# Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

Authors: Gung-Min Gie; Chang-Yeol Jung; Roger Temam
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

We study the asymptotic behavior at small diffusivity of the solutions,uε, to a convection-diffusion equation in a rectangular domain Ω. The diffusive
equation is supplemented with a Dirichlet boundary condition, which is smooth
along the edges and continuous at the corners. To resolve the discrepancy, on ∂Ω, between uε and the corresponding limit solution, u0, we propose asymptotic expansions
of uε at any arbitrary, but fixed, order. In order to manage some singular
effects near the four corners of Ω, the so-called elliptic and ordinary corner correctors
are added in the asymptotic expansions as well as the parabolic and classical
boundary layer functions. Then, performing the energy estimates on the difference
of uε and the proposed expansions, the validity of our asymptotic expansions is
established in suitable Sobolev spaces.


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## Body

## 1. Introduction

We consider a singularly perturbed convection-diffusion equation in a rectangular domainΩ=(0,1)×(0,1):
(1)Lɛuɛ:=-ɛΔuɛ-∂xuɛ=finΩ,uɛ=gon∂Ω.
Here ɛ is a small but strictly positive diffusivity parameter, and f=f(x,y) is a given smooth data with ∥Dαf∥L2(Ω)≤κα, independent of ɛ, for some α's as needed in the analysis below. The function g=g(x,y) is assumed to be continuous on ∂Ω and smooth on each edge of ∂Ω. Namely, defining the restriction of g to the edges of ∂Ω as follows:
(2)g|x=0=g1(y),g|x=1=g2(y),g|y=0=g3(x),g|y=1=g4(x),
we assume that
(3)g1(0)=g3(0),g1(1)=g4(0),g2(0)=g3(1),g2(1)=g4(1),giis  smooth  for1≤i≤4
If these compatibility conditions were not satisfied, some additional considerations would be necessary, which we do not address here.In what follows, we study the asymptotic behavior of the solutions to (1) at small diffusivity ɛ.In a very nice related earlier work, [1], the asymptotic behavior of the solutions of a convection-diffusion equation similar to (1) was discussed. More precisely, in a rectangular domain Ω~=(0,l0)×(0,l1), the authors considered
(4)-ɛ′Δuɛ′-b∂xuɛ′+cuɛ′=f~inΩ~,uɛ′=g~on∂Ω~,
where b>0, c≥0 are constants and f~ is a given smooth function in Ω~. The function g~, satisfying the analogue version of (3) in Ω~, is assumed to be continuous and piecewise smooth on ∂Ω~. By constructing, in [1], asymptotic expansions of uɛ′ with respect to a small diffusivity ɛ′, the boundary layers of (4) were analyzed in a systematic way. The validity of their asymptotic expansions was established using the maximum principle.Using a simple change of variables which mapsΩ~ to Ω, and setting uɛ′=uɛeλx with a suitable λ, one can transform (4) to (1) where the two diffusivity parameters ɛ and ɛ′, respectively, in (1) and (4) are compatible; that is, ɛ/ɛ′ is of order one. Hence, via this transformation, our analysis of (1) in this paper is applicable to (4) as well. Our motivations for conducting the present study appear below. In particular we significantly simplify the proofs of [1] and make the study suitable for more general equations or systems, in particular, those which do not satisfy the maximum principle.In the boundary layer analysis associated with the convection-diffusion equation (1) (or (4)), one of the most important and difficult points is to resolve any possible singularity near the four corners of the rectangular domain Ω. Towards this end, we simplify and improve the method used in [1]. More precisely, concerning the asymptotic expansion of uɛ, we introduce the so-called elliptic boundary layer corrector, appearing in Section 3.2, near the inflow boundary at x=1 only, while in [1], it was used near both the inflow and outflow boundaries at x=1 and x=0. This simplification in the construction of the asymptotic expansions is mainly based on an observation that the corner singularities at (0,0) and (0,1) (where the flows go out) can be handled by the ordinary corner corrector, appearing in Section 3.4 (which is much easier to analyze than the elliptic corrector). Using the energy estimates rather than the maximum principle as in [1], we prove the validity of our proposed asymptotic expansions in suitable Sobolev spaces. Here we make use of the Hardy inequality (see, e.g., [2]) in the estimates. As we said above, our energy estimation approach can be easily extended to some higher order equations or systems which do not admit a maximum principle.This paper is organized as follows. In Section2, we introduce a formal expansion of uɛ as the sum of the outer and inner expansions. The outer expansion (outside of the boundary layers), which is easy to obtain, appears in Section 2. Then, in Section 3, by performing the boundary layer analysis in a systematic way, we construct the inner expansion (inside the boundary layers). In Section 4, we state and prove the main convergence result, Theorem 9, concerning the difference between uɛ and the proposed asymptotic expansion at a given order. In addition, in the appendix, we prove Lemma 1 that contains some delicate pointwise estimates on the parabolic boundary layer correctors introduced in Section 3.1. The study of the elliptic corrector and its approximation also appears in the appendix.Throughout this paper, we use the notation conventions below.Notation 1.
κ
:
=
κ
(
f
,
g
,
Ω
,
n
) is a constant depending on the data, but independent of ε; here n≥0 is the fixed (but arbitrary) order of the asymptotic expansion, as it appears in (6).Notation 2.
Ane.s.t. is a function (or a constant) whose norm in all Sobolev spaces Hs (and thus spaces Cs) is exponentially small with, for each s, a bound of the form c1e-c2/εγ,c1,c2,γ>0, ci, γ depending possibly on s.Notation 3.
For a fixedε>0, we define the energy norm of H1(Ω),
(5)∥·∥ε:=ε∥·∥H1(Ω)+∥·∥L2(Ω).


## 2. Asymptotic Expansions

To study the asymptotic behavior ofuɛ, solution of (1), we propose an asymptotic expansion uɛ of the following type:
(6)uɛ≅∑j=0nɛj(uj+Θj).
Here, at each order of ɛj, j≥0, uj corresponds to the outer expansion (outside of the boundary layer) of uɛ. To balance the discrepancy on the boundary ∂Ω of uɛ and of the uj, j≥0, we introduce the correctors Θj, j≥0, which will contribute mainly inside of the boundary layer: Θj will be itself the sum of several boundary layer functions as we will see below. The asymptotic expansion (6) is made at order n≥0. As we will see below, the expansion itself depends on n, but n, which is set at the beginning of the study, can be chosen arbitrary large.To determine the asymptotic expansion (6), we start with the outer expansion for uɛ, uɛ≅∑j=0n‍ɛjuj. Inserting this expansion into (1), we formally find the equations for the uj:
(7)-∂xu0=f,-∂xuj=Δuj-1,1≤j≤n.
We supplement these equations with the following inflow boundary conditions (see (1)–(3)):
(8)u0=g2(y)atx=1,uj=0atx=1,1≤j≤n.
While (7) are “natural”, the choice of the boundary conditions in (8) is not obvious; it will be eventually justified by the convergence theorem below. Then, by integrating the equations in x, we find the smooth outer solutions uj in the following form:
(9)u0=∫x1‍f(x1,y)dx1+g2(y),(10)uj=∫x1‍Δuj-1(x1,y)dx1,1≤j≤n.Under this construction of the outer expansion, we notice thatuɛ≅∑j=0n‍ɛjuj satisfies the boundary condition (1)2 along the edge x=1 only, and not on the three other sides of ∂Ω, y=0, y=1, and x=0 (in general). Hence, we expect boundary layers to occur near those edges, and the so-called correctors (corresponding to the Θj) will be necessary to account for these discrepancies.In Section3, performing the boundary layer analysis, we define, for each 0≤j≤n, the corrector Θj appearing in (11) below. Then, in Section 4, using (6), (9), (10), and (11), we prove our main result which is stated as Theorem 9.


## 3. Construction of the Correctors: Boundary Layers Analysis

We want the boundary value ofuɛ to match that of its approximation. In general, the boundary values of the diffusive solution uɛ and the outer expansion ∑j=0n‍ɛjuj match only at x=1, but not on the other sides of ∂Ω, y=0, y=1, and x=0. Hence, we expect that some boundary layers will occur near those three edges. To resolve this inconsistency, we will define a number of boundary layer functions on the sides and corners of ∂Ω, following an analysis reminiscent of the theory of the Prandtl boundary layer in [3–6].For the moment, comparinguɛ with a finite sum ∑j=0n‍ɛjuj, we infer from the definition of the uj in (7)–(10) that the values of these functions coincide on the segment x=1. In general, they will not coincide anywhere else on ∂Ω. First, to correct the inconsistency on the sides y=0,1, we will define the so-called parabolic correctors φj=φlj+φuj. Then, to correct the inconsistency at x=0, we will introduce the (classical) corrector θj. However, this will not be enough because additional inconsistencies appear at the corners of Ω, and, as we shall see below, it will be necessary to introduce additional correctors ξj, ζj, and ηj to handle these inconsistencies.In summary, in the subsections below, we will define theΘj in the form,
(11)Θj=φj+ξj+θj+ζj+ηj,0≤j≤n,
where the role and location of each corrector are explained, respectively, in Box 1 and Figure 1. The stretched variables of the corrector functions, which will be introduced and used in the following subsections, are indicated in the following equation:
(12)φlj=φlj(x,yε),φuj=φuj(x,(1-y)ε),ξlj=ξlj((1-x)ε,yε),ξuj=ξuj((1-x)ε,(1-y)ε),θj=θj(xε,y),ζlj=ζlj(xε,yε),ζuj=ζuj(xε,(1-y)ε),ηlj=ηlj(xε,y),ηuj=ηuj(xε,y).Box 1:The different correctors.
φ
j
=
φ
l
j
+
φ
u
j is the parabolic boundary layer corrector near y=0 and y=1,
ξ
j
=
ξ
l
j
+
ξ
u
j is the elliptic corrector which resolves the compatibility issues, in the construction of φj, at the corners (1,0) and (1,1),
θ
j is the ordinary boundary layer corrector near x = 0,
ζ
j
=
ζ
l
j
+
ζ
u
j  is the ordinary corner layer corrector which manages the effect of φj along x=0,
η
j
=
η
l
j
+
η
u
j  is the supplementary corrector that manages the effects of θj and ζj along the edges y=0 and y=1.Figure 1
Location of the outer solutionuj and the boundary layer correctors.


### 3.1. Parabolic Boundary Layers (PBL)

At the bottom (or top) boundary, that is, aty=0 (or y=1), to balance the discrepancy of uɛ and the outer expansion ∑j=0n‍ɛjuj, (see the uj as defined in (9) and (10)), we construct below the parabolic boundary layer correctors φj, 0≤j≤n.We formally insert the asymptotic expansionuɛ≅∑j=0n‍ɛjφj into (1). Using the arguments similar to those of the Prandtl theory, we see that the thickness of the boundary layer near y=0 or y=1 should be of order ɛ1/2. Then, by collecting the terms at the same order of ɛj, 0≤j≤n, we find the equations of the parabolic boundary layer correctors:
(13)-ɛ∂y2φj-∂xφj=∂x2φj-1inΩ,0≤j≤n,
where we set φ-1=0.Along the boundariesy=0, y=1, and x=1, comparing (1)2 and the boundary values of the functions in (9) and (10), we find the boundary conditions of φj, 0≤j≤n, which read. 								(14)φj=0atx=1,0≤j≤n,φ0=g3(x)-u0(x,0)aty=0,φj=-uj(x,0)aty=0,1≤j≤n,φ0=g4(x)-u0(x,1)aty=1,φj=-uj(x,1)aty=1,1≤j≤n.We will show below how to actually constructφj. For the moment, comparing the functions uɛ and ∑j=0n‍ɛj(uj-φj) for some n≥0, we see that the boundary values of these functions coincide at x=1 and y=0,1. However this has been achieved at the price of introducing some new inconsistencies. Indeed we notice, in (14), that, for each j≥0, the boundary conditions at y=0 and y=1 are not consistent with that at x=1. That is, in general, for any i≥1,
(15)∂xiu0(1,0)-∂xig3(1)≠0,∂xiu0(1,1)-∂xig4(1)≠0,∂xiuj(1,0)≠0,∂xiuj(1,1)≠0,j≥1.
Due to this inconsistency of the boundary data, if we define φj, 0≤j≤n, as a solution of (13) with (14), some derivatives of φj, 0≤j≤n get singular at the two corners (1,k), k=0,1. Therefore, to define smooth (smoother) correctors φj, 0≤j≤n, we will modify the boundary conditions (14) as introduced in (23).


#### 3.1.1. On Modification of Boundary Conditions for Smooth Correctors aty=0,1

Equation (13) with initial and boundary conditions (14) is a parabolic problem (heat equation), in which -x is the (positive) time like variable so that the boundary conditions at x=1 are the analogue of the initial conditions for an evolution problem, and the possible inconsistencies in (15) correspond to the (analogue of) the compatibility conditions between the initial and boundary values for a parabolic equation; see [7–9] and the references therein. Due to the lack of consistency at the corners (1,0) and (1,1), the correctors constructed as in (13) and (14) might get singular derivatives at those corners. We will overcome this difficulty by considering some corner functions at x=1 and y=0,1, similar to what is done in [10, 11] or [12] for the compatibility issue in parabolic problems; see [13] as well.Definingδ(x) as a smooth cut-off function, independent of ɛ, such that
(16)δ(x)={0for0≤x≤12,1forx≥34,
we set
(17)γkj(x)=γ~kj(x)δ(x),k=0,1,
where
(18)γ~k0(x)=∑i=12n+1‍(x-1)ii![∂xig3+k(1)-∂xiu0(1,k)],γ~kj(x)=-∑i=12n+1-2j‍(x-1)ii!∂xiuj(1,k),1≤j≤n.
Note that the γ~k0,  k=0,1, depend on n, but this dependency is not made explicit to make the notations (slightly) simpler. Note also (comparing to (15)) that, at x=1, for each k=0,1,
(19)γ~kj(1)=0,0≤j≤n,∂xiγ~k0(1)=∂xig3+k(1)-∂xiu0(1,k),1≤i≤2n+1,∂xiγ~kj(1)=-∂xiuj(1,k),1≤j≤n,1≤i≤2n+1-j.We introduce the stretched variables,(20)y-=yɛ,y~=(1-y)ɛ.
Then, using (13), (14), (17), and (20), we define the parabolic corrector, φj, 0≤j≤n, as
(21)φj=φlj+φuj=φlj(x,y-)+φuj(x,y~),
where φlj and φuj (lower and upper parts of φj, resp.) are defined as the solutions of the 1D heat equations below.For0≤j≤n,
(22)-∂y-2φlj-∂xφlj=∂x2φlj-1for(y-,x)∈ℝ+×(0,1),φlj=h0j(x)aty-=0,φlj⟶0asy-⟶∞,φlj=0atx=1,-∂y~2φuj-∂xφuj=∂x2φuj-1for(y~,x)∈ℝ+×(0,1),φuj=h1j(x)aty~=0,φuj⟶0asy~⟶∞,φuj=0atx=1.
Here the hkj(x), 0≤j≤n, k=0,1, are defined by
(23)h00(x):=g3(x)-u0(x,0)-γ00(x),h0j(x):=-uj(x,0)-γ0j(x),1≤j≤n,h10(x):=g4(x)-u0(x,1)-γ10(x),h1j(x):=-uj(x,1)-γ1j(x),1≤j≤n.
Note that the smooth boundary conditions hkj(x), 0≤j≤n, k=0,1, along y- or y~=0 are compatible with the other 0 boundary (initial) conditions at x=1 in the sense that, for each 0≤j≤n,
(24)∂xihkj(1)=0,k=0,1,0≤i≤2n+1-2j.From [14–16] or [1], we recall the explicit expressions of φlj=φlj(x,y-), 0≤j≤n,
(25)φl0=2π∫y-/2(1-x)∞‍exp(-y122)h00(x+y-22y12)dy1,(26)φlj=2π∫y-/2(1-x)∞‍exp(-y122)h0j(x+y-22y12)dy1+12π∫01-x‍∫0∞‍1x1{exp(-(y--y1)24x1)+12π∫01-x‍∫0∞‍1x1-exp(-(y-+y1)24x1)}12π∫01-x∫0∞‍‍×∂x2φlj-1(x+x1,y1)dy1dx1.
The expressions of φuj=φuj(x,y~), 0≤j≤n, are identical to (25) and (26) with h00,  h0j, and y-, respectively, replaced by h10,   h1j and y~.


#### 3.1.2. Estimates on the Parabolic Boundary Layers

We now state some pointwise estimates for theφlj and φuj, 0≤j≤n, which are proved in the appendix.Lemma 1.
For each0≤j≤n and s≥0, one has the following pointwise estimates:
(27)|ys∂xi∂ymφlj|≤κɛ(s-m)/2exp(-cyɛ),0≤i+m≤2n+2-2j,(28)|ys∂xi∂ymφuj|≤κε(s-m)/2exp(-c1-yε),0≤i+m≤2n+2-2j,
for a generic constant c>0 independent of x, y, and ε.From Lemma1, it is easy to deduce the following Lp estimates.Lemma 2.
For each0≤j≤n and s≥0, one has, for 0≤i+m≤2n+2-2j,
(29)∥ys∂xi∂ymφlj∥Lp(Ω)+∥ys∂xi∂ymφuj∥Lp(Ω)κε(s-m)/2+1/2p.Using (21) and (22), we find the equations for φj=φlj+φuj, 0≤j≤n:
(30)-ɛ∂y2φj-∂xφj=∂x2φj-1inΩ,φj=h0j(x)+φuj|y=0aty=0,φj=h1j(x)+φlj|y=1aty=1,φj=0atx=1.
Thanks to Lemma 1, we notice that
(31)φuj|y=0=e.s.t.,φlj|y=1=e.s.t.,
where e.s.t. denotes an exponentially small term with respect to the (small) parameter ɛ as defined in Notation 2.


### 3.2. Elliptic Boundary Layers (EBL)

In Section3.1, to construct the consistent parabolic boundary layer correctors φj, 0≤j≤n, we considered the modified boundary conditions (23), including γkj, instead of the natural boundary conditions (14). Here, on the two sides y=0 and y=1 of Ω, to cancel γkj which is exactly the difference of (14) and (23), we introduce elliptic boundary layer correctors ξj, 0≤j≤n, which, like γkj or γ~kj, depend on the order n of the asymptotic expansion in (6).Using (17), we define
(32)ξj=ξlj+ξuj,0≤j≤n,
where ξlj and ξuj are the solutions of the elliptic systems below.For0≤j≤n,
(33)-ɛ∂x2ξlj-ɛ∂y2ξlj-∂xξlj=0inΩ,ξlj=γ0j(x)aty=0,ξlj=0atx=0,1,ory=1,-ɛ∂x2ξuj-ɛ∂y2ξuj-∂xξuj=0inΩ,ξuj=γ1j(x)aty=1,ξuj=0atx=0,1,ory=0.
The systems (33) being elliptic equations, their well-posedness are easy to verify, and hence, we omit the proof here.The equations satisfied by the elliptic correctorsξj=ξlj+ξuj read as follows.For0≤j≤n,
(34)-ɛ∂x2ξj-ɛ∂y2ξj-∂xξj=0inΩ,ξj=γkj(x)aty=k,k=0,1,ξj=0atx=0,1.In the error analysis below in Section4, we will see that no extra terms related to the ξj appear; see (65) below. This is because the ξj, as appearing in (34), satisfies the same equation as for uɛ, supplemented with the (exact) boundary conditions that we need. From this observation, as it will be justified in Section 4, we notice that our main result in Theorem 9 does not require any estimate on the ξj.As an extra information which might be useful elsewhere, but not in this paper, by performing the energy estimate on (34), we find that
(35)∥ξj∥ɛ≤κɛ1/4,0≤j≤n.
In the appendix, we introduce an explicit approximate ξ- of ξ. Then, thanks to the estimates on ξ-, one can obtain some pointwise estimates on ξ as well; see Appendix B.At this stage, the functionsuɛ and ∑j=0n‍ɛj(uj+φj+ξj) match along the sides x=1 and y=0,1 (with singular derivatives of high orders for the φj). More precisely, from (1), (8), (30), (31), and (34), we see that
(36)uɛ-∑j=0n‍ɛj(uj+φj+ξj)={g1+∑j=0n‍ɛj(uj+φj)atx=0,0atx=1,∑j=0n‍ɛjφuj=e.s.t.aty=0,∑j=0n‍ɛjφlj=e.s.t.aty=1.We will now deal with the sidex=0 and construct the ordinary boundary layer θj and the ordinary corner layer functions (correctors) ζj.


### 3.3. Ordinary Boundary Layers (OBL)

To handle the discrepancy ofuɛ and ∑j=0n‍ɛjuj at x=0, which is a part of uɛ-∑j=0n‍ɛj(uj+φj+ξj) appearing in (36), the so-called ordinary boundary layer corrector θj is introduced in this section. We insert a formal expansion uɛ≅∑j=0∞‍ɛjθj into the diffusive equation (1). Then, using the stretched variable x-=x/ɛ, we find the equations of θj:
(37)-ɛ∂x2θj-∂xθj=ɛ-1∂y2θj-2inΩ,0≤j≤n,
where we set θ-1=θ-2=0.The main role of theθj is, at each order of ɛj, to cancel the error uɛ-∑j=0n‍ɛjuj at x=0. Hence, we supplement (37) with the boundary conditions
(38)θ0=g1(y)-u0(0,y)atx=0,θj=-uj(0,y)atx=0,1≤j≤n,θj⟶0asx⟶∞,0≤j≤n.
The explicit expressions of θj, 0≤j≤n, are inductively shown to be of the form
(39)θj=Pk(xɛ,y)exp(-xɛ),j=2k,2k+1,
where Pk(x/ɛ,y) is a polynomial in x/ɛ of degree k whose coefficients, independent of ɛ, are linear combinations of the ∂suj-s(0,y)/∂ys, 0≤s≤j, and Pk(0,y)=-uj(0,y).According to (39), we deduce the following lemmas.Lemma 3.
For each0≤j≤n, one has the pointwise estimates:
(40)|xs∂xi∂ymθj|≤κɛs-iexp(-cxɛ),s,i,m≥0,
for a constant c independent of x, y, and ɛ.Lemma 4.
For each0≤j≤n, one has
(41)∥xs∂xi∂ymθj∥Lp(Ω)≤κɛs-i+(1/p),s,i,m≥0.Thanks to Lemma3, we notice that the effect of θj near the boundary x=1 is exponentially small:
(42)θj(1,y)=e.s.t.,0≤j≤n.From (36), (38), and (42), we infer that
(43)uɛ-∑j=0n‍ɛj(uj+φj+ξj+θj)={∑j=0n‍ɛjφjatx=0,∑j=0n‍ɛjθj=e.s.t.atx=1,∑j=0n‍ɛj(φuj+θj)=∑j=0n‍ɛjθj+e.s.t.aty=0,∑j=0n‍ɛj(φlj+θj)=∑j=0n‍ɛjθj+e.s.t.aty=1.


### 3.4. Ordinary Corner Layers (OCL)

To account for the value∑j=0n‍ɛjφj at x=0, which is exactly the difference of uɛ and ∑j=0n‍ɛj(uj+φj+ξj+θj) at x=0 (see (43)), we introduce the ordinary corner layer correctors ζj, 0≤j≤n, in the form
(44)ζj=ζlj+ζuj,
where the ζlj (or ζuj) are the correctors near the corner (0,0) (or (0,1)) as constructed below.To defineζlj (or ζuj), we insert a formal expansion uɛ≅∑j=0∞‍ɛjζj into (1). Then, using the stretched variables x-=x/ɛ and y-=y/ɛ near (0,0) and using x- and y~=(1-y)/ɛ near (1,1), we collect the terms of order ɛj, 0≤j≤n. As a result, we find the equations for ζlj (or ζuj):
(45)-ɛ∂x2ζkj-∂xζkj=∂y2ζkj-1inΩ,0≤j≤n,k=uorl.
Here, we set ζk-1=0 for k=l,u.At each order ofɛj, 0≤j≤n, to cancel the error φj near the corner (0,0) or (0,1), we supplement (45) with the boundary conditions
(46)ζkj=-φkj(0,y)atx=0,0≤j≤n,k=l,u,ζkj⟶0asx⟶∞,0≤j≤n,k=l,u.The explicit solutionsζlj of (45) and (46), 0≤j≤n, are inductively found to be of the following form:
(47)ζlj=Pj(xɛ,yɛ)exp(-xɛ),
where Pj(x/ɛ,y/ɛ) is a polynomial in x/ɛ of degree j whose coefficients, independent of ɛ, are linear combinations of the quantities ɛs∂2sφlj-s(0,y/ɛ)/∂y2s, 0≤s≤j. In the same manner, the explicit forms of the ζuj, 0≤j≤n can be found as well.Using Lemma1 and (47), we obtain the lemmas below.Lemma 5.
For each0≤j≤n and k,s≥0, one has the pointwise estimates:
(48)|xkys∂xi∂ymζlj|≤κɛk-i+((s-m)/2)exp(-c(xɛ+yɛ)),0≤i+m≤2n+2-2j,|xkys∂xi∂ymζuj|≤κɛk-i+((s-m)/2)exp(-c(xɛ+1-yɛ)),0≤i+m≤2n+2-2j,
for a generic constant c>0 independent of x, y, and ɛ.Lemma 6.
For each0≤j≤n and k,s≥0, one has, for 0≤i+m≤2n+2-2j,
(49)∥xkys∂xi∂ymζlj∥Lp(Ω)+∥xkys∂xi∂ymζuj∥Lp(Ω)≤κɛk-i+((s-m)/2)+(3/2p).Thanks to (46) and Lemma 5, we estimate the values of  ζj, 0≤j≤n, along the sides x=1, y=0, and y=1:
(50)ζj={e.s.t.atx=1,ζlj(x,0)+e.s.t.aty=0,ζuj(x,0)+e.s.t.aty=1.Now from (43), (46), and (50), we observe that
(51)uɛ-∑j=0n‍ɛj(uj+φj+ξj+θj+ζj)={0atx=0,∑j=0n‍ɛj(θj+ζj)=e.s.t.atx=1,∑j=0n‍ɛj(φuj+θj+ζj)=∑j=0n‍ɛj(θj+ζlj)+e.s.t.aty=0,∑j=0n‍ɛj(φlj+θj+ζj)=∑j=0n‍ɛj(θj+ζuj)+e.s.t.aty=1.


### 3.5. Supplementary Correctors

Now, our task is to handle, at each order0≤j≤n, the effects of θj and ζj along the top and bottom boundaries y=0,1. To this end, using (51), we construct the supplementary correctors ηj in the form
(52)ηj=ηlj+ηuj,0≤j≤n,
where
(53)ηlj=-(1-y)(θj+ζlj)(x,y=0),ηuj=-y(θj+ζuj)(x,y=1).From Lemmas3 and 5, it is easy to verify the lemmas below.Lemma 7.
For each0≤j≤n and s≥0, one has the pointwise estimates:
(54)|xs∂xi∂ymηj|≤κɛs-iexp(-cxɛ),0≤i+m≤2n+2-2j,
for a constant c independent of x, y, and ɛ.Lemma 8.
For each0≤j≤n and s≥0, one has
(55)∥xs∂xi∂ymηj∥Lp(Ω)≤ɛs-i+(1/p),0≤i+m≤2n+2-2j.Using (3), (14), (17), (22), (23), (38), and (46), we find that, near the corner (0, 0),
(56)(θ0+ζl0)(0,0)=g1(0)-u0(0,0)-φl0(0,0)=g1(0)-u0(0,0)-(g3(0)-u0(0,0))=0.
More generally, one can verify that
(57)(θj+ζlj)(0,0)=0,(θj+ζuj)(0,1)=0,0≤j≤n.Using (42), (50), and (57), we infer from (52) and (53) that, for 0≤j≤n,
(58)ηj={0atx=0,e.s.t.atx=1,-(θj+ζlj)aty=0,-(θj+ζuj)aty=1.Now, we finally obtain from (51) and (58) that
(59)uɛ-∑j=0n‍ɛj(uj+φj+ξj+θj+ζj+ηj)={0atx=0,∑j=0n‍ɛj(θj+ζj+ηj)=e.s.t.atx=1,∑j=0n‍ɛj(φuj+θj+ζj+ηj)=∑j=0n‍ɛj(θj+ζlj+ηj)+e.s.t.=e.s.t.aty=0,∑j=0n‍ɛj(φlj+θj+ζj+ηj)=∑j=0n‍ɛj(θj+ζuj+ηj)+e.s.t.=e.s.t.aty=1.
The e.s.t. on the right hand side of (59)2,3,4 are caused, respectively, by (θj+ζj+ηj) at x=1, (φuj+ζuj) at y=0, and (φlj+ζlj) at y=1.


## 3.1. Parabolic Boundary Layers (PBL)

At the bottom (or top) boundary, that is, aty=0 (or y=1), to balance the discrepancy of uɛ and the outer expansion ∑j=0n‍ɛjuj, (see the uj as defined in (9) and (10)), we construct below the parabolic boundary layer correctors φj, 0≤j≤n.We formally insert the asymptotic expansionuɛ≅∑j=0n‍ɛjφj into (1). Using the arguments similar to those of the Prandtl theory, we see that the thickness of the boundary layer near y=0 or y=1 should be of order ɛ1/2. Then, by collecting the terms at the same order of ɛj, 0≤j≤n, we find the equations of the parabolic boundary layer correctors:
(13)-ɛ∂y2φj-∂xφj=∂x2φj-1inΩ,0≤j≤n,
where we set φ-1=0.Along the boundariesy=0, y=1, and x=1, comparing (1)2 and the boundary values of the functions in (9) and (10), we find the boundary conditions of φj, 0≤j≤n, which read. 								(14)φj=0atx=1,0≤j≤n,φ0=g3(x)-u0(x,0)aty=0,φj=-uj(x,0)aty=0,1≤j≤n,φ0=g4(x)-u0(x,1)aty=1,φj=-uj(x,1)aty=1,1≤j≤n.We will show below how to actually constructφj. For the moment, comparing the functions uɛ and ∑j=0n‍ɛj(uj-φj) for some n≥0, we see that the boundary values of these functions coincide at x=1 and y=0,1. However this has been achieved at the price of introducing some new inconsistencies. Indeed we notice, in (14), that, for each j≥0, the boundary conditions at y=0 and y=1 are not consistent with that at x=1. That is, in general, for any i≥1,
(15)∂xiu0(1,0)-∂xig3(1)≠0,∂xiu0(1,1)-∂xig4(1)≠0,∂xiuj(1,0)≠0,∂xiuj(1,1)≠0,j≥1.
Due to this inconsistency of the boundary data, if we define φj, 0≤j≤n, as a solution of (13) with (14), some derivatives of φj, 0≤j≤n get singular at the two corners (1,k), k=0,1. Therefore, to define smooth (smoother) correctors φj, 0≤j≤n, we will modify the boundary conditions (14) as introduced in (23).


### 3.1.1. On Modification of Boundary Conditions for Smooth Correctors aty=0,1

Equation (13) with initial and boundary conditions (14) is a parabolic problem (heat equation), in which -x is the (positive) time like variable so that the boundary conditions at x=1 are the analogue of the initial conditions for an evolution problem, and the possible inconsistencies in (15) correspond to the (analogue of) the compatibility conditions between the initial and boundary values for a parabolic equation; see [7–9] and the references therein. Due to the lack of consistency at the corners (1,0) and (1,1), the correctors constructed as in (13) and (14) might get singular derivatives at those corners. We will overcome this difficulty by considering some corner functions at x=1 and y=0,1, similar to what is done in [10, 11] or [12] for the compatibility issue in parabolic problems; see [13] as well.Definingδ(x) as a smooth cut-off function, independent of ɛ, such that
(16)δ(x)={0for0≤x≤12,1forx≥34,
we set
(17)γkj(x)=γ~kj(x)δ(x),k=0,1,
where
(18)γ~k0(x)=∑i=12n+1‍(x-1)ii![∂xig3+k(1)-∂xiu0(1,k)],γ~kj(x)=-∑i=12n+1-2j‍(x-1)ii!∂xiuj(1,k),1≤j≤n.
Note that the γ~k0,  k=0,1, depend on n, but this dependency is not made explicit to make the notations (slightly) simpler. Note also (comparing to (15)) that, at x=1, for each k=0,1,
(19)γ~kj(1)=0,0≤j≤n,∂xiγ~k0(1)=∂xig3+k(1)-∂xiu0(1,k),1≤i≤2n+1,∂xiγ~kj(1)=-∂xiuj(1,k),1≤j≤n,1≤i≤2n+1-j.We introduce the stretched variables,(20)y-=yɛ,y~=(1-y)ɛ.
Then, using (13), (14), (17), and (20), we define the parabolic corrector, φj, 0≤j≤n, as
(21)φj=φlj+φuj=φlj(x,y-)+φuj(x,y~),
where φlj and φuj (lower and upper parts of φj, resp.) are defined as the solutions of the 1D heat equations below.For0≤j≤n,
(22)-∂y-2φlj-∂xφlj=∂x2φlj-1for(y-,x)∈ℝ+×(0,1),φlj=h0j(x)aty-=0,φlj⟶0asy-⟶∞,φlj=0atx=1,-∂y~2φuj-∂xφuj=∂x2φuj-1for(y~,x)∈ℝ+×(0,1),φuj=h1j(x)aty~=0,φuj⟶0asy~⟶∞,φuj=0atx=1.
Here the hkj(x), 0≤j≤n, k=0,1, are defined by
(23)h00(x):=g3(x)-u0(x,0)-γ00(x),h0j(x):=-uj(x,0)-γ0j(x),1≤j≤n,h10(x):=g4(x)-u0(x,1)-γ10(x),h1j(x):=-uj(x,1)-γ1j(x),1≤j≤n.
Note that the smooth boundary conditions hkj(x), 0≤j≤n, k=0,1, along y- or y~=0 are compatible with the other 0 boundary (initial) conditions at x=1 in the sense that, for each 0≤j≤n,
(24)∂xihkj(1)=0,k=0,1,0≤i≤2n+1-2j.From [14–16] or [1], we recall the explicit expressions of φlj=φlj(x,y-), 0≤j≤n,
(25)φl0=2π∫y-/2(1-x)∞‍exp(-y122)h00(x+y-22y12)dy1,(26)φlj=2π∫y-/2(1-x)∞‍exp(-y122)h0j(x+y-22y12)dy1+12π∫01-x‍∫0∞‍1x1{exp(-(y--y1)24x1)+12π∫01-x‍∫0∞‍1x1-exp(-(y-+y1)24x1)}12π∫01-x∫0∞‍‍×∂x2φlj-1(x+x1,y1)dy1dx1.
The expressions of φuj=φuj(x,y~), 0≤j≤n, are identical to (25) and (26) with h00,  h0j, and y-, respectively, replaced by h10,   h1j and y~.


### 3.1.2. Estimates on the Parabolic Boundary Layers

We now state some pointwise estimates for theφlj and φuj, 0≤j≤n, which are proved in the appendix.Lemma 1.
For each0≤j≤n and s≥0, one has the following pointwise estimates:
(27)|ys∂xi∂ymφlj|≤κɛ(s-m)/2exp(-cyɛ),0≤i+m≤2n+2-2j,(28)|ys∂xi∂ymφuj|≤κε(s-m)/2exp(-c1-yε),0≤i+m≤2n+2-2j,
for a generic constant c>0 independent of x, y, and ε.From Lemma1, it is easy to deduce the following Lp estimates.Lemma 2.
For each0≤j≤n and s≥0, one has, for 0≤i+m≤2n+2-2j,
(29)∥ys∂xi∂ymφlj∥Lp(Ω)+∥ys∂xi∂ymφuj∥Lp(Ω)κε(s-m)/2+1/2p.Using (21) and (22), we find the equations for φj=φlj+φuj, 0≤j≤n:
(30)-ɛ∂y2φj-∂xφj=∂x2φj-1inΩ,φj=h0j(x)+φuj|y=0aty=0,φj=h1j(x)+φlj|y=1aty=1,φj=0atx=1.
Thanks to Lemma 1, we notice that
(31)φuj|y=0=e.s.t.,φlj|y=1=e.s.t.,
where e.s.t. denotes an exponentially small term with respect to the (small) parameter ɛ as defined in Notation 2.


## 3.1.1. On Modification of Boundary Conditions for Smooth Correctors aty=0,1

Equation (13) with initial and boundary conditions (14) is a parabolic problem (heat equation), in which -x is the (positive) time like variable so that the boundary conditions at x=1 are the analogue of the initial conditions for an evolution problem, and the possible inconsistencies in (15) correspond to the (analogue of) the compatibility conditions between the initial and boundary values for a parabolic equation; see [7–9] and the references therein. Due to the lack of consistency at the corners (1,0) and (1,1), the correctors constructed as in (13) and (14) might get singular derivatives at those corners. We will overcome this difficulty by considering some corner functions at x=1 and y=0,1, similar to what is done in [10, 11] or [12] for the compatibility issue in parabolic problems; see [13] as well.Definingδ(x) as a smooth cut-off function, independent of ɛ, such that
(16)δ(x)={0for0≤x≤12,1forx≥34,
we set
(17)γkj(x)=γ~kj(x)δ(x),k=0,1,
where
(18)γ~k0(x)=∑i=12n+1‍(x-1)ii![∂xig3+k(1)-∂xiu0(1,k)],γ~kj(x)=-∑i=12n+1-2j‍(x-1)ii!∂xiuj(1,k),1≤j≤n.
Note that the γ~k0,  k=0,1, depend on n, but this dependency is not made explicit to make the notations (slightly) simpler. Note also (comparing to (15)) that, at x=1, for each k=0,1,
(19)γ~kj(1)=0,0≤j≤n,∂xiγ~k0(1)=∂xig3+k(1)-∂xiu0(1,k),1≤i≤2n+1,∂xiγ~kj(1)=-∂xiuj(1,k),1≤j≤n,1≤i≤2n+1-j.We introduce the stretched variables,(20)y-=yɛ,y~=(1-y)ɛ.
Then, using (13), (14), (17), and (20), we define the parabolic corrector, φj, 0≤j≤n, as
(21)φj=φlj+φuj=φlj(x,y-)+φuj(x,y~),
where φlj and φuj (lower and upper parts of φj, resp.) are defined as the solutions of the 1D heat equations below.For0≤j≤n,
(22)-∂y-2φlj-∂xφlj=∂x2φlj-1for(y-,x)∈ℝ+×(0,1),φlj=h0j(x)aty-=0,φlj⟶0asy-⟶∞,φlj=0atx=1,-∂y~2φuj-∂xφuj=∂x2φuj-1for(y~,x)∈ℝ+×(0,1),φuj=h1j(x)aty~=0,φuj⟶0asy~⟶∞,φuj=0atx=1.
Here the hkj(x), 0≤j≤n, k=0,1, are defined by
(23)h00(x):=g3(x)-u0(x,0)-γ00(x),h0j(x):=-uj(x,0)-γ0j(x),1≤j≤n,h10(x):=g4(x)-u0(x,1)-γ10(x),h1j(x):=-uj(x,1)-γ1j(x),1≤j≤n.
Note that the smooth boundary conditions hkj(x), 0≤j≤n, k=0,1, along y- or y~=0 are compatible with the other 0 boundary (initial) conditions at x=1 in the sense that, for each 0≤j≤n,
(24)∂xihkj(1)=0,k=0,1,0≤i≤2n+1-2j.From [14–16] or [1], we recall the explicit expressions of φlj=φlj(x,y-), 0≤j≤n,
(25)φl0=2π∫y-/2(1-x)∞‍exp(-y122)h00(x+y-22y12)dy1,(26)φlj=2π∫y-/2(1-x)∞‍exp(-y122)h0j(x+y-22y12)dy1+12π∫01-x‍∫0∞‍1x1{exp(-(y--y1)24x1)+12π∫01-x‍∫0∞‍1x1-exp(-(y-+y1)24x1)}12π∫01-x∫0∞‍‍×∂x2φlj-1(x+x1,y1)dy1dx1.
The expressions of φuj=φuj(x,y~), 0≤j≤n, are identical to (25) and (26) with h00,  h0j, and y-, respectively, replaced by h10,   h1j and y~.


## 3.1.2. Estimates on the Parabolic Boundary Layers

We now state some pointwise estimates for theφlj and φuj, 0≤j≤n, which are proved in the appendix.Lemma 1.
For each0≤j≤n and s≥0, one has the following pointwise estimates:
(27)|ys∂xi∂ymφlj|≤κɛ(s-m)/2exp(-cyɛ),0≤i+m≤2n+2-2j,(28)|ys∂xi∂ymφuj|≤κε(s-m)/2exp(-c1-yε),0≤i+m≤2n+2-2j,
for a generic constant c>0 independent of x, y, and ε.From Lemma1, it is easy to deduce the following Lp estimates.Lemma 2.
For each0≤j≤n and s≥0, one has, for 0≤i+m≤2n+2-2j,
(29)∥ys∂xi∂ymφlj∥Lp(Ω)+∥ys∂xi∂ymφuj∥Lp(Ω)κε(s-m)/2+1/2p.Using (21) and (22), we find the equations for φj=φlj+φuj, 0≤j≤n:
(30)-ɛ∂y2φj-∂xφj=∂x2φj-1inΩ,φj=h0j(x)+φuj|y=0aty=0,φj=h1j(x)+φlj|y=1aty=1,φj=0atx=1.
Thanks to Lemma 1, we notice that
(31)φuj|y=0=e.s.t.,φlj|y=1=e.s.t.,
where e.s.t. denotes an exponentially small term with respect to the (small) parameter ɛ as defined in Notation 2.


## 3.2. Elliptic Boundary Layers (EBL)

In Section3.1, to construct the consistent parabolic boundary layer correctors φj, 0≤j≤n, we considered the modified boundary conditions (23), including γkj, instead of the natural boundary conditions (14). Here, on the two sides y=0 and y=1 of Ω, to cancel γkj which is exactly the difference of (14) and (23), we introduce elliptic boundary layer correctors ξj, 0≤j≤n, which, like γkj or γ~kj, depend on the order n of the asymptotic expansion in (6).Using (17), we define
(32)ξj=ξlj+ξuj,0≤j≤n,
where ξlj and ξuj are the solutions of the elliptic systems below.For0≤j≤n,
(33)-ɛ∂x2ξlj-ɛ∂y2ξlj-∂xξlj=0inΩ,ξlj=γ0j(x)aty=0,ξlj=0atx=0,1,ory=1,-ɛ∂x2ξuj-ɛ∂y2ξuj-∂xξuj=0inΩ,ξuj=γ1j(x)aty=1,ξuj=0atx=0,1,ory=0.
The systems (33) being elliptic equations, their well-posedness are easy to verify, and hence, we omit the proof here.The equations satisfied by the elliptic correctorsξj=ξlj+ξuj read as follows.For0≤j≤n,
(34)-ɛ∂x2ξj-ɛ∂y2ξj-∂xξj=0inΩ,ξj=γkj(x)aty=k,k=0,1,ξj=0atx=0,1.In the error analysis below in Section4, we will see that no extra terms related to the ξj appear; see (65) below. This is because the ξj, as appearing in (34), satisfies the same equation as for uɛ, supplemented with the (exact) boundary conditions that we need. From this observation, as it will be justified in Section 4, we notice that our main result in Theorem 9 does not require any estimate on the ξj.As an extra information which might be useful elsewhere, but not in this paper, by performing the energy estimate on (34), we find that
(35)∥ξj∥ɛ≤κɛ1/4,0≤j≤n.
In the appendix, we introduce an explicit approximate ξ- of ξ. Then, thanks to the estimates on ξ-, one can obtain some pointwise estimates on ξ as well; see Appendix B.At this stage, the functionsuɛ and ∑j=0n‍ɛj(uj+φj+ξj) match along the sides x=1 and y=0,1 (with singular derivatives of high orders for the φj). More precisely, from (1), (8), (30), (31), and (34), we see that
(36)uɛ-∑j=0n‍ɛj(uj+φj+ξj)={g1+∑j=0n‍ɛj(uj+φj)atx=0,0atx=1,∑j=0n‍ɛjφuj=e.s.t.aty=0,∑j=0n‍ɛjφlj=e.s.t.aty=1.We will now deal with the sidex=0 and construct the ordinary boundary layer θj and the ordinary corner layer functions (correctors) ζj.


## 3.3. Ordinary Boundary Layers (OBL)

To handle the discrepancy ofuɛ and ∑j=0n‍ɛjuj at x=0, which is a part of uɛ-∑j=0n‍ɛj(uj+φj+ξj) appearing in (36), the so-called ordinary boundary layer corrector θj is introduced in this section. We insert a formal expansion uɛ≅∑j=0∞‍ɛjθj into the diffusive equation (1). Then, using the stretched variable x-=x/ɛ, we find the equations of θj:
(37)-ɛ∂x2θj-∂xθj=ɛ-1∂y2θj-2inΩ,0≤j≤n,
where we set θ-1=θ-2=0.The main role of theθj is, at each order of ɛj, to cancel the error uɛ-∑j=0n‍ɛjuj at x=0. Hence, we supplement (37) with the boundary conditions
(38)θ0=g1(y)-u0(0,y)atx=0,θj=-uj(0,y)atx=0,1≤j≤n,θj⟶0asx⟶∞,0≤j≤n.
The explicit expressions of θj, 0≤j≤n, are inductively shown to be of the form
(39)θj=Pk(xɛ,y)exp(-xɛ),j=2k,2k+1,
where Pk(x/ɛ,y) is a polynomial in x/ɛ of degree k whose coefficients, independent of ɛ, are linear combinations of the ∂suj-s(0,y)/∂ys, 0≤s≤j, and Pk(0,y)=-uj(0,y).According to (39), we deduce the following lemmas.Lemma 3.
For each0≤j≤n, one has the pointwise estimates:
(40)|xs∂xi∂ymθj|≤κɛs-iexp(-cxɛ),s,i,m≥0,
for a constant c independent of x, y, and ɛ.Lemma 4.
For each0≤j≤n, one has
(41)∥xs∂xi∂ymθj∥Lp(Ω)≤κɛs-i+(1/p),s,i,m≥0.Thanks to Lemma3, we notice that the effect of θj near the boundary x=1 is exponentially small:
(42)θj(1,y)=e.s.t.,0≤j≤n.From (36), (38), and (42), we infer that
(43)uɛ-∑j=0n‍ɛj(uj+φj+ξj+θj)={∑j=0n‍ɛjφjatx=0,∑j=0n‍ɛjθj=e.s.t.atx=1,∑j=0n‍ɛj(φuj+θj)=∑j=0n‍ɛjθj+e.s.t.aty=0,∑j=0n‍ɛj(φlj+θj)=∑j=0n‍ɛjθj+e.s.t.aty=1.


## 3.4. Ordinary Corner Layers (OCL)

To account for the value∑j=0n‍ɛjφj at x=0, which is exactly the difference of uɛ and ∑j=0n‍ɛj(uj+φj+ξj+θj) at x=0 (see (43)), we introduce the ordinary corner layer correctors ζj, 0≤j≤n, in the form
(44)ζj=ζlj+ζuj,
where the ζlj (or ζuj) are the correctors near the corner (0,0) (or (0,1)) as constructed below.To defineζlj (or ζuj), we insert a formal expansion uɛ≅∑j=0∞‍ɛjζj into (1). Then, using the stretched variables x-=x/ɛ and y-=y/ɛ near (0,0) and using x- and y~=(1-y)/ɛ near (1,1), we collect the terms of order ɛj, 0≤j≤n. As a result, we find the equations for ζlj (or ζuj):
(45)-ɛ∂x2ζkj-∂xζkj=∂y2ζkj-1inΩ,0≤j≤n,k=uorl.
Here, we set ζk-1=0 for k=l,u.At each order ofɛj, 0≤j≤n, to cancel the error φj near the corner (0,0) or (0,1), we supplement (45) with the boundary conditions
(46)ζkj=-φkj(0,y)atx=0,0≤j≤n,k=l,u,ζkj⟶0asx⟶∞,0≤j≤n,k=l,u.The explicit solutionsζlj of (45) and (46), 0≤j≤n, are inductively found to be of the following form:
(47)ζlj=Pj(xɛ,yɛ)exp(-xɛ),
where Pj(x/ɛ,y/ɛ) is a polynomial in x/ɛ of degree j whose coefficients, independent of ɛ, are linear combinations of the quantities ɛs∂2sφlj-s(0,y/ɛ)/∂y2s, 0≤s≤j. In the same manner, the explicit forms of the ζuj, 0≤j≤n can be found as well.Using Lemma1 and (47), we obtain the lemmas below.Lemma 5.
For each0≤j≤n and k,s≥0, one has the pointwise estimates:
(48)|xkys∂xi∂ymζlj|≤κɛk-i+((s-m)/2)exp(-c(xɛ+yɛ)),0≤i+m≤2n+2-2j,|xkys∂xi∂ymζuj|≤κɛk-i+((s-m)/2)exp(-c(xɛ+1-yɛ)),0≤i+m≤2n+2-2j,
for a generic constant c>0 independent of x, y, and ɛ.Lemma 6.
For each0≤j≤n and k,s≥0, one has, for 0≤i+m≤2n+2-2j,
(49)∥xkys∂xi∂ymζlj∥Lp(Ω)+∥xkys∂xi∂ymζuj∥Lp(Ω)≤κɛk-i+((s-m)/2)+(3/2p).Thanks to (46) and Lemma 5, we estimate the values of  ζj, 0≤j≤n, along the sides x=1, y=0, and y=1:
(50)ζj={e.s.t.atx=1,ζlj(x,0)+e.s.t.aty=0,ζuj(x,0)+e.s.t.aty=1.Now from (43), (46), and (50), we observe that
(51)uɛ-∑j=0n‍ɛj(uj+φj+ξj+θj+ζj)={0atx=0,∑j=0n‍ɛj(θj+ζj)=e.s.t.atx=1,∑j=0n‍ɛj(φuj+θj+ζj)=∑j=0n‍ɛj(θj+ζlj)+e.s.t.aty=0,∑j=0n‍ɛj(φlj+θj+ζj)=∑j=0n‍ɛj(θj+ζuj)+e.s.t.aty=1.


## 3.5. Supplementary Correctors

Now, our task is to handle, at each order0≤j≤n, the effects of θj and ζj along the top and bottom boundaries y=0,1. To this end, using (51), we construct the supplementary correctors ηj in the form
(52)ηj=ηlj+ηuj,0≤j≤n,
where
(53)ηlj=-(1-y)(θj+ζlj)(x,y=0),ηuj=-y(θj+ζuj)(x,y=1).From Lemmas3 and 5, it is easy to verify the lemmas below.Lemma 7.
For each0≤j≤n and s≥0, one has the pointwise estimates:
(54)|xs∂xi∂ymηj|≤κɛs-iexp(-cxɛ),0≤i+m≤2n+2-2j,
for a constant c independent of x, y, and ɛ.Lemma 8.
For each0≤j≤n and s≥0, one has
(55)∥xs∂xi∂ymηj∥Lp(Ω)≤ɛs-i+(1/p),0≤i+m≤2n+2-2j.Using (3), (14), (17), (22), (23), (38), and (46), we find that, near the corner (0, 0),
(56)(θ0+ζl0)(0,0)=g1(0)-u0(0,0)-φl0(0,0)=g1(0)-u0(0,0)-(g3(0)-u0(0,0))=0.
More generally, one can verify that
(57)(θj+ζlj)(0,0)=0,(θj+ζuj)(0,1)=0,0≤j≤n.Using (42), (50), and (57), we infer from (52) and (53) that, for 0≤j≤n,
(58)ηj={0atx=0,e.s.t.atx=1,-(θj+ζlj)aty=0,-(θj+ζuj)aty=1.Now, we finally obtain from (51) and (58) that
(59)uɛ-∑j=0n‍ɛj(uj+φj+ξj+θj+ζj+ηj)={0atx=0,∑j=0n‍ɛj(θj+ζj+ηj)=e.s.t.atx=1,∑j=0n‍ɛj(φuj+θj+ζj+ηj)=∑j=0n‍ɛj(θj+ζlj+ηj)+e.s.t.=e.s.t.aty=0,∑j=0n‍ɛj(φlj+θj+ζj+ηj)=∑j=0n‍ɛj(θj+ζuj+ηj)+e.s.t.=e.s.t.aty=1.
The e.s.t. on the right hand side of (59)2,3,4 are caused, respectively, by (θj+ζj+ηj) at x=1, (φuj+ζuj) at y=0, and (φlj+ζlj) at y=1.


## 4. Asymptotic Error Analysis: The Main Result

We define(60)ρ~≔(uɛ-∑j=0n‍ɛj(uj+Θj))|∂Ω=(the right-hand side of(59)),
where Θj=φj+ξj+θj+ζj+ηj as defined in (11).We construct an extensionρ of ρ~ in the form
(61)ρ(x,y):=r+(1-x)(ρ~-r)|x=0+x(ρ~-r)|x=1+(1-y)(ρ~-r)|y=0+y(ρ~-r)|y=1,
where
(62)r(x,y):=(1-x)(1-y)ρ~|x=y=0+x(1-y)ρ~|x=1,y=0+(1-x)yρ~|x=0,y=1+xyρ~|x=y=1.
Then one can verify that ρ|∂Ω=ρ~ and
(63)ρisane.s.t.inΩ.We set(64)wɛn:=uɛ-∑j=0n‍ɛj(uj+Θj)-ρ.
Then, using (1), (7), (11), (30), (34), (37), (45), (53), and (60), the equation of wɛn reads
(65)Lɛwɛn=ɛn+1(Δun+∂x2φn+∂y2θn+Ξ1n)+ɛn(∂y2θn-1+ɛ∂y2ζn+Ξ2n)+e.s.t.inΩ,wɛn=0on∂Ω,
where
(66)Ξ1=(1-y)∂y2θn(x,y=0)+y∂y2θn(x,y=1),Ξ2=(1-y)(∂y2θn-1+ɛ∂y2ζln)(x,y=0)+y(∂y2θn-1+ɛ∂y2ζun)(x,y=1).We multiply (65) by exwɛn and integrate over Ω. Then, using Lemmas 1, 3, and 5 and the Hardy inequality (see, e.g., [2]), we find that
(67)ɛ|wɛn|H12+1-ɛ2|wɛn|L22≤κɛn+1|∫Ω‍(Δun+∂x2φn+∂y2θn+Ξ1n)wɛndxdy|+κɛn|∫Ω‍x(∂y2θn-1+ɛ∂y2ζn+Ξ2n)(wɛnx)dxdy|≤κɛ2n+2+14|wɛn|L22+ɛ2|wɛn|H12.Thanks to the elliptic regularity theory and (67), we also find that
(68)∥wɛn∥H2(Ω)≤κ∥Δwɛn∥L2(Ω)≤κɛ-1∥∂xwɛn+lower order terms∥L2(Ω)≤κɛn-(1/2).Finally, from (63), (64), (67), and (68), we obtain the convergence result on the difference of uɛ and the proposed asymptotic expansions that we summarize in  Theorem 9.Theorem 9.
For each fixedn≥0, as the diffusivity parameter ɛ vanishes, the difference between the solution uɛ of (1) and its asymptotic expansion (6) converges to zero in the following sense:
(69)∥uɛ-∑j=0n‍ɛj(uj+Θj)∥ɛ≤κɛn+1,(70)∥uɛ-∑j=0n‍ɛj(uj+Θj)∥H2(Ω)≤κɛn-(1/2).Remark 10.
Due to the continuity ofg as appearing in (3), ξ0 can be omitted in (11) for j=0 at order ε0. Then one can verify that
(71)∥uε-(u0+φ0+θ0+ζ0+η0)∥ε≤κε3/4,
which gives a convergence estimate that a factor of ε1/4 is worse than what is stated in (69) for n=0.



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*Source: 532987-2013-05-12.xml*