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\newcommand{\bc}{\begin{center}}
\newcommand{\ec}{\end{center}}
\def \vf {\varphi}
\def \eps {\varepsilon}
\def \s {\sigma}
\def \Lh {\Lambda}
\def \om {\omega}
\def \bom {\overline {\omega}}
\def \G {\Gamma}
\def \g {\gamma}
\def \Gb {\ovl {G}}
\def \Db {\ovl {D}}
\def \Gbh {\ovl {G}_h}
\def \Dbh {\ovl {D}_h}
\def \d {\partial}
\def \ov {\over}
\def \eq {\equiv}
\newcommand{\til}{\widetilde}
\newcommand{\ovl}{\overline}
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\newcommand{\r}[1]{\mbox{\rm{#1}}}
\def\theequation{\arabic{section}.\arabic{equation}}
\parindent 3em
\begin{document}
\sloppy
\title {Grid approximations of singularly perturbed \\
systems of parabolic convection-diffusion equations \\
with counterflow \thanks{This research was supported in part
by the Russian Foundation for Basic Research under Grant
N 95-01-00039 and by the Netherlands Organization for Scientific
Research NWO, dossiernr. 047.003.017.}}
\author{ G.I.~Shishkin }
\date{}
\maketitle
\vspace{-1cm}
\bc
{\it Institute of Mathematics and Mechanics, Ural Branch of the
Russian Academy of Sciences, 620219 Ekaterinburg, Russia. ~
E-mail: Grigorii@shishkin.ural.ru}
\ec
\begin{abstract}
The first boundary value problem is considered on a strip for a
system of two singularly perturbed parabolic equations.
The perturbation parameters multiplying the
highest derivatives of each of the equations are mutually
independent and can take arbitrary values from the interval $(0,1]$.
When these parameters equal zero, the system of parabolic
equations degenerates into a system of hyperbolic first-order
equations coupled by the reaction terms. The convective terms
(i.e., their components orthogonal to the boundaries of the strip)
that are involved in the different equations have opposite
directions (that is to say, convection with counterflow). This case
brings us to the appearance of boundary layers in the neighbourhood
of both boundaries of the strip.
For this boundary value problem difference schemes, which converge
uniformly with respect to the parameters,
are constructed here using the condensing mesh method.
We also consider the construction of parameter-uniform
convergent difference schemes in the case of a system of singularly
perturbed elliptic equations which degenerate into first order
equations, if the parameter equals zero.
\end{abstract}
\begin{center}
{\large\bf Introduction}
\end{center}
The investigation of the nonstationary process of heat and mass transfer
in a moving medium, when the heat conduction and diffusion coefficients
are small, leads to the boundary-value problem solution for systems
of singularly
perturbed parabolic equations, i.e., parabolic equations with the highest
derivatives multiplied by a small parameter. If the parameter equals
zero, these equations transform into first-order equations with
convective terms.
When the parameter tends to zero, boundary layers appear
in the neighbourhood of those parts of the boundary, where the characteristics
of the reduced equations leave the domain.
For these boundary value problems it is necessary to develop
special difference schemes, for which the error in the approximate solution
is independent of the parameter and depends only on the number of
mesh points used, i.e., difference schemes that converge uniformly
with respect to the parameter.
In [4--7] schemes which converge uniformly with respect to the parameter
were constructed for a scalar equation by the method of specially
condensing meshes (described, for instance, in [3,\,4]). Note that
the fitted operator method (see description of the method, e.g., in [8,\,9])
can be used to construct special difference schemes
for some singularly perturbed equations with convective terms.
Here we consider the construction of special difference schemes
for a Dirichlet problem on a strip in the case of a system of two singularly
perturbed parabolic equations, which are coupled by the reaction terms.
When the parameter is equal to zero, the second-order equations degenerate
into first-order equations containing both space and time derivatives.
The convective terms in the different equations (i.e., their components
orthogonal to the boundaries of the strip) have opposite directions.
In this backward-flow case boundary layers appear in the neighbourhood of
both boundaries of the strip.
Using the technique of special condensing grids, we construct
difference schemes convergent uniformly with respect to the parameter.
A similar technique is used in the construction of parameter-uniformly
convergent difference schemes for systems of elliptic equations
with convective terms.
In [10] the author considered the system for equations where
convective flows, i.e., their components orthogonal to the boundary,
were directed to one and the same boundary of the strip. In that
forward-flow case, unlike what has been stated above,
a boundary layer appears only in the neighbourhood of this boundary
as the parameters multiplying the highest derivatives of the equations
tend to zero. It should be pointed out that the main terms of the
regular components in the asymptotic expansions are the solution of a
system of hyperbolic equations with the boundary conditions given
on that part of the boundary
across which the flow enters the domain,
i.e., on one side of the boundary only in the case of forward flow and
on both sides in the case of counterflow.
\bc
{\large\bf 1. Formulation of the problem}
\ec
{\bf 1.} On the strip of width $d$
$$
D = \{x:\ \ 00,\ \ b^2_1(x,t)<0, \ \
\left| b^i_1(x,t)\right| \geq b_0,\quad (x,t)\in\ovl{G},\ \ i,s=1,2,
$$
$a_{0}, \ p,\ b_0 > 0.$
The components $\eps_{i}$ of the vector-parameter $(\eps_1,\eps_2)^T$
take arbitrary values from the half-interval (0,1].
In the sequel, when investigating the solution of the boundary value problem
and its derivatives, we shall assume for simplicity that the following
condition holds
$$
c^{ii}(x,t) \ge c_{0},\ \ mc^{ii}(x,t) \ge\ \mid c^{ij}(x,t) \mid,\quad
(x,t) \in \Gb,
\eqno(1.3)
$$
$$
c_{0} > 0,\ \ i,\ j=1,2,\ i\neq j,\ \ m = m_{(1.3)} < 1.
$$
If condition (1.3) is not satisfied (e.g., when the coefficients
$c^{ii}(x,t)$ can take the values of different signs
on $\Gb$ or be negative) we pass to the function
$v(x,t) = u(x,t)\exp(\alpha t)$ and
choose $\alpha$ sufficiently large so that
condition (1.3) would be satisfied for the coefficients
in the equations for the function $v(x,t)$.
We shall also use such a matrix form for system (1.2)
$$
Lu(x,t) = f(x,t),\ (x,t) \in G,\ \ u(x,t) = \vf (x,t),\ (x,t) \in S.
\eqno(1.2\mbox{c})
$$
Here
$$
Lu(x,t) = L_{(1.2)}(\eps_{1},\eps_{2})u(x,t) \eq \left(
\begin{array}{cc}
L^{1}_{0} & 0 \\
0 & L^{2}_0
\end{array} \right)
u(x,t) -C(x,t)u(x,t)=
$$
$$
=\left\{\eps^0\!\sum_{s=1,2}A_{s}(x,t)\frac{\d^2}{\d x_s^2}+
\sum_{s=1,2}B_s(x,t)\frac{\d}{\d x_s}-P(x,t)\frac{\d}{\d t}-
C(x,t)\right\}u(x,t),
$$
$$
\eps^0=\left(
\begin{array}{cc}
\eps_1 & 0\\
0 & \eps_2
\end{array} \right)\,, \ \
A_{s}(x,t)=\left(
\begin{array}{cc}
a^1_{s}(x,t) & 0\\
0 & a^2_{s}(x,t)
\end{array} \right)\,, \ \
B_{s}(x,t)=\left(
\begin{array}{cc}
b^1_{s}(x,t) & 0\\
0 & b^2_{s}(x,t)
\end{array} \right)\,,
$$
$$
P(x,t)=\left(
\begin{array}{cc}
p^1(x,t) & 0\\
0 & p^2(x,t)
\end{array} \right)\,, \quad
C(x,t) = \left ( \begin{array}{cc}
c^{11}(x,t) & c^{12}(x,t)\\
c^{21}(x,t )& c^{22}(x,t)
\end{array} \right )\,,
$$
$u(x,t),\ f(x,t),\ \vf(x,t)$ are vector-functions (vector-columns).
Here and below by $M,\ M_{i}$ (or $m,\ m_{i}$) we denote sufficiently
large (small) positive constants that are independent of the parameters
$\eps_1$, $\eps_2$. In the case of grid problems these constants do not
depend on parameters of stencils in the difference schemes used.
The notation $m_{(j.k)}$, $M_{(j.k)}$ ($L_{(j.k)}$, $f_{(j.k)}(x,t)$)
means that these constants (or operators, functions) are first introduced
in formula $(j.k)$.
The solution of the problem is regarded as a function
$u\in C^{2,1}(G)$, which is continuous and bounded on $\Gb$,
and satisfies the differential equations (1.2a) on
$G$ and the boundary conditions (1.2b) on $S$.
Model problems of this kind arise, for example, if we study
heat transfer in counterflow heat exchangers. In such devices,
the flows of heat-conducting fluids (gases) are separated by
heat-conducting membranes; these flows having different inlet
temperatures are directed towards each other. Such mathematical
problems appear, for example, in modeling diffusion process
for multicomponent systems in moving medium, where convective flows of
certain chemically reacting components take opposite directions.
In this case the parameters multiplying the highest
derivatives characterize the diffusion coefficient of the matters, the
functions $(b^{i}_1(x,t),b^{i}_2(x,t))$ describe the velocity of
convective transport for each matter, the functions $c^{ij}(x,t)$ refer
to the direct and reverse chemical reaction rates (see, e.g. [11]).
We shall assume that the solution of the problem is sufficiently smooth
when the parameter vector (or for short, parameter) is fixed.
Sufficient conditions for the existence of a smooth solution are
considered in [12].
For $\eps_1=\eps_2=0$ the system of parabolic equations (1.2a) degenerates
into a system of hyperbolic equations. We define the positive direction of
the characteristics for the reduced equation by the vector
$$
b^i(x,t)=(-b_1^i(x,t),\,-b_2^i(x,t),\,p^i(x,t))),\ \ (x,t)\in\Gb,\ i=1,2.
$$
These characteristics of the operator $L^1_{0(1.2)}(\eps_1=0)$
(or $L^2_{0(1.2)}(\eps_2=0)$) leave the domain $G$ through the boundary
$S_1^L$ (through the boundary $S_2^L$). As the parameters tend to zero,
regular boundary layers, i.e., boundary layers described by ordinary
differential equations, appear in the neighbourhood of the boundary $S^L$; \
$S^L=S_1^L\cup S_2^L$,
$S^L_i=\G_i\times (0,T]$, $\G_1=\{x:\, x_1=0\}$, $\G_2=\{x:\, x_1=d\}$.
{\bf 2.} The technique of constructing and justifying
special finite difference schemes for problem (1.2), (1.1) is similar
to that given in [10].
We shall consider separately each of the boundary value
problems with the following variants of the component-parameters:
$$
\eps_{1}=\eps_{2}=\eps,\ \ \eps \in (0,1],
\eqno(1.4)
$$
$$
\eps_{1}=\eps,\ \ \eps_{2}=1,\ \ \eps \in (0,1],
\eqno(1.5)
$$
$$
\eps_{1}=\eps,\ \ \eps_{2}=\mu,\ \ \eps,\ \mu \in (0,1].
\eqno(1.6)
$$
For definiteness we suppose $\eps\leq \mu$.
Problem (1.2), (1.6) is more general than problems (1.2), (1.4)
and (1.2), (1.5). Singular parts of the solutions of problems (1.2), (1.4)
and (1.2), (1.5) are simpler than those of (1.2), (1.6).
This means that a priori estimates for the solutions of these first two
are more accurate and give a higher order of convergence of the special
schemes than for problem (1.2), (1.6).
{\bf 3.}
It should be noted that classical difference schemes even for one
singularly perturbed equation do not yield the approximate solutions
which converge uniformly with respect to the parameter [4,\,8].
Thus, in the case of boundary value problems for systems of
singularly perturbed parabolic equations with convective terms,
we come to the problem of constructing
special difference schemes convergent uniformly in the parameter.
\begin{center}
{\large\bf 2. A priori estimates of the solutions and derivatives}
\end{center}
{\bf 1.} We give estimates for the solutions of boundary value problems
(1.2), (1.4)--(1.6). The operators ~
$ L^{i} \eq L_{0(1.2)}^i-c^{ii}(x,t), \ \ i=1,\,2$ ~
are monotone.
We shall assume that the following condition is fulfilled:
$$
u\in C^{4+\alpha,2+\alpha/2}(\Gb), \quad \alpha\in (0,1).
\eqno(2.1)
$$
Using the maximum principle [12,\,13] we get the uniform with respect to
the parameter boundedness of the solution
$$
\mid u(x,t)\mid\ \le M,\ \ (x,t) \in \Gb,
\eqno(2.2\mbox{a})
$$
where \ \ \ \ \
$ \mid u(x,t)\mid\ = \max\limits_{i}\mid u^{i}(x,t)\mid,$ \
$M = M_{(2.2)}$,
$$
M_{(2.2)} =2(1-m^2_{(1.3)})^{-1}\max\,\left[\,c^{-1}_{0}\max_{\Gb}\mid f(x,t) \mid,
\,\max_{S}\mid \vf(x,t) \mid\ \right].
$$
To simplify finding estimates of the derivatives, we suppose the following
condition to be satisfied:
$$
a_{s}^i(x,t)\eq a_{s}^i,\ \ b_s^i(x,t)\eq b_s^i,\ \ p^i(x,t)\eq p^i,\quad
(x,t)\in\Gb,\ \ s,\,i=1,2.
\eqno(2.3)
$$
Using the majorant function technique (see, e.g. [14]), we obtain the
estimates
$$
\left| {\d ^{k+k_0} \ov \d x^{k_1}_1 \d x^{k_2}_2 \d t^{k_0}}
u(x,t) \right| \le M\left[\,\eps_{1}^{-k} + \eps_{2}^{-k}\,\right],\ \
(x,t) \in \Gb,
\ \ k+2k_0\leq 4.
\eqno(2.2\mbox{b})
$$
T\,h\,e\,o\,r\,e\,m \ 2.1. \ {\it Let the functions $u(x,t)$, the solutions
of boundary value problems $(1.2)$, $(1.4)$--$(1.6)$, satisfy the condition
$(2.1)$. Then, under condition $(2.3)$, the estimates $(2.2)$ are satisfied.}
{\bf 2.} We give a number of estimates based on the
asymptotic representations for the solution of problem
(1.2), (1.4).
We represent the function $u(x,t)=u_{(1.2,1.4)}(x,t)$, i.e.,
the solution of problem (1.2), (1.4),
as a sum of functions (vector-valued functions)
$$
u(x,t)=u_{(1.2,1.4)}(x,t) = U_1^{[0]}(x,t) +\eps U_1^{[1]}(x,t)+
V_1^{[0]}(x,t)+\eps V_1^{[1]}(x,t)+ v_1(x,t)\eq
\eqno(2.4)
$$
$$
\eq U_1(x,t)+V_1(x,t)+v_1(x,t)\eq U^{(1)}_1(x,t)+V_1(x,t),\ \ (x,t) \in \Gb,
$$
where $U^{(1)}_1(x,t)$ and $V_1(x,t)$ are the "regular" and singular parts
of the solution; the derivatives of the function $U^{(1)}_1(x,t)$ of up to
the second order with respect to the space variables are bounded
$\eps$-uniformly (see estimate (2.9)).
The components from representation (2.4) are the solutions
of the problems
$$
L_{1(2.5)}^iU_1^{[0]i}(x,t) = f^i(x,t),\quad (x,t) \in G\cup S^L_i,
\eqno(2.5)
$$
$$
U_1^{[0]i}(x,t) = \vf^i(x,t),\quad (x,t) \in S_0\cup S^{L}_{3-i};
$$
$$
L_{(1.2)}V_1^{[0]}(x,t) = 0,\quad (x,t) \in G,
$$
$$
V_1^{[0]i}(x,t) = \vf^i(x,t) - U_1^{[0]i}(x,t), \quad
(x,t) \in S_0\cup S^L_i;
$$
$$
L_{1(2.5)}^i U_1^{[1]i}(x,t)=-L_{2(2.5)}^i U_1^{[0]i}(x,t),
\ \ (x,t)\in G\cup S^L_i,
$$
$$
\begin{array}{c}
U_1^{[1]i}(x,t)=-\eps^{-1}V_1^{[0]i}(x,t),\quad (x,t) \in S^L_{3-i},\\[0.8ex]
U_1^{[1]i}(x,t)=0,\quad (x,t) \in S_0;
\end{array}
$$
$$
L_{(1.2)}V_1^{[1]}(x,t) = 0,\quad (x,t) \in G,
$$
$$
\begin{array}{c}
V_1^{[1]i}(x,t) = - U_1^{[1]i}(x,t), \quad (x,t) \in S^L_i,\\[0.8ex]
V_1^{[1]i}(x,t)=0,\quad (x,t) \in S_0;
\end{array}
$$
$$
L_{(1.2)}v_1(x,t)=-\eps^2L_{2(2.5)}U^{[1]}_1(x,t),\quad (x,t)\in G,
$$
$$
\begin{array}{c}
v_1^i(x,t)=-\eps V_1^{[1]i}(x,t),\quad (x,t)\in S^L_{3-i},\\[0.8ex]
v_1^i(x,t)=0,\quad (x,t)\in S_0\cup S_i^L; \qquad i=1,2.
\end{array}
$$
Here
$$
L_{1(2.5)}u(x,t)\eq\left\{\sum_{s=1,2}B_s(x,t)\frac{\d}{\d x_s}-P(x,t)
\frac{\d}{\d t}-C(x,t)\right\}\,u(x,t),
$$
$$
L_{2(2.5)}u(x,t)\eq \sum_{s=1,2}A_{s}(x,t)\frac{\d^2}{\d x_s^2}
\,u(x,t),
$$
the functions $V_1^{[0]}(x,t)$ and $V_1^{[1]}(x,t)$ decrease
exponentially as the point $(x,t)$ moves away from the set $S^L$.
Let the following condition be satisfied:
$$
U_1^{[0]}\in C^{K+\alpha,(K+\alpha)/2}(\Gb),\ \, U_1^{[1]},\, V_1,\, v_1\in
C^{K-2+\alpha,(K-2+\alpha)/2}(\Gb),
\ \ \alpha\in (0,1),\ \, K\geq 6.
\eqno(2.6)
$$
Then for the functions $U_1^{[0]}(x,t)$, $U_1^{[1]}(x,t)$, $V_1(x,t)$,
$v_1(x,t)$ the estimates are fulfilled:
$$
\left| {\d ^{k+k_{0}} \ov \d x^{k_{1}}_{1}\d x^{k_{2}}_{2}\d t^{k_{0}}}
U_1^{[j]}(x,t)\right| \le M, \quad j=0,\,1,
\eqno(2.7\mbox{a})
$$
$$
\mid V_1^i(x,t)\mid\ \leq M\left[\exp\left(-m\eps^{-1}r(x,\G_i)\right)+\eps
\exp\left(-m\eps^{-1}r(x,\G_{3-i})\right)\right], \quad
i=1,\,2,
\eqno(2.7\mbox{b})
$$
$$
\mid v_1(x,t)\mid\ \leq M\eps^2,\quad (x,t)\in\Gb,\ \ k+2k_0\leq 4,
\eqno(2.7\mbox{c})
$$
where $m=m_{(2.7)}$ is a sufficiently small number, $m_{(2.7)}0,\ \, b^2_1(x)<0, \ \
\left| b^i_1(x)\right| \geq b_0, \quad x\in\Db,\ \ i,s=1,2,\ \
a_0,\, b_0 > 0.
$$
The components $\eps_i$ of the vector-parameter $(\eps_1,\eps_2)^T$
take any values from the half-interval $(0,1]$.
When investigating the solution of the problem and its derivatives,
for simplicity, we shall assume that the following condition is satisfied:
$$
c^{ii}(x) \ge c_0,\ \ mc^{ii} \ge \ \mid c^{ij}(x) \mid,\quad x \in \Db,
$$
where \ $c_0>0,\ \ i,j = 1,2,\ \ i \ne j,\quad m< 1$.
The solution of problem (5.1) is regarded as a function
$u \in C^2(D)$, which is continuous and bounded on $\Db$, and satisfies
equations (5.1a) on $D$ and boundary conditions (5.1b) on $\G$.
We assume that the solution of this problem is sufficiently smooth
for fixed values of the parameters.
Conditions sufficient for the existence of a smooth
solution are considered in [15].
When the parameters tend to zero, regular boundary layers appear in the
neighbourhood of the set $\G$.
\setcounter{section}{5}
\setcounter{equation}{1}
We shall separately consider the boundary value problems for the following
variants of the component-parameters:
\begin{subequations} \label{5.2}
\begin{eqnarray}
&& \mbox{either } ~~~ \eps_1 = \eps_2 = \eps,\quad \eps \in (0,1],\\[0.5ex]
&& \mbox{or } ~~~ \eps_1 = \eps,\ \ \eps_2 = 1,\quad \eps \in (0,1],\\[0.5ex]
&& \mbox{or else } ~~~ \eps_1 = \eps,\ \ \eps_2 = \mu,\quad \eps,\, \mu \in (0,1],
\ \ \eps\leq \mu.
\end{eqnarray}
\end{subequations}
{\bf 2.} The {\it a-priori} estimates for the solutions of the above problems,
including their derivatives, can be found similarly to Section 2
for parabolic equations. We suppose that these solutions are sufficiently
smooth for each fixed value of the vector-parameter.
Actually, as a result, we can find those conditions for the data of
problems (5.1), (5.2) under which the solutions would
satisfy the {\it a-priori} estimates similar to the estimates
for boundary value problems (1.2),
(1.4)--(1.6). More precisely, the estimates like (2.2), including only
the space derivatives, hold for the solutions and their derivatives
in the case of problems (5.1), (5.2), and also
the estimates like (2.7b), (2.9),
(2.13), and (2.17), (2.21), (2.25) hold for the components from relevant
asymptotic expansions. In that case we shall say that the solutions
$u(x)$ of boundary value problems (5.1), (5.2) have the right
behaviour.
{\bf 3.} To solve problem (5.1), (5.2) we use the following difference scheme.
On the set $\ovl{D}$ we introduce the mesh
$$
\ovl{D}_h=\ovl{D}_{h(3.1)}.
\eqno(5.3)
$$
On the mesh $\ovl{D}_h$ we consider the difference scheme
$$
\Lh_{(5.4)}^i z(x)=f^i(x),\ \ x\in D_h,\quad
z^i(x)=\vf^i(x),\ \ x\in\G_h,\quad i=1,2.
\eqno(5.4)
$$
Here \ \ $D_h=D\cap\ovl{D}_h,\ \ \G_h=\G\cap\ovl{D}_h$,
$$
\Lh^i_{(5.4)}z(x)\equiv \Lh_0^i z^i(x)-\sum_{j=1,2}c^{ij}(x)z^j(x),
$$
$$
\Lh_0^iz^i(x)=\Lh_0^i(\eps_i)z^i(x)
\equiv \left\{\eps_i\sum_{s=1,2}a_{s}^i(x)\delta_{\ovl{xs}\,\widehat{xs}}+
\sum_{s=1,2}\left[\,b_s^{i+}(x)\delta_{xs}+
b_s^{i-}(x)\delta_{\ovl{xs}}\,\right]\right\}
z^i(x).
$$
The operators ~
$
\Lh^i z(x)\equiv \left\{\Lh_0^i-c^{ii}(x)\right\}z^i(x),\ \
x\in D_h,\ \, i=1,2 $ ~
are monotone.
We study the convergence of finite difference schemes for elliptic
equations similarly to the analysis done in Sections 3, 4
for parabolic equations.
For the solutions of the difference scheme (5.4), (5.3)
in the case of problems (5.1), (5.2) we obtain, respectively,
the estimates
$$
\begin{array}{lll}
\mid u(x)-z(x)\mid\ \leq M\eps^{-2}N^{-1}, &
x\in\ovl{D}_h & \mbox{ for} ~~ \eps_1=\eps_2=\eps,\\[1ex]
\mid u(x)-z(x)\mid\ \leq M\eps^{-3}N^{-1}, &
x\in\ovl{D}_h & \mbox{ for} ~~ \eps_1=\eps,\ \ \eps_2=1,\\[1ex]
\mid u(x)-z(x)\mid\ \leq M\mu\eps^{-3}N^{-1}, &
x\in\ovl{D}_h & \mbox{ for} ~~ \eps_1=\eps,\ \ \eps_2=\mu.
\end{array}
\eqno(5.5)
$$
{\bf 4.} Now we give the special difference schemes for boundary
value problems $(5.1)$, $(5.2)$.
In case ~$\eps_1=\eps_2=\eps$ ~
and ~$\eps_1=\eps,\ \eps_2=1$ ~we introduce, on the set $\ovl{D}$, the mesh
$$
\begin{array}{ll}
\ovl{D}_h^{\,c}=\ovl{D}_{h(4.2)}^{\,c} &
\mbox{for} ~~ \eps_1=\eps_2=\eps,\\[1ex]
\ovl{D}_h^{\,c}=\ovl{D}_{h(4.4)}^{\,c} &
\mbox{for} ~~ \eps_1=\eps,\ \ \eps_2=1,
\end{array}
\eqno(5.6\mbox{a})
$$
where
$$
\sigma_{(4.2)}=\sigma_{(4.2)}(\eps,N_1,m_{(5.6)}), \quad
\sigma_{(4.4)}=\sigma_{(4.4)}(\eps,N_1,m_{(5.6)}),
\eqno(5.6\mbox{b})
$$
$m_{(5.6)}$ is any number from the interval $(0,m_{0})$,
$m_0=(a^0)^{-1}b_0$. We solve these problems by scheme (5.4) on
the mesh $\ovl{D}_h=\ovl{D}_{h(5.6)}^{\,c}$.
Taking into account the {\it a-priori} estimates, we establish
the $\eps$-uniform convergence of scheme (5.4), (5.6) with the
error bounds given by
$$
\mid u(x)-z(x)\mid\ \leq MN^{-1}\ln N,\ \ x\in\ovl{D}_h \quad
\mbox{for} ~~~ \eps_1=\eps_2=\eps ~~~
\eqno(5.7)
$$
and
$$
\mid u(x)-z(x)\mid\ \leq MN^{-1}\ln N,\ \ x\in\ovl{D}_h \quad
\mbox{for} ~~ \eps_1=\eps,\ \ \eps_2=1,
\eqno(5.8)
$$
where $\ovl{D}_h=\ovl{D}_{h(5.6)}^{\,c}$.
In case ~$\eps_1=\eps,\ \eps_2=\mu$ ~ we use the difference scheme
(5.4) on the mesh
$$
\ovl{D}_h^{\,c}=\ovl{D}_{h(4.6)}^{\,c}.
\eqno(5.9\mbox{a})
$$
Here
$$
\begin{array}{l}
\sigma_{1(4.6)}=\sigma_1(\eps,N_1,m_{(5.9)}),\\[1ex]
\sigma_{2(4.6)}=\sigma_2(\eps,\mu,N_1,m_{(5.9)}),
\end{array}
\eqno(5.9\mbox{b})
$$
$m_{(5.9)}$ is any number from the interval $(0,m_{0})$.
Taking into account the {\it a-priori} estimates, we
obtain the $(\eps,\mu)$--uniform convergence of scheme (5.4), (5.9)
$$
\mid u(x)-z(x)\mid\ \leq MN^{-1/10},\ \ x\in \ovl{D}_{h(5.34)}^{\,c} \quad
\mbox{for} ~~ \eps_1=\eps,\ \ \eps_2=\mu.
\eqno(5.10)
$$
T\,h\,e\,o\,r\,e\,m \ 5.1.\ {\it Let the solutions of boundary value
problems $(5.1)$, $(5.2)$ have the right behaviour. Then the following
assertions hold: \ (i) classical difference scheme $(5.4)$, $(5.3)$
converges for fixed values of the parameters; \
(ii) \ special difference scheme $(5.4)$, $(5.6)$
in the case of problems $(5.1)$, $(5.2\r{a})$ and $(5.1)$, $(5.2\r{b})$
(for $\eps_1=\eps_2=\eps$ and $\eps_1=\eps,\ \eps_2=1$), and
also scheme $(5.4)$, $(5.9)$ in the case of problem $(5.1)$, $(5.2\r{c})$
(for $\eps_1=\eps,\ \eps_2=\mu$) converge uniformly with respect to the
parameters. The solutions of these difference schemes satisfy estimates
$(5.5)$, $(5.7)$, $(5.8)$, $(5.10)$.}
\begin{center}
{\large\bf References}
\end{center}
\small
\begin{enumerate}
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% \end{thebibliography}
\end{enumerate}
\end{document}