Corrector estimates in homogenization of a nonlinear transmission problem for diffusion equations in connected domains.

This paper is devoted to the homogenization of a nonlinear transmission problem stated in a two-phase domain. We consider a system of linear diffusion equations defined in a periodic domain consisting of two disjoint phases that are both connected sets separated by a thin interface. Depending on the field variables, at the interface, nonlinear conditions are imposed to describe interface reactions. In the variational setting of the problem, we prove the homogenization theorem and a bidomain averaged model. The periodic unfolding technique is used to obtain the residual error estimate with a first-order corrector.

INTRODUCTION

We consider coupled linear parabolic equations describing the diffusion of two species in two different phases of one physical domain separated by a thin periodic interface. The coupling of the species arises via nonlinear transmission conditions at the interface, which model surface reactions. Nonlinear interface reactions are relevant, for instance, in electrochemistry, see, eg, Landstorfer et al1 for adsorption and solvation effects at metal‐electrolyte interfaces, and Efendiev et al2 for electro‐chemical reactions in lithium‐ion batteries.

The characteristic length scale of the periodic cell is given by the homogenization parameter ε>0. The main objective is to derive a macroscopic model for vanishing ε, where both phases are connected sets. The limit bidomain model is given via two coupled parabolic equations defined in the macroscopic domain describing the diffusion of the two species in each phase and reactions at the interface. In the case of connected‐connected domains, we exploit the existence of a continuous extension operator from the periodic domain to the whole domain following.3, 4

A qualitative homogenization result for reaction‐diffusion systems with nonlinear transmission conditions has recently been obtained in Gahn et al.5 The limit in the microscopic equations is derived rigorously in the sense of the two‐scale convergence, however, without corrector estimates. There also exists a vast literature on transmission problems with linear interface conditions, eg, Donato et al6 and Donato and Monsurro.7 See references therein for the case of elliptic equations as well as the extensions of the homogenization result to parabolic equations in Jose8 and to nonlinear monotone transmission conditions in Donato and Le Nguyen.9 For the treatment of oscillating third boundary conditions, we refer to Belyaev et al10 and Oleinik and Shaposhnikova.11

Within elecktrokinetic modeling (see Allaire et al12), in previous studies,13, 14, 15, 16 there were considered generalized Poisson‐Nernst‐Planck (PNP) models over two‐phase domains accounting for interface reactions. The corresponding PDE system obeys a structure of the gradient flow; see, eg, other works.17, 18, 19 The paper20 considers the homogenization over a two‐phase domain for static PNP equations and homogeneous interface conditions. In Kovtunenko and Zubkova,21 residual error estimates for the averaged monodomain solution with first‐order correctors were justified under the simplifying assumption that the flux across the interface is of order O(ε 2).

In this paper, however, we are mainly interested in quantitative asymptotic results supported by corrector estimates. There exist many articles on the derivation of error estimates for different classes of reaction‐diffusion systems, eg, other works,22, 23, 24, 25 exploiting a higher regularity of the limit solution and the continuous extension operator from a perforated domain. Moreover, unfolding‐based error estimates have been proven for linear, elliptic transmission problems in Reichelt,26 for reaction‐diffusion systems with linear boundary conditions in perforated domains in Muntean and Reichelt,27 and for systems with nonlinear interface conditions in a two‐phase domain in Fatima et al.28 The latter results are based on the quantification of the periodicity defect for the periodic unfolding operator in Griso,29, 30 and they hold without assuming higher regularity for the corrector problem.

Our approach uses the periodic unfolding method introduced in Cioranescu et sl31 and further refined in Franců32 and Mielke and Timofte.33 To make our error estimates rigorous, we have to assume higher regularity for the limit solutions as well as for the correctors solving the local cell problems. This additional regularity for the limit problem is in line with established homogenization results by, eg, literature.34, 35, 36 Our result provides residual error estimates with a first‐order corrector of order , which is (generally) optimal for H 1‐estimates up to an Lipschitz boundary, whereas in Fatima et al,28 the error is of order ε 1/4. For this task, we apply the Poincaré inequality in periodic domains (see Lemma 2) and the uniform extension in connected periodic domains (see Lemma 3).

The paper is structured as follows: In Section 2, we formulate the transmission problem and all relevant assumptions. In Section 3, we prove the existence of solutions to our model and provide a priori estimates. In Sections 4 and 5, we define the periodic unfolding operator and provide important properties as well as first asymptotic results. In Section 6, we state and prove our main result on the residual error estimates.

SETTING OF THE TRANSMISSION PROBLEM

For a fixed homogenization parameter ε>0, we consider a macroscopic domain Ω consisting of two subsets , , which are disjoint by a thin interface Γ. The both components are assumed to be connected such that . By , we mean the surface measure of points where the boundaries of and Ω will meet.

We make the following geometric assumptions.

The reference domain is a d‐dimensional hyperrectangle, , ie, it is This assumption suffices to split Ω into periodic cells in (D3).

The unit cell Y=(0,1) consists of two open, connected subsets Y 1 and Y 2, which have Lipschitz continuous boundaries ∂Y 1, ∂Y 2 and are disjoint by the interface Γ=∂Y 1∩∂Y 2. We assume the reflection symmetry, ie, for k=1,…,d, i=1,2. This assumption allows us to define periodic functions on Y in (29). Let n 1 and n 2 denote the unit normal vectors at the respective boundaries ∂Y 1 and ∂Y 2. Every normal is chosen outward from the domain, and it does not depend on scaling by ε.

For ε>0, we introduce the decomposition of a point as into the floor part and the fractional part . According to (1), let the set of integer vectors denote the numbering of local cells inside Ω. We call ε an admissible parameter, if the reference domain Ω from (D1) can be partitioned periodically into the local cells as follows: For a treatment of small boundary layers, see Reichelt.37, lemma 2.3.3

As a consequence of (D1) to (D3), the periodic components and and their interface Γ are determined via By this, the outward normal vectors at coincide with the normal vectors n at ∂Y for i=1,2 and do not depend on the scaling ε. The interface Γ is a Lipschitz continuous manifold.

For admissible ε>0, time t∈(0,T) with the final time T>0 fixed, the space variable in the two‐component domain, we consider a nonlinear transmission problem for , i=1,2, such that The notation ∂ stands for the time derivative, ∇ for the spatial gradient, and “  ·  for the scalar product in . Below, we explain in detail the terms entering the system (4). We note that |Γ|=O(1/ε); therefore, the scaling ε in (4b) appears naturally just compensating the longer interface.

The diffusivity matrices , i=1,2, are symmetric, uniformly bounded and elliptic: There exist such that

The matrices entering (4a) to (4c) are defined as according to the notation (1) and are assumed to be periodic.

In the transmission conditions (4b), the functions , i=1,2, describe interface reactions and are assumed to satisfy

the uniform growth condition: there exists K g>0 such that

the Lipschitz continuity: There exists such that for all , i=1,2.

The linear diffusion equations (4a) are supported by the standard, homogeneous Dirichlet boundary conditions (4c) and the initial data (4d) for given , i=1,2.

We introduce the variational formulation of the problem (4) as follows: find , i=1,2, in the search (solution) space satisfying the initial condition (4d) and the nonlinear equation for all test functions v from the test space The notation in stands for the topologically dual space to , and denotes the duality between them.

WELL‐POSEDNESS

This section provides the existence of weak solutions in the sense of variational formulation for the microscopic problem (8).

(Well‐posedness)

The unique solution to the nonlinear transmission problem (8) exists and satisfies the following a priori estimate: uniformly in ε∈(0,ε 0) for ε 0>0 sufficiently small.

Under assumptions on positivity of the initial data everywhere in , the solution is positive at least locally in time, and at any time under the assumption of the positive production rate from RoubÍček38: where stands for the negative part of the function.

□

To prove existence of the solution, we apply the Tikhonov‐Schauder fixed point theorem. We iterate (8) starting with the suitable initialization , , i=1,2.

For m>m 0, , a solution can be found, which satisfies the initial data (4d) and the linearized equations for all test functions , using the notation for short. We can test (11) with leading to We estimate the integral in the right‐hand side of (12) applying weighted Young inequality with a weight , the trace theorem (25) below, and the growth condition (6): where with a constant K tr from the trace theorem (25) and K g from (6). Expressing the first term in the left‐hand side of (12) by the chain rule as , using the uniform ellipticity (5) of and the estimate (13), this follows For δ<α, applying Grönwall inequality, we obtain and taking in (14) the supremum over t∈(0,T), we conclude Hence, using (6) from (12), it follows uniformly with respect to m→∞ and ε→0, and the continuous embedding of the solution in holds; see Dautray and Lions.39, p509

Therefore, the mapping defined when solving (11) has compact image, and hence, there exists an accumulation point , i=1,2, and a subsequence still denoted by m such that as m→∞ The continuity of in the weak topology is justified using the Lipschitz continuity of the nonlinear term g in (7). Applying the fixed point theorem40, section 4.8, theorem 8.1, p293 and the a priori estimate (9) proves the existence of a weak solution of problem (8).

To prove uniqueness, we consider the difference , i=1,2, of two solutions of (8) with the test function :

The integral is estimated due to the Lipschitz continuity (7) as Then, collecting the expressions (16) and (17), applying the Cauchy‐Schwarz and Grönwall inequalities, we get and hence conclude , which proves .

To prove the nonnegativity of , we decompose the solution into the positive and the negative parts as: and substitute it in the Equation (8) with the test function . The assumption of the positive production rate (10) together with the uniform ellipticity (5) of and the nonnegativity of the initial data lead to the estimate: hence, and . If everywhere in , then at least for t sufficiently small, which follows by the continuity of the solution. This completes the proof.

We note that Theorem 1 can be extended for inhomogeneous diffusion equations (4a), where the uniform upper bound is proved in Gurevich and Reichelt41 for reaction functions distributed over domains .

PERIODIC UNFOLDING TECHNIQUE

Following Cioranescu et al,42 we recall the technique based on the periodic unfolding and averaging operators providing continuous mappings between the components and , i = 1,2, up to the boundaries.

For , the unfolding operator , i=1,2, in the domain is defined by and for , the operator , i=1,2, is performed on the boundary by For φ(x,y)∈L 2(Ω;L 2(Y )), the averaging operator , i=1,2, in the domain is defined by and for φ(x,y)∈L 2(Ω;L 2(∂Y )), the operator , i=1,2, on the boundary is expressed by

Abusing the notation is used for a left inverse operator of T according to Lemma 1 (i), which is also right inverse in the special cases accounting in Lemma 1 (ii). For those functions that belong to , the restriction of the unfolding operator T is well‐defined as the mapping , and for functions in L 2(Ω;H 1(Y )), the restriction of the averaging operator is well‐defined as , where is from (3). We note that the spaces and do not coincide because functions from are discontinuous while they can have jumps across the interface Γ.

The operator properties are collected below in Lemma 1.

(Properties of the operators For arbitrary and (x,y)↦φ(x,y)∈L 2(Ω;H 1(Y )∩L 2(∂Y )), i=1,2, and the extension by zero: for , otherwise for , the following properties hold:

invertibility of T : ;

invertibility of :

for x∈Ω, if φ(y) is a constant or periodic function of the argument y∈Y ,

for x∈Ω, where is the average ;

composition rule: for any elementary function ;

chain rules: εT (∇u)(x,y)=∇(T u)(x,y), and for and φ∈H 1(Ω×Y );

integration rules:

boundedness of T :

□

The property (iib) follows in a straightforward manner from the calculation of for x∈Ω and z∈Y: and the fact that as a consequence of the definition (19a) if for all φ(x,y)∈L 2(Ω;H 1(Y )). The proof of the other properties can be found in other studies.20, 21, 31, 42, 43

ASYMPTOTIC ANALYSIS

In this section, we collect some auxiliary tools used later in the derivation of the residual error estimates.

(Poincaré inequality in periodic domains) For , the following Poincaré inequality holds (see, eg, Cioranescu et al42, 43):

□

We recall the Poincaré inequality for a function φ(y)∈H 1(Y ) in the unit cell with connected subsets Y for i=1,2: Integrating (23) over Ω yields for all φ∈L 2(Ω;H 1(Y )). Choosing φ=T u gives For the left‐hand side, we use the composition rule (iii) as well as . For all (x,y)∈Ω×Y , we have while noting that for all y∈(0,1). This shows, in particular, that is constant for a.e. x∈Ω.

We recall the trace theorem in unit cells for a function φ∈L 2(Ω;H 1(Y )): with K tr>0. After the substitution of φ=T u for the function , there follows (see, eg, Monsurrò44): In particular, repeating the arguments in the proof of Lemma 2, the trace inequality in periodic domains can be shown:

(Uniform extension in connected periodic domains) For , there exists a continuous extension from the connected set to Ω such that in and If u=0 on , then exists satisfying (27).

□

Indeed, the assertion holds in accordance with previous studies,3, 4, 45, chapter 4 and the zero trace at the boundary ∂Ω is argumented in Höpker.46, theorem 3.5

Below, we recall the auxiliary result from Fellner and Kovtunenko20, lemma 2 and Kovtunenko and Zubkova.21, lemma 4.1

(Asymptotic restriction from Ω to For given functions u,v∈H 1(Ω) (which have no jumps across the interface Γ), the asymptotic estimate holds as ε→0 for i=1,2.

Based on the geometric assumptions (D1) to (D4), we define the space of periodic functions in the cells Y by We set the standard cell problem determining , i=1,2, from where the last line in (30c) implies that for i=1,2 and k=1,…,d. In (30), the notation for y∈Y stands for the matrix of derivatives with entries , k,l=1,…,d, and denotes the identity matrix. The system (30) admits the weak formulation: find vector‐functions such that for all test functions . A solution of (31) exists, and it is defined up to a constant in Y .

Based on the solution N of the cell problem (31), the diffusivity matrices A admit the following asymptotic representation formulated in the lemma below; see Fellner and Kovtunenko20 and Kovtunenko and Zubkova.21

(Asymptotic formula for periodic diffusivity matrices)

For the solution N of the cell problem (31), the following representation holds: with given by the averaging and it is a symmetric d‐by‐d matrix: The d‐by‐d matrix B (y) is periodic and has the following divergence form in the cell Y : Its components are skew‐symmetric: the matrix B is divergence‐free: and the average . At the interface, the condition holds:

Assume that N ∈W 1,(Y ). For varying function and fixed , the following integral form corresponding to the averaged equation (50): with the help of the corrector is approximated as follows:

□

For the vector‐valued solution N of (31), the representation (32) follows from the Helmholtz theorem; see Zhikov et al.36, section 1.1 The interface condition (35) is obtained after substitution of (32) into (30b).

Let and be given. To prove (37), we rewrite in (36) in virtue of the integration rules from Lemma 1 in the microvariable y:

For the constant matrix, the identity holds. Then, expressing from (32), using the product rule the chain rule , and the notation of the corrector , we rearrange the following terms: Taking into account this formula, is performed equivalently by with the integral is written component‐wisely as follows: Recalling the definition of B and the fact that it is divergence‐free, the term is integrated by parts as follows: After substitution of (40) in (39), the integral over Γ disappears due to the interface condition (35). The integral over ∂Y ∖Γ vanishes after rewriting the integral again in macrovariables because of v =0 on and because jumps across the cell boundary of v and are zero (by assumed H 3‐, hence, C 1‐smoothness of ), while B is periodic.

The integral over Ω×Y in (40) can be rewritten using the zero average as follows: where We rewrite and in the macrovariable x in all local cells using the integration rules (20) and (21) and then apply to the result the Cauchy‐Schwarz inequality and the Poincaré inequality (23).

Below, the indices k,l,m will refer to both x as well as y coordinates. We are starting from where it is for all : with . First, there are some constants such that Similarly, there exists K 3>0 such that We substitute in (39) the expression of from (40) and use (35), such that Rewriting the integrals in microvariables with the help of the integration rules (20) and (21), the following estimate takes place with K 4>0: Using the estimates (41) and (42), from (44) after integration over time, it follows (37) that proves the assertion of Lemma 5.

With these preliminaries, in the next section, we homogenize the nonlinear transmission problem (8) as ε→0.

THE MAIN HOMOGENIZATION RESULT

We state the averaged bidomain diffusion problem determining the functions , i=1,2, in the time‐space domain (0,T)×Ω from where the effective matrices are defined in (33). It implies the variational formulation: find in the space such that it satisfies the initial condition (45c) and the following nonlinear equation: for all text functions . In (46), the notation ⟨·,·⟩Ω implies the duality between H 1(Ω) and its topologically dual space H 1(Ω)∗.

The solvability of (46) can be obtained in the same way as for (8) due to the uniform boundedness (6) and the continuity (7) of the nonlinear term g . Moreover, the a priori estimate like (9) holds (for i=1,2): In Theorem 2, we need smoothness of the macroscopic solution and the uniform boundedness of N and of its gradient in order to prove the residual error estimate, which is a standard assumption for cell problems; see, ie, Zhikov et al.36, section 5.6, theorem 5.10 These assumptions might be weekend just to get a two‐scale convergence to the homogenized problem.

(Residual error estimate) Let the cell problem (31) obey the Lipschitz continuous solution N ∈W 1,(Y ), and the macroscopic solution be such that , , i=1,2. Then the solution of the inhomogeneous problem (8) and the first‐order corrector to the solution of the averaged problem (46) given by where is a periodic extension of N to Y, satisfy the residual error estimate: where Err12 is determined in (66).

We start with derivation of an asymptotic equation for the difference (see (51)). Multiplying the diffusion equation (45a) with a test function , integrating it over , it follows the variational equation in two subdomains for i=1,2: The integration by parts in (49) due to the Dirichlet condition (45b) leads to

We choose and . With a special choice of v , it can be equal to v. For test functions , i=1,2, we subtract (50) from the inhomogeneous equation (8): and gather the terms as follows: where the following notation was used Err0 is given by the formula (37) from Lemma 5, and other residual error functions Err, k=1,2,3, in the right‐hand side of (51) will be introduced and estimated next.

We use the Cauchy‐Schwarz inequality and the expansion of the time‐derivative of the corrector implying that Applying to the restriction operator from Lemma 4, then using the boundedness (6) and the Lipschitz continuity (7) for g leads to and the further error function (with K 7=|Γ|L g)

In the following, we aim at substitution of v by piecewise constant average for , j=1,2. For this task, we decompose I in (52) as follows: with the terms defined as We apply the integration rule (20b) to the first term of Err4 and rewrite the third term using . Based on the boundedness (6) of g , from the Cauchy‐Schwarz inequality, it follows the error estimate where . Here, we have used the Poincaré inequality (22), following the trace inequality in periodic domains (26) such that Applying Young inequality to J implies that Due to the Lipschitz continuity (7) of g , using the mean inequality application of the integration rule (21c) and the trace inequality (25) proceeds further because of (see Cioranescu et al43, proposition 2.17) where First, we estimate Err5 in (57). Since , according to Griso,29, formula (3.4) the auxiliary estimate for the term in Err5 holds: Therefore, from the trace theorem (24) in Ω×Y and (21b), we have and the term Err5(v,ε) is estimated by

Let η Ω(x) be a smooth cutoff function with a compact support in Ω and equals one outside an ε‐neighborhood of the boundary ∂Ω such that and . For further use, we employ the following functions expressed equivalently in two ways as where is the uniform extension of according to Lemma 3.

We will derive the estimates for with the help of substitution of the test function v=w from (59) into the expressions for Err(v,ε), k=0,1,…,5. This implies the following structure of the bounds: where the terms are defined by means of According to the uniform estimate (9) in Theorem 1 and the continuous extension (27), we have following that all α =O(1) and U =O(1) for k=0,1,…,5.

The asymptotic equation (51) tested with the function v=w from (59) leads to with the following two terms: We note that M is not an error term; in contrary, it enters with the factor −δ 1 the left‐hand side of the estimate (65) following later.

Err6 is estimated by integration by parts with respect to time after using Young inequality and the continuous embedding which implies that where The term is evaluated by Young inequality with the weight δ 1>0 and using the boundedness property of A with the upper bound β from (5) as where It follows where We note that U 7=O(1), in particular, because 1−η Ω≠0 on a O(ε)‐set using the fact that 1−η Ω≠0 on a set of measure O(ε), thus compensating ∇η Ω=O(ε −1) here.

Therefore, using the inequality (57) for J (⟨T w ⟩) and the uniform positive definiteness (33) of A with the lower bound α>0, from (62), we arrive at the estimate where , and After summation over i=1,2 we rearrange the terms such that where , , and the error Err9 implies After taking the supremum over time, using the embedding theorem (63), we estimate the first term in the left‐hand side of (65) by the lower bound We continue the estimate (65) by taking δ 1 small enough such that γ>0. Therefore, applying Grönwall inequality leads to As a consequence, from (65) and the embedding theorem (63), we conclude with the estimate which finishes the proof.

DISCUSSION

Compared with previous results in the literature on multiscale diffusion equations, in the paper, we derived the macroscopic bidomain model that is advantageous for numerical simulation; we first proved the homogenization result supported by residual error estimate of the asymptotic corrector due to the nonlinear transmission condition at the microscopic level, which appears to describe interface chemical reactions.

For further generalization of the obtained result, we suggest to consider the case of connected‐disconnected domains and . While in the connected domain the uniform extension is applicable, the disconnected domain allows a discontinuous Poincaré estimate (see Kovtunenko and Zubkova21).

CONFLICT OF INTEREST

This work does not have any conflicts of interest.