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\def \eps {\varepsilon}
\def \ov {\over}
\def \vf {\varphi}
\def \d {\partial}
\def \Lh {\Lambda}
\def\om{\omega}
\def \oom {\overline {\omega}}
\def \bom {\overline {\omega}}
\def \eq {\equiv}
\def \s {\sigma}
\def \G {\Gamma}
\def \Gb {\ovl {G}}
\def \Db {\ovl {D}}
\def \Gbh {\ovl {G}_h}
\def \Dbh {\ovl {D}_h}
\def \dtb { \delta_{ \ovl {t}}}
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\begin{document}
\sloppy
%\large
%\baselineskip 20pt
\noindent
\date{\empty}
\title{Grid Approximation of Singularly Perturbed
Systems of Elliptic and Parabolic Equations with Convective Terms \thanks{This
work was supported by the Russian Foundation for Basic Research,
Grant N 95-01-00039}}
\author{G. I. Shishkin \thanks{Institute of Mathematics and Mathematics,
Ural Branch of the Russian Academy of Sciences, Ekaterinburg 620219,
Russia. E-mail: Grigorii@shishkin.ural.ru}}
\maketitle
\vspace{-0.5cm}
%\normalsize
%\baselineskip 12pt
\begin{abstract}
The third boundary value problem with mixed boundary conditions is
considered on a strip for a system of two singularly perturbed parabolic
equations with convective terms. The perturbation parameters $\eps_i,\,
i=1,2$ multiplying the highest derivatives of the $i$-th equation 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 unknown solutions (i.e., by the reaction terms).
Using the condensing mesh method, we construct finite difference schemes,
which converge uniformly with respect to the parameters.
A similar technique with a proper modification can be used
for the case of a system of singularly perturbed elliptic equations
with convective terms, considered on the same strip.
%\medskip
%{\bf AMS} (1991): 65N06, 65N12
\end{abstract}
%\thispagestyle{myheadings}
%\markboth
%{\sc G. I. Shishkin}
%{\sc Singularly Perturbed Systems of Parabolic Equations}
%\large
%\baselineskip 20pt
\section*{Introduction}
The investigation of nonstationary processes of heat and mass transfer in
moving medium, when heat conduction and diffusion coefficients are small,
leads to the numerical solution of boundary value problems for systems
of singularly perturbed parabolic equations with convective terms.
If the parameters equal zero, these equations degenerate into
hyperbolic equations.
When the parameters tend to zero, a boundary layer appear
in the neighbourhood of that part of the boundary where the characteristics
of reduced equations leave the domain.
The use of classical difference schemes \cite{marc,sama} for these problems
gives rise to difficulties due to restricted smoothness of the solution
for small values of the parameter.
Therefore, it is necessary to develop special difference schemes
for which the error in the approximate solutions
is independent of the parameter and depends only on the number of
the mesh nodes, i.e., schemes convergent uniformly
with respect to the parameter.
Schemes which converge uniformly with respect to the parameter
were constructed in \cite{shi}--\cite{mill} for a stationary scalar equation
by the use of the method of special condensing meshes (see, e.g.,
\cite{shi,mill,bakh} for description of this method).
In this paper we consider parameter-uniform discrete approximations
for a system of two singularly perturbed parabolic equations.
This system degenerates into a system of hyperbolic equations
as the parameter is equal to zero.
Using the technique of special condensing grids, difference schemes
convergent uniformly with respect to the parameter are constructed.
\newsection{Problem statement}
\setcounter{section}{1}
{\bf 1.} On the strip ~$D = \{x:\ \ 0 0$; the coefficients $\alpha^i(x)$ depend on $x_1$ only,
i.e., $\alpha^i(x)=\alpha^i(x_1),\ x_1=0,d$, moreover,
$\alpha^i(0)=0$, $\alpha^i(d)$ is an arbitrary number from [0,1].
The components $\eps_{i}$ of the vector-parameter $(\eps_1,\eps_2)^T$
take arbitrary values in the half-interval (0,1].
In the sequel, for simplicity, we assume that the following
condition is true\footnote{
Here and below $M$ (or $m,\ m^0$) denote sufficiently
large (small) positive constants independent of $\eps_1$, $\eps_2$.
In the case of grid problems these constants do not
depend on the mesh parameters.
The notation $m_{(j.k)}$ ($L_{(j.k)}$, $D_{(j.k)}$)
means that this constant (or operator, set) is introduced
in the formula $(j.k)$.}:
$$
c^{ii}(x,t) \ge c_{0},\ \ mc^{ii}(x,t) \ge\ |\, c^{ij}(x,t) \,|,\ \
(x,t) \in \Gb,
\eqno(1.2)
$$
where $
c_{0} > 0,\; i,\,j=1,2,\ i\neq j,\ \ m = m_{(1.2)} < 1$.
If condition (1.2) is not fulfilled, e.g., the coefficients
$c^{ii}(x,t)$ can take values of different signs
or be negative on the set $\Gb$, we pass to the function
$v(x,t) = u(x,t)\exp(\alpha t)$ and
choose the value of $\alpha$ sufficiently large so that
condition (1.2) holds
for the coefficients in the equations for the function
$v(x,t)$.
The solution of the problem is regarded as a function which is continuous
and bounded on $\Gb$ and satisfies the differential equations (1.1a) on
$G$ and the boundary conditions (Eqs. (1.1b) and (1.1c)) on $S$.
Model problems of this kind arise, for example, when the
diffusion process studied for multycomponent systems in moving medium
is complicated by chemical reactions. The parameters multiplying the highest
derivatives characterize the diffusion coefficients of the involved matters, the
functions $b^{i}_s(x,t)$ describe the velocity of convection transport
for each of the matters, the functions $c^{ij}(x,t)$ refer to the direct and
reverse chemical reaction rates (see, e.g., \cite{brai}).
We shall assume that the solution of the problem is sufficiently smooth
for a fixed value of the vector-parameter (or, shortly, of the parameter) $\eps$.
Conditions sufficient for the existence of a smooth solution are
considered in \cite{lady}. Necessary {\it a~priori } estimates of the
solution and its derivatives can be found in the Appendix.
For $\eps_1=\eps_2=0$ the system of parabolic equations (1.1a) degenerates
into a system of hyperbolic equations. We define the positive direction of
characteristics for the reduced equation by the vector\\[1.2ex]
\centerline{$
b^i(x,t)=(-b_1^i(x,t),\,-b_2^i(x,t),\,p^i(x,t))),\quad (x,t)\in\Gb,\ \
i=1,2$.}\\[1.3ex]
Depending on the behaviour of these characteristics of the
reduced equation in a neighbourhood of the boundary $S_1$,
this boundary is splitted onto subsets.
By $S_1^-$ (or $S_1^+$) we denote that part of $S_1$ through which the
characteristics exit (enter in) the domain $G$. If condition (1.1d) holds,
we have:
$S_1^-=\{(x,t):\ x_1=0,\ x_2\in R,\ t\in (0,T]\}$,
$S_1=S_1^-\cup S_1^+$.
As $\eps\to 0$, regular boundary layers, i.e., boundary
layers described by ordinary differential equations, appear in the
neighbourhood of the side $S_{1}^-$.
\setcounter{equation}{2}
{\bf 2.} We shall consider separately each of the boundary value
problems with the following variants of the component-parameters:
\begin{eqnarray}
&\eps_{1}=\eps_{2}=\eps,\ \ \eps \in (0,1],&\\[0.7ex]
&\eps_{1}=\eps,\ \ \eps_{2}=1,\ \ \eps \in (0,1],&\\[0.7ex]
&\eps_{1}=\eps,\ \ \eps_{2}=\mu,\ \ \eps,\ \mu \in (0,1].&
\end{eqnarray}
For definiteness, we suppose $\eps\leq \mu$.
Note that problem (1.1), (1.5) is more general than problems (1.1), (1.3)
and (1.1), (1.4). Singular parts of the solutions of problems (1.1), (1.3)
and (1.1), (1.4) are simpler than that of problem (1.1), (1.5).
This allows us to get more accurate {\it a priori} estimates for the solutions
of the first two problems, in comparison with (1.1), (1.5),
which provide a higher order of convergence of the special
difference schemes.
\newsection{Classical difference schemes}
{\bf 1.} We wish to construct difference schemes for problems (1.1),
(1.3)--(1.5). We shall assume that the solutions of the boundary value
problems satisfy the estimates given in Theorem 4.1 (see the Appendix).
On the set $\Gb$ we introduce the rectangular grid
$$
\Gbh = \Dbh \times \oom_{0} = \oom_{1} \times \om_{2} \times
\oom_{0}.
\eqno(2.1)
$$
Here $\oom_{1}$ is a mesh in the interval $[0,d]$, generally speaking,
non-uniform, $\om_{2}$ and $\oom_{0}$ are uniform meshes, respectively,
on the $x_{2}$-axis with step $h_{2}$ and
in the interval $[0,T]$ with step $h_{t}$. We set
$h^{i}_{1} = x^{i+1}_{1} - x^{i}_{1}$,
$x^{i}_{1},\ x^{i+1}_{1} \in \oom _{1}$,
$h_{1} = \max_{i}h^{i}_{1}$,
$h = \max_{s}h_{s}$,
$s = 1,2$.
We denote by $N_{1}+1$ and $N_{0}+1$ the number of nodes in the meshes
$\oom _{1}$ and $\oom_{0}$, respectively, by $N_{2}+1$ the number of
nodes in the mesh $\oom_{2}$ on the segment with unit length. Let
$N = \min_{s}N_{s}$, $s = 1,2,$ $h \le MN^{-1}$.
On the grid $\Gb _{h(2.1)}$, we associate problem (1.1) with the
difference scheme
$$
\begin{array}{l}
\Lh^{i}z(x,t)= f^{i}(x,t),\ \ (x,t) \in G_{h},\quad
\lambda^iz^{i}(x,t) =\psi^{i}(x,t),\ \ (x,t) \in S_{1h},\\[0.7ex]
~~z^{i}(x,t)=\vf ^{i}(x,t),\quad (x,t) \in S_{0h}, \qquad i=1,2.
\end{array}
\eqno(2.2)
$$
Here \ \ \ $G_{h} = G \bigcap \Gbh,$\ \ $S_h=S\bigcap \Gbh,$
$$
\Lh ^{i}z(x,t)=\Lh ^{i}_{(2.2)}(\eps_i)z(x,t)\eq
\Lh ^{i}_{0}z^{i}(x,t) - \sum _{j=1,2}c^{ij}(x,t)z^{j}(x,t), \quad
\Lh ^{i}_{0}z^{i}(x,t)\eq
$$
\vspace{-5mm}
$$
\eq
\left\{\eps_i \sum _{s=1,2} a^{i}_{s}(x,t)\delta_{\ovl{xs}\,\widehat{xs}}
+\!\sum_{s=1,2} [b_s^{i+}(x,t)\delta_{xs}+
b_s^{i-}(x,t)\delta_{\ovl{xs}}]-
p^i(x,t)\dtb \right\} z^i(x,t),
$$
$$
\lambda^iz^i(x,t)=
\lambda^i_{(2.2)}(\eps_i)z^i(x,t)\!\equiv\!
\left\{
\begin{array}{lc}
\left\{-\eps_i\alpha^i(x)\delta_x +(1-\alpha^i(x))\right\}z^i(x,t), &
x_1=0,\\[1ex]
\left\{\eps_i\alpha^i(x)\delta_{\ovl{x}}+(1-\alpha^i(x))\right\}z^i(x,t), &
x_1=d,
\end{array} \right.
$$
$
v^+(x,t) = 2^{-1}(v(x,t) + |v(x,t)|),\ v^-(x,t) = 2^{-1}(v(x,t) -
|v(x,t)|),
$
$\delta_{\ovl{xs}\,\widehat{xs}}\, v(x,t) = v_{\ovl{xs}\,\widehat{xs}}(x,t)$
is the second difference derivative on non-uniform meshes,
$\delta_{xs}\,v(x,t)=v_{xs}(x,t)$ and
$\delta_{\ovl{xs}}\,v(x,t)=v_{\ovl{xs}}(x,t)$,
$\delta_{\ovl{t}}\,v(x,t)=v_{\ovl{t}}(x,t)$ are the first difference
(forward and backward) derivatives, $z(x,t) = (z^1(x,t),z^2(x,t))^T.$
{\bf 2.} We now investigate the convergence of difference scheme
(2.2), (2.1). The operators
$\Lh^i_0 - c^{ii}(x,t),\ \ (x,t) \in G_h,\ i=1,2$
are monotone \cite{sama}.
Taking into account the {\it a priori} estimates of Theorem 4.1,
we evaluate the \mbox{quantities}
$(\Lh^i_{(2.2)} - L^i_{(1.1)})u(x,t)$, $i=1,2$.
Using the maximum principle \cite{sama}
for boundary value problem (1.1), (1.3), we obtain the estimate
$$
\mid u(x,t) - z(x,t) \mid \le M[\eps ^{-2}N^{-1} + N^{-1}_0],\ \ (x,t)
\in \Gbh
\eqno(2.3)
$$
with $\eps_1\!=\!\eps_2\!=\!\eps$, and, for problems
(1.1), (1.4) and (1.1), (1.5), we have
$$
\mid u(x,t) - z(x,t) \mid \le M[(\eps_1^{-3} + \eps_2^{-3})N^{-1} +
N^{-1}_0],\ \ (x,t) \in \Gbh.
\eqno(2.4)
$$
{\bf Theorem 2.1. } {\it Let for the solutions of boundary value
problems $(1.1)$, $(1.3)$--$(1.5)$ the estimates of Theorem $4.1$ (in the
Appendix) hold.
Then the difference scheme $(2.2)$, $(2.1)$ converges for fixed values
of the parameters. The solutions of the difference
schemes satisfy estimates $(2.3)$ and $(2.4)$ in the case of problem
$(1.1)$, $(1.3)$, and problems $(1.1)$, $(1.4)$ and $(1.1)$, $(1.5)$,
respectively.}
\newsection{Special difference schemes}
{\bf 1.} We will construct $\eps$-uniformly convergent schemes
for boundary value problems (1.1), (1.3) and (1.1), (1.4).
We consider the system of difference equations
$$
\begin{array}{l}
\Lh^i_{(3.1)}z(x,t) = f^i(x,t),\ \ (x,t) \in G_h,\quad
\lambda^i_{(3.1)}z^i(x,t) = \psi^i(x,t),\ \ (x,t) \in S_{1h},\\[0.7ex]
\mbox{\hspace{0.5cm}}~~z^i(x,t) = \vf^i(x,t),\quad (x,t) \in S_{0h},
\qquad i=1,2,
\end{array}
\eqno(3.1)
$$
where
$\Lh^i \eq \Lh^i_{(2.2)}(\eps_1\!=\!\eps_2\!=\!\eps)$,
$\lambda^i \eq \lambda^i_{(2.2)}(\eps_1\!=\!\eps_2\!=\!\eps)$
in the case of problem (1.1), (1.3), and
$\Lh^i \eq \Lh^i_{(2.2)}(\eps_1\!=\!\eps,\,\eps_2\!=\!1)$,
$\lambda^i \eq \lambda^i_{(2.2)}(\eps_1\!=\!\eps,\,\eps_2\!=\!1)$
in the case of problem (1.1), (1.4), on the special mesh
$$
\Gbh = \Db^{\,c}_{h(3.2)} \times \bom_0 = \Gb^{\,c}_{h(3.2)},
\eqno(3.2)
$$
which is refined (condensed) in the neighbourhood of the set $S_1^-$.
Here $\Gb_{h(3.2)}=
\Gb_{h(2.1)}$ with $\bom_1 = \bom^{\,c}_1(\s)$, $\bom^{\,c}_1(\s)=
\bom^{\,c}_{1(3.2)}(\s)$ is a mesh with piecewise constant step-size
(cf. \cite{shi})
condensing in the neighbourhood of the left-end point of $[0,d]$,
$\s$ is a parameter depending on $\eps$ and $N_1$ and satisfying
$\s \le \vartheta$, where $\vartheta $ is a positive number independent of
$\eps$ and $N_1, N_2, N_0$; $\vartheta \le 2^{-1}d$. The
mesh $\ovl{\om}^{\,c}_1(\s)$ is constructed as follows.
The set $[0,d]$ is divided into two subsets $[0,\s]$, $[\s,d]$.
On each of the subsets the mesh step is constant
and equal to $h_{(1)} = \s(\vartheta d^{-1}N_1)^{-1}$ on $[0,\s]$
and to $h_{(2)} = (d-\s)((d-\vartheta)d^{-1}N_1)^{-1}$ on $[\s,d]$.
We choose the value $\s$ from the condition~
$ \s = \s(\eps,N_1,m) = \min\,[\,\vartheta,\eps m^{-1}\ln N_1\,]$, ~
where $m$ is an arbitrary number from the interval $(0,m^0)$,
$m^0=(a^0)^{-1}b_0$.
{\bf 2.}
Taking into account the estimates of Theorem 4.2 (which is found in the
Appendix) and using the technique
described in \cite{shi}, we establish the $\eps$-uniform convergence
of scheme (3.1), (3.2) to the solution of problem (1.1), (1.3)
$$
\mid u(x,t) - z(x,t) \mid \le M\,[\,N^{-1}\ln N + N^{-1}_0\,],\ \
(x,t) \in \Gbh^{\,c}.
\eqno(3.3)
$$
{\bf Theorem 3.1. } {\it Let for the solution of boundary value
problem $(1.1)$, $(1.3)$ the estimates of Theorem $4.2$ (in the Appendix)
hold.
Then the difference scheme $(3.1)$, $(3.2)$ converges uniformly with
respect to the parameter. For the solution of the difference scheme
the estimate $(3.3)$ is valid.}
{\bf 3.} In the case of problem (1.1), (1.4) the estimates of
Theorem 4.3 lead to
$$
\mid u(x,t) - z(x,t) \mid \le M\,[\,\eps^2+N^{-1}\ln N + N^{-1}_0\,],\ \
(x,t) \in \Gbh^{\,c}.
\eqno(3.4)
$$
This and estimate (2.4) imply the $\eps$-uniform convergence
of difference scheme (3.1), (3.2) to the solution
of problem (1.1), (1.4)
$$
\mid u(x,t) - z(x,t) \mid \le M\,[\,N^{-2/5} + N^{-1}_0\,],\ \
(x,t) \in \Gbh^{\,c}.
\eqno(3.5)
$$
{\bf Theorem 3.2.} {\it Let for the solution of boundary value
problem $(1.1)$, $(1.4)$ the estimates of Theorems $4.1$ and $4.3$ (in the
Appendix) hold.
Then the difference scheme $(3.1)$, $(3.2)$
converges uniformly with respect to the parameter. For the solution of the
difference scheme the estimates $(3.4)$ and $(3.5)$ are valid.}
{\bf 4.} Let us give a special difference scheme and error bounds
for problem (1.1), (1.5).
We construct this scheme on the grid
condensing in the boundary layer
$$
\Gbh = \Db^{\,c}_{h(3.6)} \times \bom_0 = \Gb^{\,c}_{h(3.6)}
\eqno(3.6)
$$
The condensing rule is now defined by the two parameters $\eps$ and $\mu$.
Here $\Gb_{h(3.6)} = \Gb_{h(2.1)}$ with $\bom_1 = \bom^{\,c}_1(\s)$,
$\bom^{\,c}_1(\s)=\bom^{\,c}_{1(3.6)}(\s)$ is a mesh with piecewise constant
step-size, $\s=(\s_1,\s_2)$ is a parameter depending on $\eps$, $\mu $ and
$N_1$, and also $\s_1 \le 2^{-1}\vartheta$, $\s_2\le \vartheta$, where
$\vartheta\le 2^{-1}d$.
The mesh $\bom^{\,c}_1$ is constructed as follows. The set $[0,d]$ is
divided into three subsets $[0,\s_1]$, $[\s_1,\s_2]$, $[\s_2,d]$.
On each of the subsets
the mesh step is constant and equal to
$h_{(1)} = \s_1(2^{-1}\vartheta d^{-1}N_1)^{-1}$ on $[0,\s_1]$,
to $h_{(2)} = (\s_2 - \s_1)(2^{-1}\vartheta d^{-1}N_1)^{-1}$ on
$[\s_1,\s_2]$, and to $h_{(3)} = (d-\s_2)((d-\vartheta)d^{-1}N_1)^{-1}$
on $[\s_2,d]$. The values $\s_1,\ \s_2$ are chosen to satisfy the
condition: ~
$\s_1 = \s_1(\eps,N_1,m) = \min\,[\,2^{-1}\vartheta,\eps m^{-1}\ln N_1\,]$,
$\s_2 = \s_2(\eps,\mu,N_1,m) = \min\,[\,\vartheta,(\eps+\mu)m^{-1}\ln N_1]$, ~
where $m$ is an arbitrary number from the interval $(0,m^0)$.
On the grid $\Gb_{h(3.6)}$, we associate problem (1.1), (1.5) with the
system of difference equations\re
$$
\begin{array}{l}
\Lh^i_{(3.7)}z(x,t) = f^i(x,t),\ \ (x,t) \in G_h,\quad
\lambda^i_{(3.7)} z^i(x,t) = \psi^i(x,t),\ \ (x,t) \in S_{1h},\\[0.7ex]
\mbox{\hspace{0.5cm}}~~z^i(x,t) = \vf^i(x,t),\quad (x,t) \in S_{0h},
\qquad i=1,2.
\end{array}
\eqno(3.7)
$$
Here \ $\Lh^i \eq \Lh^i_{(2.2)}(\eps_1=\eps,\,\eps_2=\mu)$,
\ $\lambda^i \eq \lambda^i_{(2.2)}(\eps_1=\eps,\,\eps_2=\mu)$.
Taking into account the {\it a priori} estimates obtained on the base of
asymptotic expansions of the solution for problem (1.1), (1.5),
we find the estimates
$$
\begin{array}{l}
\mid u(x,t) - z(x,t) \mid ~\le M\,[\,\eps+\mu+N^{-1}\ln N + N^{-1}_0\,],
\ \ (x,t)\in \Gbh^{\,c},\\[1.2ex]
\mid u(x,t) - z(x,t) \mid ~\le M\,[\,\mu^{-2}(\eps+N^{-1}\ln N) + N^{-1}_0\,],
\ \ (x,t) \in \Gbh^{\,c}.
\end{array}
$$
These estimates and (2.4)
imply the $(\eps,\mu)$-uniform convergence of
scheme (3.7),\,(3.6)\\[1ex]
\centerline{$
\mid u(x,t) - z(x,t) \mid \le M\,[\,N^{-1/10} + N^{-1}_0\,],\quad (x,t) \in
\Gbh^{\,c}.$}
\begin{thebibliography}{99}
\small
%\baselineskip 12pt
\bibitem{marc} Marchuk, G. I.: {\rm Methods of Numerical Mathematics.
Springer, New York (1982)}\re
\bibitem{sama} Samarskii, A. A.: {\rm Theory of Difference Schemes.
Nauka, Moscow (1989) [in Russian]}\re
\bibitem{shi} Shishkin, G. I.: {\rm Grid Approximations of Singularly
Perturbed Elliptic and Parabolic Equations. Ural Branch of Russian
Academy of Sciences, Ekaterinburg (1992) [in Russian]}\re
\bibitem{shi1} Shishkin, G. I.: {\rm Grid approximation of a singularly
perturbed boundary value problem for a quasi-linear elliptic equation
degenerating into the first-order equation. {\em Soviet J.
Numer. Anal. Math. Modelling} {\bf 6} (1991) 61--81}\re
\bibitem{shi2} Shishkin, G. I.: {\rm Methods of constructing grid
approximations for singularly perturbed boundary value problems.
{\em Russian J. Numer. Anal. Math. Modelling} {\bf 7} (1992)
537--562}\re
\bibitem{mill} Miller, J. J. H., O'Riordan, E., Shishkin, G. I.:
Fitted Numerical
Methods for Singular Perturbation Problems. World Scientific, Singapore
(1996)\re
\bibitem{bakh} Bakhvalov, N. S.: {\rm On the optimization of methods for
solving boundary value problems in the presence of a boundary layer.
{\em Zh. Vychisl. Mat. Mat. Fiz.} {\bf 9} (1969) 841--859
[in Russian]}\re
\bibitem{brai} Brainina, Kh. Z., Neuman, Ye. Ya.:
{\rm Solid-Phase Reactions in
Electroanalytic Chemistry. Khimiya, Moscow (1982) [in Russian]}\re
\bibitem{lady} Ladyzhenskaya, O. A., Solonnikov, V. A., Ural'tseva, N.N.:
{\rm Linear and Quasi-linear Equations of Parabolic Type}. {\em Transl. of
Math. Monographs} {\bf 23}. AMS, Providence, RI (1968)\re
\end{thebibliography}
\section*{Appendix}
\setcounter{section}{4}
\subsection{A priori estimates of the solutions and their derivatives}
\small
%\baselineskip 16pt
The technique of deriving a priori estimates for problem (1.1) is similar
to that used in \cite{shi}. Here we omit the derivation of these bounds
in view of its cumbersome presentation and abundance of technical details.
{\bf 1.} We assume that the following condition holds: \vspace{-1mm}
$$
u\in C^{6,3}(\Gb).
\eqno(4.1)
$$
Using the maximum principle, we derive the $\eps$-uniform
boundedness of the solution
$$
\mid u(x,t)\mid \ \le M,\ \ (x,t) \in \Gb,
\eqno(4.2\mbox{a})
$$
where ~$|\, u(x,t)\,| = \max\limits_{i}\mid u^{i}(x,t)\mid, \ \ \,
M = 2(1-m^2_{(1.2)})^{-1}\max\,[\,c^{-1}_{0}\,
\max\limits_{\Gb}|\, f(x,t)\,|,\,\max\limits_{S_0}|\, \vf(x,t)\,|$,\\
$
[\min(1,m^0_{(4.2)})]^{-1}\max\limits_{S_1}|\,\psi(x,t)\,|\,], \quad
m^0_{(4.2)}=(a^0)^{-1}b_0$.
\medskip
For simplicity, we assume that the following
conditions are fulfilled:
$$
\begin{array}{l}
a_{s}^i(x,t)\eq a_{s}^i,\ \ b_s^i(x,t)\eq b_s^i,\ \ p^i(x,t)\eq p^i,\quad
(x,t)\in\Gb,\\[1ex]
\alpha^i(x)\equiv 0,\ \ \psi^i(x,t)=\vf^i(x,t),\quad (x,t)\in \ovl{S}_1,
\qquad s,\,i=1,2.
\end{array}
\eqno(4.3)
$$
Using the majorant function technique, we find
the estimates
$$
\left|\, {\d ^{k+k_0} \ov \d x^{k_1}_1 \d x^{k_2}_2 \d t^{k_0}}
u(x,t) \,\right| \ \le M\,[\,\eps_{1}^{-k} + \eps_{2}^{-k}\,],
\ \ (x,t) \in \Gb,\ k+2k_0\leq 4.
\eqno(4.2\mbox{b})
$$
{\bf Theorem 4.1.} {\it Let for the functions $u(x,t)$,
the solutions of problems $(1.1)$, $(1.3)$--$(1.5)$,
the condition $(4.1)$ be fulfilled.
Then, under condition $(4.3)$, the estimates $(4.2)$ hold.}
{\bf 2.}
We now estimate the solution of problem (1.1), (1.3)
on the base of asymptotic expansions. For this
we represent the function
$u(x,t)$ as a sum of vector-valued functions
$$
u(x,t)=U_1^{[0]}(x,t) +\eps U_1^{[1]}(x,t)+ V_1(x,t)+
v_1(x,t)\eq
U^{(1)}_1(x,t)+V_1(x,t),\ \ (x,t) \!\in\! \Gb,
\eqno(4.4)
$$
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 w.r.t. the space variables are bounded
$\eps$-uniformly.
Assume that the following condition is fulfilled:
$$
\begin{array}{c}
U_1^{[0]}\in C^{K+\alpha}(\Gb),\ \ U_1^{[1]}\in C^{K-2+\alpha}(\Gb),\ \
v_1\in C^{K-3+\alpha}(\Gb),\\[0.8ex]
V_1\in C^{K-1+\alpha}(\Gb),\ \ \alpha\in (0,1),\ \ K\geq 7.
\end{array}
\eqno(4.5)
$$
Then, for the components from (4.4), the following estimates are fulfilled:
$$
\left|\, {\d ^{k+k_{0}} \ov \d x^{k_{1}}_{1}\d x^{k_{2}}_{2}\d t^{k_{0}}}
U_1^{[j]}(x,t)\,\right| \ \le M, \quad j=0,1,
\eqno(4.6\mbox{a})
$$
$$
\mid V_1(x,t)\mid\ \leq M\exp(-m\eps^{-1}r),
\eqno(4.6\mbox{b})
$$
$$
\mid v_1(x,t)\mid\ \leq M\eps^2,\quad (x,t)\in\Gb,\ \ k+2k_0\leq 4,
\eqno(4.6\mbox{c})
$$
where $m$ is a sufficiently small number, $m0$; ~ $\eps_i\in (0,1]$,
$i\!=\!1,2$.
We construct and justify $\eps$-uniformly convergent schemes for
problem (4.11) similarly to the constructions done for the case
of problem (1.1).
The results,
likewise those for parabolic equations, hold for the
difference schemes developed here.
\end{document}