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Abstract
The paper provides a rigorous homogenization of the Poisson–Nernst–Planck problem stated in an inhomogeneous domain composed of two, solid and pore, phases. The generalized PNP model is constituted of the Fickian cross-diffusion law coupled with electrostatic and quasi-Fermi electrochemical potentials, and Darcy's flow model. At the interface between two phases inhomogeneous boundary conditions describing electrochemical reactions are considered. The resulting doubly non-linear problem admits discontinuous solutions caused by jumps of field variables. Using an averaged problem and first-order asymptotic correctors, the homogenization procedure gives us an asymptotic expansion of the solution which is justified by residual error estimates.
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1. Introduction
The paper is devoted to the mathematical study of homogenization of a non-linear diffusion model in a two-phase domain.
The Poisson–Nernst–Planck (PNP) model extends the diffusion law due to electro-kinetic phenomena. Namely, we consider cross-diffusion of multiple charged species coupled with an overall electrostatic potential. Motivated by the physical nature, species concentrations satisfy the total mass balance and the positivity conditions. Following [Citation1–4], this approach generalizes the classic PNP model.
The problem under consideration is characterized by the following issues.
We describe a two-phase medium with a micro-structure consisting of solid and pore phases which are separated by a thin interface. The corresponding geometry is represented by a disconnected domain. Therefore, field variables defined in the two-phase domain allow discontinuity with jumps across the interface.
A special interest of our consideration is the interface between the two phases because of electrochemical reactions that occur here. At the interface we state mixed, inhomogeneous Neumann and Robin-type conditions. Diffusion fluxes and the electric current are assumed continuous across the phase interface. The key issue is that the inhomogeneous boundary fluxes are to be described by non-linear functions of the field variables.
From a mathematical point of view, we examine a mixed system of partial differential equations of the parabolic-elliptic type. The governing equations are non-linear, coupled, and differ on the two phases. The non-linearity is due to the presence of electrochemical potentials in the model. The solvability of classic PNP systems was studied in [Citation5,Citation6]. Based on a general approach from [Citation7,Citation8], in the previous works [Citation9–11], we proved existence theorem for the generalized PNP problem and derived a-priori estimates.
Homogenization of diffusion equations is widely studied in the literature, see, for instance [Citation12–17] for adopted approaches. Most of the asymptotic results concern either linear equations, or homogeneous Neumann conditions excluding interface reactions, which are of primary importance in electro-chemistry. For possible transmission conditions stated at the interface we refer to [Citation18–20]. Homogenization of classic PNP equations was studied in [Citation21–23]. A homogenization procedure in a two-phase domain for steady-state Poisson–Boltzmann equations and homogeneous Neumann boundary conditions was investigated in [Citation24]. In the present work we continue this approach to the inhomogeneous conditions in the dynamic case. We rely on hydrostatic setting of the non-stationary problem, which is typical, e.g. for modelling of Li-ion batteries [Citation25]. For homogenization accounting for velocity fields, we refer to [Citation26,Citation27].
The difficulty of the homogenization procedure is caused by the two-phase domain. Typically, homogenization problems are considered in a perforated domain. In contrast, we describe a discontinuous prolongation from the perforated domain inside solid particles following the approach of [Citation28]. In this respect, the two-phase homogenization procedure differs from a perforated domain case. To describe jumps of the field variables across the interface and interface reaction terms, we will specify their suitable asymptotic orders.
To derive an averaged model, typically, the two-scale convergence is applied. As an advantage, we endow our asymptotic expansion with residual error estimates.
As the result of homogenization of the PNP model, we obtain an averaged model consisting of linear parabolic-elliptic equations and supported by first-order correctors. The correctors appear due to oscillating and interface data expressed by solutions of auxiliary cell problems in a unit cell. Respectively, there are three correctors given with respect to:
the periodic boundary function of the electric current at the phase interface;
the periodic matrix of permittivity;
the periodic matrices of diffusivity.
The paper has the following structure. Section 2 contains a brief description of the unfolding method: definitions and main properties. In Section 3, we formulate the PNP problem and describe its solution. Section 4 accounts for auxiliary cell problems. In Section 5, a homogenization procedure is introduced and proved rigorously. By this, the averaged problem is formulated and supported by error estimates of the corrector terms.
2. Unfolding technique
Let Ω be a domain in , where
, with the smooth boundary
and the unit normal vector ν, which is outward to Ω. We consider the unit cell
consisted of the isolated solid part
and the complementary pore part
such that
and
. The interface
is assumed to be a smooth continuous manifold with a unit normal vector ν. We set ν outward to ω, thus inward to Π.
For a small every spacial point
can be decomposed as follows
(1)
(1) into the floor part
and the fractional part
. There exists a bijection
implying a natural ordering, and its inverse is
. Based on (Equation1
(1)
(1) ), we can determine a local cell
with the index
, such that
, and
are the local coordinates with respect to the cell
.
Let be the set of indexes of all periodic cells contained in Ω, and
be the union of these cells. For every index
, after rescaling
, the local coordinate
determines the solid particle such that
with the smooth boundary
. Its complement composes the pore
by analogy with
.
Gathering over all local cells, we define the multi-component domain of periodic particles (the solid phase) denoted by with the union of boundaries
and the unit normal vector ν to each of
. The Hausdorff measure
of the interface
is of the order
due to
and the cardinality
. We denote
, which is a perforated domain. Adding a thin layer
, possibly attached to the external boundary
, composes the pore phase
.
For fixed , a two-phase medium associated to the disconnected domain
with the external boundary
and the interface
is considered, see an example geometry in Figure .
Figure 1. A two-phase domain consisted of solid particles and the pore space
with the phase interface
.
![Figure 1. A two-phase domain consisted of solid particles ωε and the pore space Qε with the phase interface ∂ωε.](/cms/asset/46772f24-2264-4c2c-b36f-2ee8a2b9bea1/gapa_a_1600676_f0001_oc.jpg)
Following [Citation29,Citation30] and based on the decomposition (Equation1(1)
(1) ), we introduce two linear continuous operators: the unfolding operator
, defined by
(2)
(2) and its left-inverse operator
called the averaging operator:
(3)
(3) where
stands for the Hausdorff measure of the set Y in
. We note that
in (Equation3
(3)
(3) ) is discontinuous across
and
. In the homogenization theory, usually x refers to as a macro-variable, y as a micro-variable, and
as the two-scale variables.
Lemma 2.1
Properties of the operators ![](//:0)
and ![](//:0)
in the domain
For arbitrary functions the following properties hold:
(4a)
(4a)
(4b)
(4b)
(4c)
(4c)
(4d)
(4d)
(4e)
(4e)
(4f)
(4f)
Proof.
(i) For and
, we calculate straightforwardly
. For
and
according to (Equation1
(1)
(1) ), the definitions (Equation2
(2)
(2) ) and (Equation3
(3)
(3) ) with
we have
since
, hence (Equation4a
(4a)
(4a) ) holds. The assertion for
can be checked.
(ii) The identity (Equation4b(4b)
(4b) ) is obvious.
(iii) The proof of (Equation4c(4c)
(4c) ) is known (see [Citation29, Section 2]). In the boundary layer, we derive straightforwardly (Equation4d
(4d)
(4d) ) from (Equation2
(2)
(2) ) and (Equation3
(3)
(3) ).
(iv) Taking first q=f=h, then in (Equation4c
(4c)
(4c) ) and (Equation4d
(4d)
(4d) ), summing them, and using
due to the chain rule
, we arrive at (Equation4e
(4e)
(4e) ) and (Equation4f
(4f)
(4f) ). This completes the proof.
A function given in the two-phase domain allows discontinuity across the interface
, see zoom in Figure . In each local cell
we distinguish the negative face
as the boundary of the particle
, and the positive face
as the opposite part of the boundary of the pore
. Gathering over all local cells establishes the positive and negative faces of the interface as
. We set the interface jump of f across
by
(5)
(5) where the corresponding traces of f at
are well defined, see [Citation31, Section 1.4]. Analogously, we define the interface jump for a function
in the unit cell as
.
Motivated by the traces, we extend to the interface the unfolding operator
by
(6)
(6) and similarly the averaging operator
,
(7)
(7) Their properties are stated below in the manner of Lemma 2.1.
Lemma 2.2
Properties of the operators ![](//:0)
and ![](//:0)
at the interface
For arbitrary functions the following properties hold:
(8a)
(8a)
(8b)
(8b)
(8c)
(8c)
(8d)
(8d)
Proof.
The proof of assertions (i) and (ii) is similar to the proof in Lemma 2.1. The proof of (Equation8c(8c)
(8c) ) is known (see [Citation29, Section 4]). Taking q=f in (Equation8c
(8c)
(8c) ) immediately follows formula (Equation8d
(8d)
(8d) ) in (iv).
The geometric construction of the operators and
in this section will be used further for homogenization over
and
as
.
3. Problem formulation
We formulate a generalized Poisson–Nernst–Planck system depending on a fixed parameter , see [Citation9–11]. We consider the number n of charged species with specific charges
, molar masses
, volume factors
, and unknown concentrations
for
and
. By
we denote the overall electrostatic potential. The two-phase medium introduced in Section 2 will be characterized below separately in the pore phase
and the solid phase
.
For the time-space variables in
with a fixed final time
, we consider the following governing equations for species
:
(9a)
(9a)
(9b)
(9b)
(9c)
(9c)
(9d)
(9d)
(9e)
(9e)
The indicator function
is equal to 1 in
, and 0 in
. The Equation (Equation9c
(9c)
(9c) ) contains the Boltzmann constant
, the temperature Θ, the Avogadro constant
, and
in (Equation9c
(9c)
(9c) ) allows us to average the non-linear diffusion fluxes (see (Equation71
(71)
(71) )). The fluxes contain the flow velocity following e.g. [Citation3,Citation4], and the dependence of potentials on the fluid pressure is due to the works by Dreyer (see [Citation1,Citation2]). The Equations (Equation9b
(9b)
(9b) )–(Equation9d
(9d)
(9d) ) will be not solved with respect to electro-chemical potentials
, flow velocity vector
with the drug coefficient η, and the pressure
, but rather reduced within a weak formulation (see (Equation22
(22a)
(22a) )). Conversely, after finding
and
, all the entropy variables
,
,
can be restored from the Equations (Equation9c
(9c)
(9c) ) and (Equation9d
(9d)
(9d) ) supported by suitable boundary conditions.
In (Equation9e(9e)
(9e) ) and (Equation9b
(9b)
(9b) ) the d-by-d matrices A and
for
imply the electric permittivity and diffusivity, respectively. They can be discontinuous in the two-phase unit cell
and satisfy the following assumptions.
for
is uniformly bounded and symmetric positive definite (spd) matrix:
(10)
(10)
for
are uniformly bounded and elliptic matrices: there exist
such that
the mass balance needs a symmetric positive definite (see (Equation13
(13)
(13) ) below) matrix
for
such that:
(11)
(11)
It is worth noting that conditions (Equation11(11)
(11) ) together with (Equation14a
(14a)
(14a) ) below are sufficient to conserve the mass within the laws (Equation9b
(9b)
(9b) )–(Equation9d
(9d)
(9d) ) as follows:
For homogenization reason, we assume that the diffusivity matrices
from (Equation11
(11)
(11) ) admit the asymptotic decomposition as follows
(12)
(12) with d-by-d matrices
,
and a d-by-d uniformly bounded, symmetric positive definite matrix D such that
(13)
(13) The oscillating matrices
and
in the Equations (Equation9b
(9b)
(9b) ) and (Equation9e
(9e)
(9e) ) are defined in Ω, and they are periodic in
.
A constant C>0 in (Equation9c(9c)
(9c) ) stands for the summary concentration. For the physical consistency, species concentrations need to satisfy in pores
:
(14a)
(14a)
(14b)
(14b)
The system (Equation9(9a)
(9a) ) is supported by the initial condition for
:
(15)
(15) where the initial data satisfy the relations in the manner of (Equation14
(14a)
(14a) ) in pores
:
(16)
(16) For given functions
and
the Dirichlet boundary conditions are:
(17)
(17) with the boundary data satisfying the similar relations and compatibility:
(18)
(18) The most delicate part of modelling is the interface conditions on
:
(19a)
(19a)
(19b)
(19b)
where the jump across
is defined in (Equation5
(5)
(5) ). The notation
and
implies the pair of traces at the phase interface
. The function
denotes the electric current through the interface in the unit cell, and
in (Equation19b
(19b)
(19b) ) is periodic at
. The capacitance density
. The equality in (Equation19b
(19b)
(19b) ) implies that the potential jump is asymptotically small
in the electric double layer. The factor
in (Equation19a
(19a)
(19a) ) is used in Theorem 5.1 for averaging of the nonlinear, thus non-periodic interface data (see (Equation72
(72)
(72) )), and the factor
in (Equation19b
(19b)
(19b) ) will be explained later in (Equation24
(24)
(24) ). For modelling and numerical simulations of data for scaling of potentials, interface and boundary conditions, we refer to [Citation25].
In (Equation19a(19a)
(19a) ), the functions
,
,
, describing the boundary fluxes of species with respect to the traces
and
of the variables
and ϕ, should satisfy
(20a)
(20a)
(20b)
(20b)
(20c)
(20c)
The example of
satisfying all assumptions (Equation20
(20a)
(20a) ) can be found in [Citation9,Citation10], e.g.
and
.
A weak formulation of the generalized PNP problem is the following one: Find and
such that for
:
(21a)
(21a)
(21b)
(21b)
which satisfy the Dirichlet boundary conditions (Equation17
(17)
(17) ), the initial conditions (Equation15
(15)
(15) ), the total mass balance and positivity conditions (Equation14
(14a)
(14a) ), and fulfil the equations:
(22a)
(22a)
(22b)
(22b)
for all test functions
and
such that
on
and
on
. In (Equation22a
(22a)
(22a) ) the following notation was used for short:
(23)
(23) The time-derivative in (Equation22a
(22a)
(22a) ) is understood in the weak sense such that
The factor
in the left-hand side of (Equation22b
(22b)
(22b) ) comes from the discontinuous Poincaré inequality, see [Citation28, Lemma 3.3], that holds for
with f=0 on
:
(24)
(24)
Under the assumptions made here, the following theorem is based on [Citation9,Citation10].
Theorem 3.1
Well-posedness
(i) There exists a solution (Equation21(21a)
(21a) ) of the generalized Poisson–Nernst–Planck problem (Equation22
(22a)
(22a) ) satisfying the total mass balance (Equation14a
(14a)
(14a) ). The positivity (Equation14b
(14b)
(14b) ) is guaranteed locally at least for small
for all
where the uniform bound is provided by the local in time positivity
of the limit solution of (Equation64
(64a)
(64a) ). Moreover, if instead of (Equation11
(11)
(11) ) the stronger assumption
is imposed, then the non-negativity
is guaranteed globally for all
.
(ii) The solution satisfies the following a-priori estimates, which are uniform in for
sufficiently small, with constants
(25a)
(25a)
(25b)
(25b)
4. Asymptotic analysis
We aim to homogenize the generalized PNP problem (Equation22(22a)
(22a) ) and to get residual error estimates. This task needs the asymptotic analysis as
.
In the following, the Poincaré and trace inequalities will be used. For functions defined in a connected domain
, there exists
such that
(26)
(26) In the particles
, applying to (Equation26
(26)
(26) ) with
the averaging operator
such that
and using the integration rules (Equation4e
(4e)
(4e) ) and (Equation4f
(4f)
(4f) ) provides
(27)
(27) In the pore phase, for
, f=0 on
, the Poincaré inequality holds
(28)
(28) In the following, we write a unique Poincaré constant
in (Equation26
(26)
(26) )–(Equation28
(28)
(28) ) for short.
For a discontinuous across the interface function
, the trace theorem provides the following estimate with a constant
:
(29)
(29) For
in the two-phase domain such that
, applying the trace theorem and the integration rules (Equation4e
(4e)
(4e) ), (Equation4f
(4f)
(4f) ), and (Equation8d
(8d)
(8d) ), from (Equation29
(29)
(29) ) it follows
(30)
(30)
Based on [Citation13,Citation24], we formulate an auxiliary lemma for homogenization over the pore part
of the reference domain Ω.
Lemma 4.1
Asymptotic formula for restriction to pores
For given functions which are continuous over the interface
the asymptotic representation in the pore space
with the porosity
holds as
(31)
(31)
4.1. Cell problems
For homogenization of the periodic function g and periodic matrices A and D, three auxiliary problems below are formulated in the two-phase unit cell .
First, for the interface data g we set the cell problem for as follows:
(32a)
(32a)
(32b)
(32b)
(32c)
(32c)
Using the space of periodic functions
we get the weak formulation of (Equation32
(32a)
(32a) ): Find
such that
(33)
(33) for all test functions
. Based on the standard elliptic theory, there exists a solution Λ defined up to a constant value in the cell Y .
Lemma 4.2
Asymptotic formula for periodic interface data
For a given function and fixed
a periodic function
defined in (Equation33
(33)
(33) ) satisfies the following asymptotic relation:
(34)
(34)
for all test functions
such that
on
.
Proof.
For such that
on
, we multiply (Equation32a
(32a)
(32a) ) with
and integrate by parts for
using (Equation32b
(32b)
(32b) ) such that
After integration of this relation over
, using the periodicity in (Equation32c
(32c)
(32c) ) for
on
, we get
(35)
(35)
Adding to the first integral over
in the left-hand side of (Equation35
(35)
(35) ) the term in
, which is of the order
, we apply to (Equation35
(35)
(35) ) the integration rules (Equation4f
(4f)
(4f) ) and (Equation8c
(8c)
(8c) ) from Section 2. The resulting integral in the right-hand side of (Equation35
(35)
(35) ) is integrated by parts in
using
on
such that
where the factor
is cancelled according to (Equation4f
(4f)
(4f) ), and
. It follows (Equation34
(34)
(34) ) and finishes the proof.
Based on Lemma 4.2, the corrector will appear in expansion (Equation66b
(66b)
(66b) ) of the solution
of the inhomogeneous equation (Equation22b
(22b)
(22b) ) after homogenization.
Second, for the permittivity matrix we formulate the following boundary value problem for a vector-function
in the two-phase unit cell:
(36a)
(36a)
(36b)
(36b)
(36c)
(36c)
In (Equation36
(36a)
(36a) ), the divergence
is taken for every
, the notation
for
stands for the matrix of derivatives with entries
for
, and
is the identity matrix.
The weak form of (Equation36(36a)
(36a) ) implies: Find
such that
(37)
(37) for all
. A solution Φ exists up to a constant in the cell Y .
Based on Φ, another corrector will appear in the asymptotic expansion (Equation66b(66b)
(66b) ) as argued in the following lemma.
Lemma 4.3
Asymptotic formula for periodic permittivity matrix
For the solution Φ of the cell problem (Equation37
(37)
(37) ) the following representation holds:
(38)
(38) where the constant d-by-d matrix
is given in the cell Y by the averaging
it is symmetric positive definite:
(39)
(39) The d-by-d matrix
in (Equation38
(38)
(38) ) has the form in
(40)
(40) which components are skew-symmetric:
(41)
(41) the average
and the matrix
is divergence-free as follows
(42)
(42) and satisfies the following conditions at the interface:
(43)
(43)
Assume that the solution of (Equation36
(36a)
(36a) ) is such that Φ and
are uniformly bounded in
. For given functions
and
the following asymptotic formula holds with an arbitrary weight
(44)
(44) where the notation
, and
is a smooth cut-off function supported in
and equals one outside an ϵ-neighbourhood of
.
Proof.
(i) For the vector-valued solution Φ of (Equation37(37)
(37) ), the representation (Equation38
(38)
(38) ) with properties (Equation39
(39)
(39) )–(Equation42
(42)
(42) ) follows from the Helmholtz theorem, see [Citation17, Section 1.1]. The interface conditions (Equation43
(43)
(43) ) are obtained after substitution of (Equation38
(38)
(38) ) into (Equation36b
(36b)
(36b) ) because of
.
(ii) Let and
be given. To prove (Equation44
(44)
(44) ), we rewrite
in virtue of the integration rules (Equation4f
(4f)
(4f) ) and (Equation8c
(8c)
(8c) ) in the micro-variable y:
(45)
(45)
For the constant matrix
holds. Then, expressing
from (Equation38
(38)
(38) ), using the product rule
, the chain rule
, and the notation
, we rearrange the following terms:
Taking into account this formula,
in (Equation45
(45)
(45) ) is equivalent to:
(46)
(46)
with the integral
written component-wisely as follows:
Recalling
from (Equation40
(40)
(40) ), we integrate by parts
and use the fact that
is divergence-free according to (Equation42
(42)
(42) ) such that
to get
(47)
(47)
After integration by parts the second time and rearranging the mixed derivatives
such that
because
is skew-symmetric as written in (Equation41
(41)
(41) ), we proceed (Equation47
(47)
(47) ):
where
.
Substituting the expression of into (Equation46
(46)
(46) ) and using the formula at
:
following from (Equation43
(43)
(43) ) and
, with the help of the integration rules (Equation4f
(4f)
(4f) ) and (Equation8c
(8c)
(8c) ) we rewrite
again with respect to the macro-variable x in the form:
(48)
(48)
where the last two terms in the integral over
have the asymptotic order
, and
is transformed to the integral over
such that
Here, the factor ϵ appears due to the integration rule over the boundary
analogously to (Equation8c
(8c)
(8c) ), the chain rule gives
and
, while in the second term ϵ appears since
(49)
(49) By this, the factor
is cancelled by division by
in (Equation46
(46)
(46) ).
We estimate the interface term in the integral over in the right-hand side of the Equation (Equation48
(48)
(48) ) by Young's inequality with a weight
as follows:
(50)
(50)
since
. Applying Green's formula in the boundary layer
and using
on
leads to the asymptotic expansion of the boundary term:
(51)
(51)
Here the ϵ-order is due to the fact that
, the uniform boundedness of
and the chain rule
according to (Equation49
(49)
(49) ).
Gathering in (Equation48(48)
(48) ) the asymptotic terms of the same order ϵ and accounting for formulas (Equation50
(50)
(50) ) and (Equation51
(51)
(51) ), the following estimate takes place with some K>0:
(52)
(52)
For a cut-off function
supported in
we set
such that
in
, the jump
at
, and
(53)
(53) From (Equation52
(52)
(52) ) and (Equation53
(53)
(53) ) if follows (Equation44
(44)
(44) ) and the assertion of Lemma 4.3.
Third, for a diffusivity matrix D corresponding to the assumption (Equation12(12)
(12) ) in Theorem 5.1 below, in analogy with (Equation36
(36a)
(36a) ), we establish the cell problem for
:
(54a)
(54a)
(54b)
(54b)
(54c)
(54c)
The system (Equation54
(54a)
(54a) ) differs from (Equation36
(36a)
(36a) ) by the interface condition and implies the following weak formulation: Find a vector-function
such that
(55)
(55) for all test functions
. A solution of (Equation55
(55)
(55) ) exists and is defined up to a piecewise constant in
. Moreover, since
is assumed, this fact follows that N=−y and
in ω. Based on N, the following lemma justifies the use of the corrector
in the formula (Equation66a
(66a)
(66a) ).
Lemma 4.4
Asymptotic formula for periodic diffusivity matrix
For the solution N of the cell problem (Equation55
(55)
(55) ) the following representation holds:
(56)
(56) where the d-by-d matrix
is constant in the cell Y and given by
it is symmetric positive definite:
(57)
(57) The d-by-d matrix
has the following form in
(58)
(58) Its components
are skew-symmetric,
and
is divergence-free in the manner of (Equation41
(41)
(41) ) and (Equation42
(42)
(42) ). At the interface the conditions hold
(59)
(59)
Assume
. For
such that
on
and arbitrary
the following asymptotic formula with
holds
(60)
(60)
Proof.
The proof is analogous to those from the previous Lemma 4.3 until (Equation47(47)
(47) ). Indeed, we derive similar to (Equation45
(45)
(45) ) and (Equation46
(46)
(46) ) formulas in micro-variables:
(61)
(61)
with
and
. Likewise (Equation47
(47)
(47) ), integration by parts of
follows that
(62)
(62)
After substitution of (Equation62
(62)
(62) ) in (Equation61
(61)
(61) ), the integral over
disappears due to the interface condition (Equation59
(59)
(59) ).
Returning to the micro-variables x with the help of the chain rule , the second term in the integral over
in (Equation61
(61)
(61) ) has the asymptotic order
. The integral over
in (Equation62
(62)
(62) ) divided by
is transformed to the integral over
with the factor
, and after integration by parts in the boundary layer
, it is of the order
, too.
The principal difference from Lemma 4.3 consists in estimation of the domain integral in .
By adding and subtracting the averaged values, we rewrite equivalently
using the property
, and
We rewrite
and
in the macro-variable x in all local cells using the integration rules (Equation4c
(4c)
(4c) ) and (Equation8c
(8c)
(8c) ), applying the chain rule
to
and to
(see (Equation49
(49)
(49) )), then apply to the result the Cauchy–Schwarz inequality and the Poincaré inequality (Equation27
(27)
(27) ). First, there are some constants
and
such that
where we have used the fact that the integral over the boundary layer
of
is zero due to the definition of the operator
in
. Similarly, there exists
such that
Finally, we integrate the estimate of
over the time
for further use.
The functions and
will associate the averaged solution in the homogenization problem presented in the next section.
5. The main homogeneous result
In this section, we establish the averaged PNP equations for the functions in the time-space domain
as follows:
(63a)
(63a)
(63b)
(63b)
which are supported by the Dirichlet boundary and initial conditions:
(63c)
(63c)
In (Equation63
(63a)
(63a) ), the averaged matrices
and
are from Lemma 4.3 and Lemma 4.4, the matrix D is from (Equation12
(12)
(12) ), the vectors N and Φ are the solutions of the two-phase cell problems (Equation55
(55)
(55) ) and (Equation37
(37)
(37) ), respectively.
From the standard existence theorems on elliptic and parabolic systems, the solution and
of the linear problem (Equation63
(63a)
(63a) ) exists and fulfils the following variational equations:
(64a)
(64a)
(64b)
(64b)
for all test functions
and
.
The main result of this paper is the following theorem.
Theorem 5.1
Averaged problem and correctors
Let the solutions N, Φ of the two-phase cell problems (Equation55(55)
(55) ), (Equation37
(37)
(37) ), and
be uniformly bounded in
the averaged solutions
and
. Then a solution
of the inhomogeneous PNP problem (Equation22
(22a)
(22a) ) and the solution
of the homogeneous PNP problem (Equation64
(64a)
(64a) ) satisfy the residual error estimates:
(65)
(65) with the norm
defined in (Equation25a
(25a)
(25a) ), and the approximate functions are
(66a)
(66a)
(66b)
(66b)
In (Equation66
(66a)
(66a) ), the vector Λ is a solution of the two-phase cell problem (Equation33
(33)
(33) ), and
is the cut-off function from Lemmas 4.3 and 4.4.
Proof.
Based on the asymptotic results of Section 3, we will prove the error estimates (Equation65(65)
(65) ). In particular, this will justify the averaged problem (Equation63
(63a)
(63a) ).
Estimate of . We start with derivation of an asymptotic equation for
as
. We apply to
Green's formulas on the pore phase:
(67a)
(67a) for all
such that
on
, and on the solid phase:
(67b)
(67b) for all
. Summing up the Equations (Equation67
(67a)
(67a) ), using the diffusion equation (Equation63a
(63a)
(63a) ) and the continuity of
across
, the variational problem (Equation64a
(64a)
(64a) ) in Ω can be expressed equivalently over the two-phase domain as follows:
(68)
(68)
for all discontinuous over
test functions
such that
on
. Further, we employ the asymptotic arguments as
.
We apply to the left-hand side of (Equation68(68)
(68) ) the asymptotic formula (Equation60
(60)
(60) ) from Lemma 4.4, which implies:
(69)
(69) where
is defined in (Equation66a
(66a)
(66a) ). In virtue of the relation
then (Equation69
(69)
(69) ) can be rewritten in terms of
in the asymptotically equivalent form:
(70)
(70) We continue with an asymptotic expansion of the perturbed problem (Equation22a
(22a)
(22a) ). Due to the assumption (Equation12
(12)
(12) ) on the diffusivity matrices and the uniform estimate
, which follows that
for
, the Equation (Equation22a
(22a)
(22a) ) is expressed in the asymptotic form:
(71)
(71)
Since
, the interface integral over
in (Equation71
(71)
(71) ) is estimated by Young's inequality due to the boundedness property (Equation20c
(20c)
(20c) ) and the trace theorem (Equation30
(30)
(30) ):
(72)
(72)
Next, we subtract the Equation (Equation70
(70)
(70) ) from (Equation71
(71)
(71) ) and utilize (Equation72
(72)
(72) ) to obtain that
(73)
(73) Integrating by parts over time in the first term in (Equation73
(73)
(73) ) implies
(74)
(74)
The initial difference here
. Using the uniform positive definiteness (Equation13
(13)
(13) ) of D, after taking the supremum over
and summing up (Equation74
(74)
(74) ) over
we arrive at the first estimate in (Equation65
(65)
(65) ):
(75)
(75)
In particular, applying the triangle inequality for
given by the sum in (Equation66a
(66a)
(66a) ), due to the uniform boundedness of N,
, and
, from (Equation75
(75)
(75) ) it follows the estimate which will be used further in (Equation82
(82)
(82) ):
(76)
(76)
Estimate of . Similarly to (Equation67
(67a)
(67a) ), we apply to the term
the following Green's formulas on the both phases
and
:
(77a)
(77a)
(77b)
(77b)
for test functions
such that
at
, and
, respectively. We sum up the Equations (Equation77
(77a)
(77a) ), use the Poisson equation (Equation63b
(63b)
(63b) ) and the continuity of
across the interface
. Applying the asymptotic formula (Equation31
(31)
(31) ) from Lemma 4.1 we rewrite (Equation64b
(64b)
(64b) ) over the two-phase domain as follows:
(78)
(78)
for all test functions
such that
at
.
Applying the inequality (Equation44(44)
(44) ) from Lemma 4.3 with
proceeds the expansion (Equation78
(78)
(78) ) with some K>0 as
(79)
(79)
Next, we add to (Equation79
(79)
(79) ) the Equation (Equation34
(34)
(34) ) describing Λ from Lemma 4.2 and use the definition of
to get
(80)
(80)
The subtraction of (Equation80
(80)
(80) ) from the perturbed equation (Equation22b
(22b)
(22b) ) implies that
(81)
(81)
After substitution in (Equation81
(81)
(81) ) the test function
, which is zero at
, using Young's inequality with a weight
and applying the asymptotic bound (Equation76
(76)
(76) ) of
, we obtain the asymptotic inequality for
such that
for
:
(82)
(82)
where
. For
chosen small enough, using the uniform positive definiteness of A in (Equation10
(10)
(10) ) and the lower bound (Equation24
(24)
(24) ), taking the supremum over
in (Equation82
(82)
(82) ) follows the second estimate in (Equation65
(65)
(65) ) and finishes the proof.
6. Discussion
Passing to the limit in (Equation14(14a)
(14a) ), we derive the total mass balance and the non-negativity for the averaged species concentrations
.
According to the governing relations (Equation9c(9c)
(9c) ) and (Equation9d
(9d)
(9d) ), we can introduce the entropy variables
,
, and
corresponding to the solution of the averaged problem (Equation63
(63a)
(63a) ) as follows:
We observe the following technical assumptions used for the homogenization:
the asymptotic factor
,
, in the electrochemical potentials
in (Equation9c
(9c)
(9c) );
the asymptotic factor
by the interface reactions
in (Equation19a
(19a)
(19a) );
asymptotic decoupling of the diffusivity matrices
in (Equation12
(12)
(12) ).
Our future work is pointed towards possible relaxing these assumptions.
Acknowledgements
The authors thank two referees for the comments which helped to improve the manuscript.
Disclosure statement
No potential conflict of interest was reported by the authors.
Additional information
Funding
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