate the fact that we are working in that set, although the essential integral has to be posed in Ω. In fact, (4.16) now becomes At this stage, we see that the bilinear formā W 1 k v k 0 ,φ is non-negative certainly due to the presence of the gradient. Furthermore, it holds for a.e.
which implies > 0. (4.21) This essentially contradicts the equality (4.19) and thus the assumption W 1 k > 0 does not hold. Equivalently, it means that for a.e. x ∈ Ω, it holds u k In the same vein, we can prove that u k 0 − u k−1 0 ≤ D k by setting (4.22) and employing the test functionφ = v k 0 − D k + where f + := max {f, 0}. For brevity, we omit the details and leave it to the reader.
Henceforward, we obtain In view of the fact that ḡ Thanks to Lemma 2, one then deduces It suffices to take into account the second term on the right-hand side of (4.24). Observe that we can bound it from above by using the standard inequality 1 + x ≤ exp(x) for x ∈ R. Indeed, it is aided by the sum and integral We now employ the elementary inequality (j + 2) κ ≤ 2 κ−1 (j κ + 2 κ ) ≤ Cj κ for j ≥ 2 to arrive at by virtue of κ > δ0 η+δ1 and 1 < k+2 k+1 < 2. Hence, this way we complete the proof of the theorem.
Cf. [13], if f and R γ k are smooth functions (and so is the functionḡ γ k (t) = R γ k (t) − (η + δ 1 ) t defined above), the limit function u k 0 can be sufficiently smooth to guarantee the essential boundedness of its high-order derivatives. Eventually, by adapting the classical two-scale asymptotic expansion u k ε (x) = u k 0 (x) + εu k 1 x, x ε + ε 2 u k 2 x, x ε + . . . one obtains the following error estimates.
Theorem 6. Assume (A 1 ) holds and suppose that the internal source f and the regularization R γ k are smooth.
Then there exists C > 0 independent of ε such that It is worth noting that the constant C involved in Theorem 6 can depend on k, although the smoothness of f and R γ k may overcome this dependence by a high-order regularity of u k 0 in some sense, e.g. the case of strong or classical solution. This matter will be fully explored in the future, but in principle, this dependence is hard to estimate in the numerical aspect. Therefore, it is practically not necessary to take k very large to get a fine approximation of u ε and it avoids being time-consuming from the computational standpoint, additionally.

Corollary 2.
Under the assumptions of Theorem 3 and Theorem 6, the following rate of convergence holds

Numerical implementation
In this section, we numerically investigate the potential of the developed iterative method in approximating nonlinear elliptic problems, which are described on complex porous domains. Here, we focus on the case α = 0 since this case not only remains nonlinear in the macroscopic problem, but also can be considered as a paradigm for treating other problems of interest. Besides, the case, α > 0, is simple and can be handled in a more straightforward manner since the reaction term converges to 0 as ε tends 0.
For the purpose of illustration, let us consider the problem (1.1) in a two-dimensional unit square Ω = (0, 1) × (0, 1). The highly oscillatory diffusion coefficient is chosen as , and we take R(u) as in (3.15) with p = 2, and the source term is f = 1.
In order to demonstrate the L 2 estimate of the error between u ε and u k ε , we first solve the problem (4.4) for u k ε , which involve solving (4.3) and (4.5) for the cell functions χ i , i = 1, 2, and for the effective diffusion coefficient, A 0 , respectively. We consider (4.3) in the unit cell Y = (0, 1) × (0, 1) with a hole of radius r = 0.4 and porosity, |Y l | = 1 − πr 2 . Moreover, we take the constants η = 0.4 and δ 1 = 1 for the stabilization constant M , cf. Theorem 1. Note that the regularization R is given, according to (3.15), by Eventually, by plugging the cell solutions χ i , i = 1, 2 in (4.5), we can compute the homogenized diffusion coefficient, as follows:  The original problem (1.1) for u ε is solved using the Newton-Raphson method and the P1 standard finite elements on a non-uniform mesh discretization with a size h, satisfying h < ε. Since the mesh is non-uniform, we denote by max h the largest value of the mesh size to qualify the condition h < ε. As ε decreases from 0.5, max h also decreases as shown in Table 1. The iterative scheme for problems (3.4) and (4.4) are solved until the difference between the L 2 estimates of successive iterations is close to zero.
To assess the efficiency of the linearization of (1.1) in (3.4), we take into account the following relative error estimates: The relative error between u k ε,h and u h 0,k are tabulated in Table 2. It indicates that the discrepancy between the solutions of the linearized microscopic problem and the linearized macroscopic problem is good. However, the convergence rate of the Newton's iteration for u ε,h is fast when compared with the convergence rate of the linearization algorithm for u k ε,h . One key result, as demonstrated in Tables 1 and 2, is that the proposed linearization algorithm conveniently promotes the passage from the nonlinear microscopic description to the corresponding macroscopic description, without the need of performing Taylor's expansion for nonlinear terms as is customary in the classical homogenization theory. We also remark that the Newton's iteration usually needs a fine initial guess to attain convergence, while the choice is arbitrary for our linearization scheme.
As to our numerical results, by fixing ε = 0.1 the relative L 2 error between u ε,h and u k ε,h with k = 2 gives 0.0033% and we have illustrated these solutions in Figure 2.  Tables 1 and 2, shows the consistency of our method with the classical homogenization theory, i.e. the convergence of u ε to u 0 , for some k in the iterative scheme.

Concluding remarks
In this work, we have proposed a regularization-and linearization-based scheme to construct efficient approximations of both microscopic and macroscopic problems. Although, for example, in power-law nonlinearity, a geometric regularization parameter is needed to prove the well-posedness of microscopic problem, in practice one can utilize the harmonic progression at the macro-scale to get convergence from the scheme without paying attention to the rate. Note that the harmonic choice of the regularization parameter is rather well-suited to polynomial approximations as deduced in (3.3). We emphasize that the arguments in this paper can be typically applied to single out a reliable approximation of the macroscopic equation if the weak solvability of the microscopic scenario is mathematically known. This approach could be helpful for engineering needs, among several types of linearization methods.
Furthermore, in upcoming works we will attempt to adopt the so-called boundary layers correctors (see [27]) to our context to improve the error estimates in the numerical perspective.

Proof of Lemma 2
The proof is essentially done by induction. Indeed, it trivially holds for k = 3. For any k = n, one now suppose that p n + q n ≤ a n + Our aim is to prove that it still holds true for the case k = n + 1, i.e.