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\begin{document}
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\bc
{\large\bf ON FINITE DIFFERENCE SCHEMES FOR SINGULARLY
PERTURBED PARABOLIC EQUATIONS WITH \\
AN\, INITIAL PARABOLIC LAYER }\\[3ex]
Grigorii I. Shishkin\\[1.5ex]
Institute of Mathematics and Mechanics, Ural Branch of the
Russian Acad. Sci.,\\
Ekaterinburg, 620219, Russia
\ec
\vspace{3ex}
\noindent
{\bf ABSTRACT.}
A method to construct grid approximations for singularly perturbed
problems is considered. As is known, the error in the approximate
solutions by classical finite difference schemes depends here on the
perturbation parameters and may be comparable with the
exact solution. On a segment, we consider a class of boundary value
problems for singularly perturbed parabolic equations with
convective terms. The time derivative and also the derivatives with respect
to the space variables, which enter the equation, are multiplied by
small parameters taking arbitrary values in the half-interval
(0,1]; a union of these perturbation parameters is the vector-parameter
$\eps$. The solutions of such problems have both boundary
and initial layers, in particular, layers of parabolic type.
We show issues arising when we use fitted operator methods to construct
schemes convergent, in the discrete maximum norm, uniformly with respect
to the parameters. It is shown that there exist
no schemes from the natural class of fitted operator methods that converge
uniformly in each of the perturbation parameters (or $\eps$-uniformly).
Thus, we conclude that for the above class of boundary value problems
a condensing mesh technique is necessary
to construct $\eps$-uniformly convergent schemes.\\[1.5ex]
{\bf AMS Subject Classification.} \ 65M06, 65M10
\bc
{\bf 1. INTRODUCTION}
\ec
Mathematical modeling of heat and mass transfer in moving medium leads
us to a sufficiently wide class of boundary value problems for
singularly perturbed equations with convective terms in that case
when the process duration is short and the coefficients of temperature
conductivity or diffusion are small. Such equations contain small
parameters multiplying the space derivatives, including
the highest derivatives, and/or the time derivative. The small values
of these perturbation parameters (or a part of them) give rise to
boundary and/or initial layers. For such problems,
the error in the approximate solutions by classical finite difference
schemes, i.e., schemes which have been developed for regular problems,
is commensurable with the exact solution of the problem (see, e.g.,
Il'in [1], Doolan et al [2] and Shishkin [3] in the case of equations with
a small parameter multiplying the highest derivatives with respect to
the space variables and Section 3 for equations with a small parameter
at the time derivative). Therefore there is a need to develop special
difference schemes whose solutions converge to the solution of the
boundary value problem $\eps$-uniformly.
\baselineskip 15.7pt
There exist two different methods, both widely used for the construction
of special difference schemes: (a) a fitted operator method
(see its description, for example, in Il'in [1], Doolan et al [2]) and
(b) a condensing mesh method (its description can be seen, e.g., in
Shishkin [3], Bakhvalov [4] and Miller et al [5]; in this last work
these methods are referred to as fitted mesh methods).
However, in the presence
of parabolic (both boundary and initial) layers, the fitted operator method
is inapplicable for the construction of $\eps$-uniformly convergent schemes.
The proof of this result can be seen in Shishkin [3], Miller et al [5] and
Shishkin [6], [7] in the case of boundary layers and in Section 4
in the case of an initial layer.
Grid approximations for a parabolic equation with a small parameter
multiplying the time derivative were not previously considered,
and they were examined only sporadically for the case of small parameters
multiplying the derivatives in space (see, e.g., Titov and Shishkin [8], and
Shishkin [9] where the fitted operator method was used for the construction
of special schemes).
In the present paper we consider the Dirichlet problem for singularly
perturbed parabolic equations on a segment; these equations contain
convective terms. The time derivative and also the derivatives with respect
to the space variables, which enter the equation, are multiplied by
parameters taking arbitrary values from the half-interval
(0,1]. As the parameters (or one of them) tend to zero, boundary and
initial layers appear. Depending on the relations between the parameters,
these layers can be both regular and parabolic, or hyperbolic, ones.
Previously hyperbolic layers were not observed in the case of
singularly perturbed parabolic equations.
The issues are shown that arise when we attempt to solve such problems
using classical difference approximations and also if we use fitted
operator methods for constructing difference schemes which converge
uniformly in the perturbation parameters. It follows from this analysis
that for the stated class of boundary value problems to construct
$\eps$-uniformly convergent schemes, the use of meshes condensing
in the layers is necessary.
\baselineskip 16pt
\bc
{\bf 2. POSING THE PROBLEM}
\ec
\setcounter{equation}{0}
{\bf 2.1.} On the domain $G=D\times(0,T]$ with the boundary
$S=S(G)=\ov{G} \setminus G$, where
$ D=\{\,x:\ 00, \quad (x,t)\in \ov{G}
\eqno (2.3\r{a}) $$
if \ $\eps_2 \geq m(\eps_1+\eps_3+\eps_4)$, \ or the condition
$$
c(x,t)\geq c_0>0, \quad (x,t)\in \ov{G}
\eqno (2.3\r{b}) $$
if $\eps_4 \geq m(\eps_1+\eps_2+\eps_3)$.
Using the comparison theorems in the case of conditions
(2.2) and (2.3), due to the maximum principle we find
that
$$
|u(x,t)|\!\le M[(\eps_1+\eps_2+\eps_3+\eps_4)^{-2}\max_{\ov{G}}|\F|+
\max_S|\f(x,t)|], \ (x,t)\in \ov{G}. ~~~
\eqno (2.4) $$
The estimate $(2.4)$ is sharp with respect to the entering parameters
$\eps_i,\;i=1,\ldots, 4.$ Bounding the function, we use the technique
similar to that in Shishkin [3], Ladyzhenskaya et al [10] and
Friedman [11].
In the case of Eqs. (2.2) and (2.3), estimate (2.4) implies the
boundedness of the solution uniformly with respect to the parameters
$\eps_i$ (or, in short, $\eps$-uniform boundedness) if the function
$\F$ satisfies the condition
$$
\vert\F\vert\leq M(\eps_1+\eps_2+\eps_3+\eps_4)^2, \ \ (x,t)\in\ov{G}.
\eqno (2.5\r{a}) $$
\renewcommand{\theequation}{2.5c}
We shall assume that the function $\F$ has the form
$$
\F=(\eps_1+\eps_2+\eps_3+\eps_4)^2 f(x,t),\ \ (x,t)\in\ov{G},
\eqno (2.5\r{b}) $$
where $f(x,t)$ is a sufficiently smooth function. The differential equation
(2.1a) can be written in the form
\beq
L_{(2.5)} u(x,t)\! \equiv \!\left\{\te_1^{\,2} a(x,t)\frac{\pa^2}{\pa x^2} +
\te^{\,2}_2b(x,t)\frac{\pa}{\pa x}-\te_3^{\,2}p(x,t)\frac{\pa}{\pa t}-
\te_4^{\,2}c(x,t)\,\right\}\!u(x,t)\!= && \no\\[0.7ex]
=f(x,t),\quad (x,t)\in G. \hspace{4cm} &&
\eeq
Here
$\te_i=\eps_i(\eps_1+\eps_2+\eps_3+\eps_4)^{-1},\;i=1,2,3,4$ are
reduced ("dimensionless") parameters;
$ \te_1,\, \te_3\in (0,1],$
$ \te_2,\, \te_4\in [0,1],$ moreover,
$\te_1+\te_2+\te_3+\te_4=1$.
It is convenient for the further analysis to decompose the domain
of varying $\eps_i$, $i=1,2,3,4$ (the four-dimensional unit cube
that does not contain the hyperplanes
$\eps_1=0$ and $\eps_3=0$, on which Eq. (2.1a) ceases to be
parabolic) onto subdomains where one of the parameters would
predominate over the other ones:
$$
\eps_3\ge m^0(\eps_1+\eps_2+\eps_4),
\eqno (2.6) $$
$$
\eps_1\ge m^0(\eps_2+\eps_3+\eps_4),
\eqno (2.7) $$
$$
\eps_2\ge m^0(\eps_1+\eps_3+\eps_4),
\eqno (2.8) $$
$$
\eps_4\ge m^0(\eps_1+\eps_2+\eps_3)
\eqno (2.9) $$
with $m^0$ being arbitrary numbers, $m^0<1$. These conditions
(2.6)--(2.9) can be rewritten in the form: \
$\te_i\ge m^0(1+m^0)^{-1}$, $i=1,2,3,4.$
In Section 3, if $\eps_1,\ \eps_3\in (0,1],\ \eps_1+\eps_3\geq m$ and
$\eps_2=0$ or $\eps_2=1$, we show some computational difficulties
that arise from the numerical solution of problem (2.1) with the use of
classical difference schemes. The error in the discrete solution
depends on the parameters $\eps_i$ and may be comparable with the
exact solution for small values of $\eps_1$ and $\eps_3$. This leads to
the problem of developing special difference schemes that converge
$\eps$-uniformly.
It is known that even if $\eps_1\in (0,1],\ \eps_2=\eps_3=1$
the fitted operator methods get inapplicable to construct those
special schemes (see, e.g., Shishkin [3], [6]). In Section 4 it is shown that
these methods are also inapplicable in case
$\eps_3\in (0,1],\ \eps_1=\eps_2=1$.
The structure of the boundary and initial layers (and the corner ones as
well) and also the applicability of the fitted operator method under
different relations between the parameters are discussed in Section 5.
\bc
{\bf 3. CLASSICAL DIFFERENCE SCHEMES}
\ec
\renewcommand{\theequation}{3.1}
{\bf 3.1.} Let us discuss some issues arising from the numerical
solution of the boundary value problem (2.1) with the model
example
\beq
L_{(3.1)} u(x,t)\equiv
\left\{\eps_1^2\frac{\pa^2}{\pa x^2}+\frac{\pa}{\pa x}
-\eps_3^2\frac{\pa}{\pa t}\right\}\! u(x,t)\!=\!f(x,t),\;(x,t)\in G,&& \\[1ex]
u(x,t)=\vp(x,t), \quad (x,t) \in S. \hspace{6cm} && \no
\eeq
Here \vspace{-1mm}
$$
G=D\times (0,T],\ \ \ov{D}=[-d,d],
\eqno (3.2) $$
$f(x,t),\ \vp(x,t)$ are sufficiently smooth functions on the set
$\ov{G}$ and on the sides of $S$, respectively,
$\vp\in C(S)$, the parameters are assumed to satisfy the condition
$$
\eps_1,\ \eps_3\in(0,1],\ \ \eps_1+\eps_3\ge m.
\eqno (3.3) $$
We shall say that grid approximations of the boundary value problem
are classical if for these we use discretisations of the problem that are
constructed on the base of finite difference schemes, finite element or
finite volume techniques on meshes with arbitrary,
independent of $\eps$, distribution of the nodes (e.g., uniform
meshes).
\renewcommand{\theequation}{3.4}
To solve the problem we use the classical difference scheme (cf. Samarsky
[12])
\beq
\Lambda_{(3.4)} z(x,t)\equiv
\{\eps_1^2\delta_{x\ov{x}}+\delta_x-\eps_3^2\delta_{\ov{t}}\}z(x,t)=f(x,t),\;
(x,t) \in G_h, && \\[0.5ex]
z(x,t)=\vp(x,t), \quad (x,t) \in S_h. \hspace{5cm} \no
\eeq
Here \vspace{-1mm}
$$
\ov{G}_h=\ov{D}_h\times\ovo_0, \; G_h=G \cap \ov{G}_h, \; S_h=S\cap\ov{G}_h,
\eqno (3.5) $$
$\ov{D}_h$ and $\ovo_0$ are uniform grids on the segments $\ov{D}$ and $[0,T]$
with the step-sizes $h=2dN^{-1} $ and $ h_t=TN_0^{-1} $, respectively.
By $ N+1$ and $N_0+1 $ we denote respectively the number of nodes in the grids
$\ov{D}_h$ and $\ovo_0$; \
$\delta_{x\ov{x}}\,z(x,t)=z_{x\ov{x}}(x,t)$ and $\delta_x\, z(x,t)=z_x(x,t)$,
$\delta_{\ov{t}}\,z(x,t)=z_{\ov{t}}(x,t)$ are the second
and first (forward and backward) difference derivatives.
Under the condition \vspace{-1mm}
$$
\eps_1\in (0,1],\ \ \eps_3\ge m
\eqno (3.6) $$
the solution of problem (3.4) has a regular boundary layer
(for small $\eps_1$), that is, a layer described by an ordinary
differential equation; the solution of the finite difference scheme
(3.4), (3.5) does not converge to the solution of the boundary value
problem $\eps_1$-uniformly (see, for example, Il'in [1] and Shishkin [3]).
Thus, in the case of condition (3.6) the grid approximations,
including monotone finite difference schemes (cf. Samarsky [12]) and
also schemes on the base of finite element and finite volume
methods (cf. Johnson [13]), does not converge $\eps_1$-uniformly.
\begin{Lem}
Let for the solution of boundary value problem $(3.1),\ (3.2)$
the classical finite difference schemes be used. Then the approximate
solution does not converge to the
exact one $\eps_1$-uniformly under condition $(3.6)$.
\end{Lem}
\renewcommand{\theequation}{3.8}
{\bf 3.2.} Let the parameters $\eps_1,\ \eps_3$ satisfy the condition
$$
\eps_3\in (0,1],\ \ \eps_1\geq m.
\eqno (3.7) $$
We will analyze the behaviour of the error in the approximate solution
for the problem
\beq
L_{(3.8)} u(x,t) &\equiv& \left\{\frac{\pa^2}{\pa x^2}+\frac{\pa}{\pa x}-
\eps_3^2\frac{\pa}{\pa t}\right\}u(x,t)=0, \ \ (x,t)\in G,\\[1ex]
u(x,t)&=&\vp(x,t), \quad (x,t)\in S,\no
\eeq
where $\ov{G}=\ov{G}_{(3.2)},\;$ $\eps_3\in (0,1].$
We define the function $\vp(x,t)$ by\\[1ex]
\centerline{$
\vp(x,t)=\eta(x)v_0(x),\;(x,t)\in S,\ \
v_0(x)=2^{-1}\pi^{-1/2}\exp(-4^{-1}x^2),\;x\in R$.}\\[1.5ex]
Here $\eta(x)=\eta(x,d)$ is a cut-off smooth function such that
$0\le \eta(x)\le 1$, $\eta(x)=1$ if $|x|\le 2^{-1}d,\;$ $\eta(x)=0$ if
$|x|\ge 3/4\,d,\;$ and aslo
$\eta(x)=\eta(-x),\;$ $\eta^{\prime}(x)\le 0,\;$ $x\ge 0$.
In the neighbourhood of the set $S_0$ a parabolic initial layer
appears as $\eps_3\to 0$, that is, a layer described by a
parabolic equation.
\renewcommand{\theequation}{3.9}
Note, the function
\beq
v(x,t)&=&v(x,t;\eps_3)=2^{-1}\pi^{-1/2}\eps_3
(t+\eps_3^2)^{-1/2}\times\\[0.5ex]
&&\hspace{-6mm}\times\exp\left(-4^{-1}\eps_3^2(x+\eps_3^{-2}t)^2 \,
(t+\eps_3^2)^{-1}\right),\ \
x\in R,\;t\ge 0\no
\eeq
is the solution of the problem
\renewcommand{\theequation}{3.10}
\begin{equation}
L_{(3.8)}u(x,t)=0,\ \ x\in R,\ \ t\in(0,T],\quad
u(x,0)=v_0(x).
\end{equation}
The solution of problem (3.8) satisfies the estimate
$$
|u_{(3.8)}(x,t)-v(x,t)|\le m,\ \ (x,t)\in \ov{G},\;t\le 4^{-1}\eps_3^2,
\eqno (3.11) $$
where the constant $m$ depends on $d$ and can be arbitrarily small if
we choose $d$ to be sufficiently large:
$m\!=\!m(d)\!\to 0 \r{ for } d\to \infty$.
\renewcommand{\theequation}{3.12}
For the solution of problem (3.8), (3.2) we use the
difference scheme
\begin{equation}
\Lambda_{(3.12)}z(x,t)=0,\ \ (x,t)\in G_h,\quad
z(x,t)=\vp(x,t),\ \ (x,t)\in S_h,
\end{equation}
where $\Lambda_{(3.12)}=\Lambda_{(3.4)}$ for $\eps_1=1$, i.e.,
$\Lambda_{(3.12)}=\delta_{x\ov{x}}+\delta_x-\eps_3^2\delta_{\ov{t}}$,
$\ov{G}_h=\ov{G}_{h(3.5)}.$
Let us estimate the difference of the functions
$u_{(3.8)}(x,t)-z_{(3.12)}(x,t)$ in an $m$-neighbourhood of the point (0,0).
As a preliminary, we make auxiliary constructions.
From the variables $x,t$ we pass to the variables
$x,\tau;\;\tau=\tau(t,\eps_3)=\eps_3^{-2}t$. In these new variables
the boundary value problem (3.8), (3.2) and difference scheme (3.12), (3.5)
take the form
\beqv
L^0_{(3.8)}u^0(x,\tau)&\equiv&
\left\{\frac{\pa^2}{\pa x^2}+\frac{\pa}{\pa x}-\frac{\pa}{\pa \tau}\right\}
u^0(x,\tau)=0,\ \ (x,\tau)\in G_{\tau},\\[1ex]
u^0(x,\tau)&=&\vp^0(x,\tau),\quad (x,\tau)\in S_\tau;\\[2ex]
\Lambda^0_{(3.12)}z^0(x,\tau)&\equiv&
\left\{\delta_{x\ov{x}}+\delta_x-\delta_{\ov{\tau}}\right\} z^0(x,\tau)=0,\ \
(x,\tau)\in G_{h\tau},\\[1ex]
z^0(x,\tau)&=&\vp^0(x,\tau),\quad (x,\tau)\in S_{h\tau};
\eeqv
the sets $G_\tau^0$ in the variables $x,\tau$ correspond to the sets
$G^0\subseteq \ov{G}$. We use the notation $w(x,t(\tau))=w^0(x,\tau).$
For the function $v^0_{(3.9)}(x,\tau)$ the following relations are
fulfilled
$$
\Lambda^0_{(3.12)}\left(v^0(x,\tau)-z^0_{(3.12)}(x,\tau)\right)=
\left(\Lambda^0_{(3.12)}-L^0_{(3.8)}\right)v^0(x,\tau),
$$
$$
\left|\left(\Lambda^0_{(3.12)}-L^0_{(3.8)}\right)v^0(x,\tau)-
\left(\frac{\pa}{\pa \tau}-\delta_{\ov{\tau}}\right)
v^0(x,\tau)\right|\ \le Mh,
$$
$$
\left(\frac{\pa}{\pa \tau}-\delta_{\ov{\tau}}\right)v^0(x,\tau)\ge 2^{-1}h_\tau
\min_{\tau-h_{\tau}\le \theta \le \tau}\frac{\pa^2}{\pa\tau^2}v^0(x,\theta),
\quad (x,\tau)\in G_{h\tau},
$$
$$
\frac{\pa^2}{\pa\tau^2}v^0(0,0)=3\cdot 8^{-1}\pi^{-1/2},
$$
where $h_\tau=\eps^{-2}_3h_t.$ One can find such sufficiently
small number $h_1$ and sufficiently small neighbourhood of the point
$(0,0)$, belonging to $\ov{G}_{\tau}$:
$$
G^0_{\tau}=\left\{(x,\tau): |x|\le m_1,\;0<\tau\le m_0\right\},
\eqno (3.13) $$
that the inequality
$$
\Lambda^0_{(3.12)}\left(v^0(x,\tau)-z^0_{(3.12)}(x,\tau)\right)\ge m h_\tau,\ \
(x,\tau)\in G^0_{h\tau}
\eqno (3.14) $$
is satisfied for any $h\le h_1,\;h_\tau=m_0.$ Here \ $m_{(3.14)}\le 8^{-1}\pi^{-1/2}.$
Let the function $w(x,\tau)$ satisfy the relations
\beqv
\Lambda^0_{(3.12)}w(x,\tau)&\ge& mh_\tau,\quad (x,\tau)\in G^0_{h\tau},\\[0.2ex]
w(x,\tau)&=&0,\quad (x,\tau)\in S^0_{h\tau},\qquad m=m_{(3.14)}.
\eeqv
Then, using the comparison theorems, we obtain
$$
\min_{\ov{G}^0_{h\tau}}w(x,\tau)\le w(0,m_0)\le -mh_\tau.
\eqno (3.15) $$
From this inequality it follows that for the difference of the functions
$v^0(x,\tau)-z^0_{(3.12)}(x,\tau)$ at least one of the following inequalities
is satisfied:
$$
\max_{\ov{S}^0_{h\tau}}\left|v^0(x,\tau)-z^0_{(3.12)}(x,\tau)\right|\
\ge 2^{-1}m_{(3.15)}h_\tau,
$$
$$
\max_{G^0_{h\tau}}\left|v^0(x,\tau)-z^0_{(3.12)}(x,\tau)\right|\
\ge 2^{-1}m_{(3.15)}h_\tau.
$$
Consequently,
$$
\max_{\ov{G}_{h\tau}}\left|v^0(x,\tau)-z^0_{(3.12)}(x,\tau)\right|\
\ge 2^{-1}m_{(3.15)}h_\tau
\eqno (3.16) $$
for any $h,\;h_\tau$ such that $h\le h_1\;$ and $\;h_\tau= m_{0(3.13)}$.
Under the condition \vspace{-1mm}
$$
\eps_3=\eps_3(h_t)=\left(m_{0(3.13)}\right)^{-1/2}\left(h_t\right)^{1/2}
\eqno (3.17) $$
inequality (3.16) implies the estimate
$$
\max_{\ov{G}_h}\left|v(x,t)-z_{(3.12)}(x,t)\right|\ge m,
\eqno (3.18) $$
where $m=2^{-1}m_{(3.15)}m_{0(3.13)}$, for any $h\le h_1$ and $h_t\le m_0$.
The value $d$ is chosen sufficiently large such that the inequality
$m_{(3.11)}(d)\le 2^{-1}m_{(3.18)}$ is true.
Then under condition (3.17) we get that
$$
\max_{\ov{G}_h}\left|u_{(3.8)}(x,t)-z_{(3.12)}(x,t)\right|\ge 2^{-1}m_{(3.18)}
$$
for any $h\le h_1,\; h_t\le m_0$.
Thus, the solution of difference scheme (3.12), (3.5) for $N,N_0\ra 0$
does not converge to the solution of boundary value problem (3.8), (3.2)
$\eps_3$-uniformly.
\begin{Lem}
Let for the solution of boundary value problem $(3.1),\ (3.2)$
the classical finite difference schemes (Eqs. $(3.4)$) be used.
Then, under condition $(3.7)$, the approximate solution
does not converge to the exact one $\eps_3$-uniformly.
\end{Lem}
\renewcommand{\theequation}{3.19}
Difficulties in the numerical solution, similar to those stated above,
arise in the case of the problem
\beq
L_{(3.19)} u(x,t)\equiv\left\{\eps_1^2\frac{\pa^2}{\pa x^2}-
\eps_3^2\frac{\pa}{\pa t}\right\}\!u(x,t)\!=\!f(x,t),\;
(x,t)\in G, && \\[1ex]
u(x,t)=\vp(x,t), \quad (x,t) \in S \hspace{5cm} && \no
\eeq
provided that condition (3.3) is true. In the case of conditions (3.6) and
(3.7), respectively, parabolic boundary and initial layers appear.
\begin{Theor}
Let for the solution of boundary value problem $(2.1)$
the classical finite difference scheme be used.
Then the approximate solution does not converge to the exact one
$\eps$-uniformly; provided that $\eps_2=0$ or $\eps_2=1$ the approximate
solutions does not converge $\eps_1$-uniformly in the case of condition
$(3.6)$ and $\eps_3$-uniformly in the case of condition $(3.7)$.
\end{Theor}
\bc
\bf 4. SCHEMES BASED ON A FITTED OPERATOR METHOD
\ec
For the construction of special difference schemes, which converge
$\eps$-uniformly, fitted operator methods are more attractive as
they allow us to use the simplest (e.g., uniform) meshes.
In [14] for problem (3.1), (3.2) under condition (3.6) (in the case of a
regular boundary layer) a fitted scheme was constructed to converge
$\eps_1$-uniformly. However, even for problem (3.19), (3.2)
under condition (3.6) (with a parabolic boundary layer)
there do not exist fitted operator schemes, whose solutions converge to the
exact solution $\eps_1$-uniformly (see Shishkin [3], [6] and [7]).
Let us show issues arising in the construction of a scheme based on a
fitted operator method for problem
(3.1), (3.2) under condition (3.7).
\renewcommand{\theequation}{4.1}
{\bf 4.1.} Suppose that we are interested in the grid approximation of
the Cauchy problem for the singularly perturbed equation
\beq
L_{(4.1)} u(x,t)\equiv \left\{\frac{\pa^2}{\pa x^2}+\frac{\pa}{\pa x}
-\eps_0^2\frac{\pa}{\pa t}\right\}\!u(x,t)\!=\!f(x,t),\; (x,t)\in G, && \\[1ex]
u(x,t)=\vp(x,t), \quad (x,t) \in S. \hspace{5cm} && \no
\eeq
Here \vspace{-1mm}
$$
G=R\times(0,T],
\eqno (4.2) $$
$f(x,t),\ (x,t)\in \ov{G}$ and $\vp(x,t),\ (x,t)\in S$ are sufficiently
smooth functions, $\eps_0\in (0,1]$. In the neighbourhood of the set $S_0$
an initial layer appears as $\eps_0\ra 0$.
Let $z(x,t),\ (x,t)\in \ov{G}_h$ be the solution of some finite
difference scheme on the grid set \
$ \ov{G}_h=\omega\times\ov{\omega}_0$. \
Here $\omega$ and $\ov{\omega}_0$ are, generally speaking, nonuniform grids
on the axis $x$ and the segment $[0,T]$, respectively. By $N_0+1$ and $ N+1$
we denote the number of nodes in the grid $\ov{\omega}_0$ and the minimal
number of nodes in the grid $\omega$ on the segment with unit length;
$h_0\le MN_0^{-1},\; h\le MN^{-1}$, where $h$ and $h_0$ are the maximal
step-sizes of $\omega$ and $\ov{\omega}_0$.
We say that the solution of this difference scheme converges
$\eps_0$-uniformly (in the discrete maximum norm) if for the function
$z(x,t)$ the following estimate is fulfilled:\\[0.3ex]
\centerline{$
\max\limits_{\ov{G}_h}|u(x,t)-z(x,t)|\le \lambda(N^{-1},N_0^{-1})$,}\\[0.5ex]
where as $N,N_0\ra \infty$ we have $\lambda(N^{-1},N_0^{-1})\ra 0$ uniformly
with respect to the parameter $\eps_0$.
Let us describe a class of finite difference schemes (class {\bf A}) defined
by sufficiently natural conditions, in which we shall construct the fitted
operator scheme for such a particular problem
$$
L_{(4.1)}u(x,t)=0,\quad (x,t)\in G,
\eqno (4.3\r{a}) $$
\vspace{-6mm} $$
u(x,t)=\vp(x),\quad (x,t)\in S.
\eqno (4.3\r{b}) $$
For example, if
$$
\vp(x)=\vp^n(x)=\sin(n\pi^{-1}x),
\eqno (4.4) $$
where $n$ is a parameter, an integer,
the solution of this problem is the function
$W^n(x,t)=\sin(n\pi^{-1}(x+\eps_0^{-2}t))\exp(-n^2\pi^{-2}\eps_0^{-2}t),\;
(x,t)\in \ov{G}$, which is an initial layer type function.
We construct the finite difference scheme on the uniform rectangular grid
$$
\ov{G}_h=\omega\times\ovo_0,
\eqno (4.5) $$
where $\omega \ii \ovo_0$ are uniform grids with step-sizes $h=N^{-1} \ii
h_0=TN_0^{-1}$, respectively; here we use a four-point stencil of implicit
finite difference schemes in the general form \vspace{-2.5mm}
$$
\Lambda_{(4.6)}z(x,t)\equiv \left\{\delta_{x\ov{x}}+\delta_x-P\delta_{\ov{t}}\right\}z(x,t)=0,\;(x,t)\in G_h,
\eqno (4.6) $$
where the coefficient $P$ is a functional of the coefficients of Eq.
(4.3a) and depends on $x, t, h, h_t, \eps_0;\;h_t=h_0$.
From the variables $x,t$ we pass to the variables $x,\tau;\;
\tau=\tau(t,\eps_0)=\eps_0^{-2}t$. The singularly perturbed equation (4.3a)
transforms to the regular equation
$$
L_{(4.7)}^0 u^0(x,\tau) \equiv \left\{\frac{\pa^2}{\pa x^2}+\frac{\pa}{\pa x}
-\frac{\pa}{\pa \tau}\right\} u^0(x,\tau)=0,\ \ (x,\tau)\in G_\tau;
\eqno (4.7\r{a}) $$
$G_\tau^0$ is an image of the set $G^0\subseteq \ov{G}$;
we denote $v(x,t(\tau))=v^0(x,\tau)$. On the boundary $S_\tau$ the function
$u^0(x,\tau)$ takes the given values
$$
u^0(x,\tau)=\vp^n(x),\ \ (x,\tau)\in S_\tau.
\eqno (4.7\r{b}) $$
Such a relation $u_{(4.3)}(x,t)=u^0(x,\eps_0^{-2}t)$ is true, where
$u_{(4.3)}(x,t)$ is the solution of problem (4.3).
On the grid $G_{h\tau}$, Eqs. (4.6) take the form
$$
\Lambda_{(4.8)}^0z^0(x,t)\equiv \left\{\delta_{x\ov{x}}+
\delta_x-\gamma^0\delta_{\ov{\tau}}\right\}z^0(x,\tau)=0,\ \
(x,\tau)\in G_{h\tau},
\eqno (4.8) $$
where $z^0(x,\tau)=z(x,\eps_0^2\tau),\;\gamma^0(x,\tau,h,h_\tau,\eps_0)=
\eps_0^{-2}P(x,\eps_0^2\tau,h,\eps_0^2h_\tau,\eps_0),\;
h_\tau=\eps_0^{-2}h_t.$
Note that the parameter $\eps_0$ does not enter into the formulation
of problem (4.7) (in the variables $x,\tau$).
The grid sets $G_{h\tau}$ and $S_{h\tau}$ are independent of the
parameter $\eps_0$; these sets are defined by the step-sizes
$h,\;h_\tau$ only.
Because problem (4.7), and consequently its solution, and also the grid
$\ov{G}_{h\tau}$ are independent of the parameter $\eps_0$, therefore
it is natural to look for the coefficients of the finite difference equation
(4.8), corresponding to differential equation (4.7a), in the form
independent of the parameter $\eps_0$
$$
\Lambda^0_{(4.9)}z^0(x,\tau)\equiv\left\{\delta_{x\ov{x}}+
\delta_x-\gamma^0(x,\tau,h,h_\tau)\delta_{\ov{\tau}}\right\}z^0(x,\tau)\!=\!0,
\ \ (x,\tau)\in G_{h\tau} ~~~
\eqno (4.9) $$
Equations (4.6) on the grid $G_{h(4.5)}$ in the variables $x,t$ and Eqs.
(4.9) on the grid $G_{h\tau}$ in the variables $x,\tau$ are equivalent.
The condition of pointwise approximation on smooth functions of the
operator $L^0_{(4.7)}$ by the operator $\Lambda^0_{(4.9)}$ \ [12] brings
us to the relation
$$
\left|\gamma^0(x,\tau,h,h_{\tau})-1\right|\le \m,
\eqno (4.10\r{a}) $$
where $\m \ra 0$ as $h,h_\tau \ra 0$ at a point $(x,\tau)\in G_{h\tau}$,
the value $\m$ does not depend on the parameter $\eps_0$.
We shall say that the operator $\Lambda^0_{(4.9)}$ approximates the operator
$L^0_{(4.7)}$ uniformly on the set $G^*_\tau \subset \ov{G}_\tau$, if
$\m$ does not depend on $x,\tau$ for $(x,\tau)\in G^*_\tau$, that is,
$$
\m=\lambda(h,h_\tau) \ \r{ for }\ (x,\tau)\in G_\tau^*\cap G_{h\tau}.
\eqno (4.10\r{b}) $$
We shall assume that condition (4.10) is satisfied, where $G^*_\tau$
belongs to an $m$-neighbourhood of the set $S_{0\tau}=S_\tau$.
In this class $\bf A$ we seek to construct fitted operator schemes.
The following theorem is valid.
\begin{Theor}
In the class $\bf A$ of finite difference schemes there does not exist a
scheme, whose solution converges as $h,\;h_t\ra 0$ to the solution
of boundary value problem $(4.3),\ (4.2)$ $\eps_0$-uniformly.
\end{Theor}
The proof of this theorem is given in Subsection 4.3.
{\it Remark\, 1.}
Let the coefficient $\gamma_0$ in Eq. (4.9) depend on the parameter $\eps_0$.
In this case the statement of Theorem 2 remains valid.
{\it Remark\, 2.} Suppose that fitted operator schemes are
constructed on four-point stencils of implicit difference schemes with
a few fitted coefficients, and also the coefficients of these schemes
in the variables $x,\tau$ approximate the coefficients of Eq. (4.7a) on
some subset from an $m$-neighbourhood of the set $S_{0\tau}$ uniformly.
Let the difference scheme be monotone. In this class there do not exist
$\eps_0$-uniformly convergent finite difference schemes.
The results of Theorem 2 and the subsequent remarks can be explained
as follows. All the solutions of particular problem
(4.3), (4.2), (4.4) are singular ones.
The singular solutions of problem (4.3), (4.2), where
$\vp(x),\;x\in R$ is a smooth function, cannot be represented as a linear
combination of a finite sum of some fixed basis functions of an
initial layer type. Therefore by using a finite number of fitted
coefficients, it is impossible to satisfy the difference equations
for all basis functions.
{\bf 4.2.} In the case of problem (4.1), (3.2) we consider a class of
finite difference schemes (the class {\bf B}) which belongs to the class
{\bf A}. This class {\bf B} is defined by the condition
$\ov{G}=\ov{G}_{(3.2)}$. The theorem similar to Theorem 2 is valid.
\begin{Theor}
In the class ~\mbox{{\bf B}}~ of finite difference schemes there does not
exist a scheme, whose solution converges to the solution of boundary
value problem $(4.1),\ (3.2)$ $\eps_0$-uniformly as $h,\;h_t\ra 0$.
\end{Theor}
The proof of Theorem 3 is like that of Theorem 2.
Thus, in the case of problem (3.1), (3.2) under condition (3.3), in the
natural classes of difference schemes there does not exist a fitted
operator scheme (namely, under condition (3.7)) whose solution converges
to the exact one $(\eps_1,\eps_3)$--uniformly. For problem (3.19), (3.2) under condition
(3.3) there does not also exist such fitted operator schemes from the
natural classes of difference schemes (neither under condition (3.6), see
Shishkin [3] and [6], nor under condition (3.7); the proof of the last
statement is similar to the proof of the analogous statement
for problem (3.1), (3.2) under condition (3.7)),
that converge $(\eps_1,\eps_3)$--uniformly.
This means that fitted operator methods are not suitable for singularly
perturbed problems whose solution has a parabolic initial layer, unless
the mesh itself is fitted. So in the case of problem (2.1) we come
to such a conclusion:
{\it for singularly perturbed boundary value problems with a
parabolic initial layer
to construct $\eps$-uniformly convergent schemes,
the use of special meshes
condensing in a neighbourhood of the initial
layer is necessary.}
{\bf 4.3.} The {\it proof} \ of Theorem 2 is performed by the contradiction
method according to a plan of proving the non-existence of
fitted operator schemes convergent $\eps_1$-uniformly in the case of
problem (3.19), (3.2) under condition (3.6) (see Shishkin [3],\,[6] and
[7]). Assume that on the grid $\ov{G}_{h(4.5)}$ there exists a finite
difference scheme convergent $\eps_0$-uniformly.
Let us study this scheme.
From condition (4.10) such an important property of
the function $\g{}{0} $ follows (we name it by the property ($\star$)).
Let $(x_0,\tau_0)\in G_\tau$ be some point from the neighbourhood
of $G^*_\tau$, and also the set
$\ov{G}_{0\tau} = \mbox{$[x_0-\delta,x_0+\delta]$}\times [0,\tau_0]$,
$\delta>0$ belong to $\ov{G}^*_\tau.$ For any sufficiently small value
$ \delta_0>0$ one can find an $\delta^0=\delta^0(\delta_0)$\,-\,neighbourhood
of the point $(x_0,0)$, belonging to $\ov{G}_{0\tau}$
(namely, the set $G^0_\tau=(x_1^0,x_2^0)\times(0,\tau^0)$,
where $x_1^0=x_0-\delta^0$, $x^0_2=x_0+\delta^0$,
$\tau^0=2^{-1}\delta^0$, \ $\ov{G}^0_\tau\subseteq\ov{G}_{0\tau})$, such
that for any $(x_1,\tau)$, $(x_2,\tau)\in G_\tau^0$
\ and any $h,\,h_\tau\le m_1$, $m_1=m_1(\delta_0)$ the
following inequalities are satisfied
$$
\begin{array}{c}
\left|\g{1}{0}-1\right|\le m_2,\\[1ex]
\left|\g{1}{0}-\g{2}{0}\right|\le \delta_0,\quad
(x_1,\tau),\;(x_2,\tau)\in G_\tau^0.\no
\end{array}
\eqno (4.11) $$
The property ($\star$) ensures the validity of the maximum principle
for a Dirichlet problem on the set
$\ov{G}^0_{h\tau}=\ov{G}^0_\tau\cap\ov{G}_{h\tau}$
in the case of Eqs. (4.9) (on the set $\ov{G}^0_{h}=\ov{G}^0\cap\ov{G}_{h}$
for Eqs. (4.6)).
To prove Theorem 2 we estimate the functions
$w^i(x,t)=u^i(x,t)-z^i(x,t),\;
i=1,2,$ where $ u^i(x,t)=\Phi_0^i(x,t)$ is the solution of problem (4.3),
$z^i(x,t)$ is the corresponding solution of the difference scheme;
$\Phi_0^i(x,t)=\Phi^i(x,\eps_0^{-2}t)$ are auxiliary functions.
The functions $\Phi^{ i}(x,\tau),\;i=1,2$ are defined by\\[1ex]
\centerline{$
\Phi^i(x,\tau)=\Phi^i(x,\tau;x^*)=\sin(i(x-x^*+\tau))\exp(-i^2\tau),\quad
(x,\tau)\in \ov{G}_\tau,\; i\!=\!1,2$,}\\[1ex]
where $x^*$ is a parameter chosen from the segment $[x_1^0,x_2^0];$
here we set $x^*=x_2^0$.
The functions $\Phi^i(x,\tau)$ satisfy the regular differential equation
(Eq. (4.7a)). We define the auxiliary "fitted coefficients"
$\g{}{1}$, $\g{}{2}$ and the function $\g{}{*}$, i.e.,
the mean value of these coefficients, by
$$
\begin{array}{c}
\left(\delta_{x\ov{x}}+\delta_x - \gamma^i\delta_{\ov{\tau}}\right)\Phi^i(x,\tau)=0,
\ \ i=1,2,\\[1ex]
\g{}{*}=2^{-1}\left(\g{}{1}+\g{}{2}\right), \quad (x,\tau) \in G_{h\tau}.
\end{array}
$$
The value $\delta^0$ is chosen sufficiently small such that the following
condition is fulfilled
$$
\delta_{x\ov{x}}\Phi^i(x,\tau)+\delta_x\Phi^i(x,\tau)\ge m,\quad
(x,\tau)\in G^0_{h\tau},\ \ i=1,2.
$$
\renewcommand{\theequation}{4.12}
For the functions $\Phi^j(x,\tau)$ and the coefficients $\g{}{i}$
the following relations are valid
\begin{equation}
\delta_{x\ov{x}}\Phi^j(x,\tau)=\frac{\pa^2}{\pa x^2}\Phi^j(x,\tau)+\frac{1}{4!}
\frac{\pa^4}{\pa x^4} \Phi^j(x_1,\tau)h^2+
\frac{1}{4!}\frac{\pa^4}{\pa x^4}\Phi^j(x_2,\tau)h^2, \ \ \
\end{equation}
\centerline{$x-h \le x_1,\; x_2\le x+h$,}
\beqv
\delta_x\Phi^j(x,\tau)&=&\frac{ \pa}{ \pa x}\Phi^j(x,\tau)+
\frac{1}{2}\frac{ \pa^2}{ \pa x^2}\Phi^{j}(x,\tau)h+
\frac{ 1}{ 6}\frac{ \pa^3}{ \pa x^3}\Phi^j(x_3,\tau)h^2,\no\\[1ex]
&&\hspace{6mm}x \le x_3 \le x+h,\no\\[2ex]
\delta_{\ov{\tau}}\Phi^j(x,\tau)&=&\frac{ \pa}{ \pa\tau}\Phi^j(x,\tau)-
\frac{1}{2}\frac{ \pa^2}{ \pa \tau^2}\Phi^{j}(x,\tau)h_{\tau}+
\frac{ 1}{ 6}\frac{ \pa^3}{ \pa\tau^3}\Phi^j(x,\tau_1)h_\tau^2,\no\\[1ex]
&&\hspace{6mm}\tau-h_{\tau} \le \tau_1 \le \tau,\no
\eeqv
\vspace{-3mm}
$$
\begin{array}{c}
|\,\delta_{\ov{\tau}}\Phi^j(x,\tau)-j\{\cos(j\nu)[1+j^2h_\tau]-
\hspace{4cm}\\[1.5ex]
\mbox{\hspace{2cm}} -j\sin(j\nu)[1+2^{-1}(j^2-1)h_\tau]\}
\exp(-j^2\tau)\,|\le M h_\tau^2,\\[2ex]
\left|\left(\delta_{x\ov{x}}+\delta_x\right)\Phi^j(x,\tau)-j\left\{\cos(j\nu)-
j\sin(j\nu)\left[1+2^{-1}h\right]\right\}\exp(-j^2\tau)\right|\le Mh^2,\\[2ex]
|\,\g{}{i}-\left\{1-i^2\cos(i\nu)h_\tau- \right.
\hspace{3cm}\\[1.5ex]
\left. \mbox{\hspace{2cm}} -2^{-1}i\sin(i\nu)
\left[h-(i^2-1)h_\tau\right]\right\}\,|
\le M_1\left[h^2+h_\tau^2\right],\\[2ex]
\left|\,\g{}{1}-\g{}{2}-3h_\tau\,\right|\le M_2\left[h^2+h^2_\tau+\delta^0\left(h+h_\tau\right)\right],\\[1.5ex]
\nu=\nu(x,\tau)=x+\tau-x^*,\ \
|x-x_0| \le \delta_0,\ \ \tau \le \delta_0, \ \ i,j=1,2.
\end{array}
$$
We see that the values of $\gamma^1 \ii \gamma^2$ differ by a positive
quantity of the order of the value $\beta_{(4.12)}=3h_\tau$ for
sufficiently small values of $h,\ h_\tau \ii \delta^0$. We choose
the constants $h_1,\;h_{\tau1} \ii \delta_1^0$ sufficiently small so
that
$$
M_3\left[h_1^2+h_{\tau1}^2+\delta^0_1\left(h_1+h_{\tau1}\right)\right]\le
8^{-1}m_{(4.13)},
\eqno (4.13) $$
where $m_{(4.13)}=3h_{\tau_1},\; M_3=2(M_{1(4.12)}+M_{2(4.12)})$.
The value $\delta^0$ is chosen sufficiently small such that
the inequalities $\delta^0\le \delta_1^0$ and $\delta_0\le 8^{-1}m_{(4.13)}$,
inequalities (4.11), and also the following inequality are true
$$
\left|\g{1}{i}-\g{2}{i}\right|\le \delta_0,\quad (x_1,\tau),\;
(x_2,\tau)\in \ov{G}^0_\tau,\;i=1,2.
$$
The constant $h_{\tau2}$ is chosen to satisfy the condition: \
$h_{\tau2}\le h_{\tau1}=3^{-1}m_{(4.13)},\ \ h_{\tau2}\le\delta^0$
and further it is fixed.
We construct the set $G^0_\tau$
$$
G^0_\tau=\left\{(x,\tau): \ |x-x_0|<\delta^0,\;0<\tau\le h_{\tau 2}\right\},
\eqno (4.14) $$
on which we shall study the fitted scheme. Note that $h_{\tau 2}$ and
$\delta^0$ are independent of the parameter $\eps_0$.
For $x=x_0,\;\tau=h_{\tau 2}$ at least one of the following
inequalities is fulfilled
$$
\gamma^0(x_0,h_{\tau 2},h,h_{\tau 2})\ge \gamma^*(x_0,h_{\tau2},h,h_{\tau2}),
\eqno (4.15\r{a}) $$
$$
\gamma^0(x_0,h_{\tau 2},h,h_{\tau 2})\le \gamma^*(x_0,h_{\tau2},h,h_{\tau2}).
\eqno (4.15\r{b}) $$
Suppose that inequality (4.15a) is valid. Then
$$
\gamma^0(x,h_{\tau2},h,h_{\tau2})\ge \gamma^2(x,h_{\tau2},h,h_{\tau2})+
8^{-1}m_{(4.16)},\quad
(x,t)\in G^0_{h\tau},
\eqno (4.16) $$
where $G^0_\tau=G_{\tau(4.14)},\; m_{(4.16)}=3h_{\tau 2}$. In this case
the function $w^{02}(x,\tau)=w^2(x,t(\tau))$ satisfies the relation
$$
\Lambda^0_{(4.9)}w^{02}(x,\tau)\!=\!(\gamma^2(x,\tau,h,h_{\tau 2})-
\gamma^0(x,\tau,h,h_{\tau 2}))\delta_{\ov{\tau}}\Phi^2(x,\tau)\ge
m^1_{(4.17)}, \ \ \
\eqno (4.17\r{a}) $$
\centerline{$(x,\tau)\in G^0_{h\tau}$.}\\[0.5ex]
In the case of inequality (4.15b) for the function
$w^{01}(x,\tau)=w^1(x,t(\tau))$ we get
$$
\Lambda^0_{(4.9)}w^{01}(x,\tau)\le -m^2_{(4.17)},\quad
(x,\tau)\in G^0_{h\tau}.
\eqno(4.17\r{b}) $$
Let us assume that the difference scheme converges $\eps_0$-uniformly
$$
\left|w^i(x,t)\right|\le \lambda(h,h_t),\ \ (x,t)\in \ov{G}_h,\;
\eps_0\in(0,1],\;i=1,2.
\eqno (4.18) $$
This implies that on the set $S^0_{h\tau},\;S^0=\ov{G}^0\setminus G^0$,
the following inequality is satisfied
$$
\left|w^{0i}(x,\tau)\right|\le \lambda(h,\eps_0^2h_{\tau 2}),\ \
(x,\tau)\in S^0_{h\tau},\;i=1,2,
\eqno (4.19) $$
where $\lambda(h,\eps_0^2h_{\tau 2})\ra 0$ for
$\eps_0,\, h \to 0,\ h_{\tau 2}\!=\!\r{const}$.
Taking into account (4.17),\,(4.19) it is shown using the
maximum principle that, at least for one value $i=j$,
the next inequality is true
$$
\left|u^{0j}(x_0,h_{\tau 2})-z^{0j}(x_0,h_{\tau 2})\right| \ge m ~~
\mbox{ for } ~~ h\le h_1,\;\eps_0\in (0,\eps_0^1],
\eqno (4.20) $$
where $h_1$ and $\eps_0^1$ are sufficiently small numbers.
From (4.20) it follows that
$$
\max_{\ov{G}_h}\left|u^j(x,t)-z^j(x,t)\right|\ge m
\eqno(4.21)
$$
for any $h\le m_{1(4.11)},\ h_t\le \eps_0^2m_{1(4.11)},\
h_t\le(\eps_0^1)^2h_{\tau 2},\
\eps_0^1\!=\!\eps_{0(4.20)}^1,\ h\!\le\! h_{1(4.20)}$;
\ $h_t=\eps_0^2h_{\tau 2}$.
Inequality (4.21) contradicts assumption (4.18) for
$\eps_0\in (0,\eps_0^1]$. This completes the proof of Theorem 2.
\bc
\bf 5. STRUCTURE OF SINGULAR PARTS OF THE SOLUTION
\ec
\baselineskip 15.5pt
In the case of problem (2.1) we give certain conditions for the
perturbation parameters, under which there exist no schemes based on
the fitted operator method.
The singular parts of the solution for problem (2.1)
(i.e., the principal terms in its asymptotic expansion) exhibit the
behaviour depending on the ratio between the parameters.
The boundary value problems corresponding to these singular solutions
can be written out by using the techniques of asymptotic expansions
(see, e.g., [15]).
Let us give a brief description of the singular components of the
solution in the case of conditions (2.6)--(2.9).
{\it Notation.} \ Assume $S_1=S^L$, $S_i=\Gamma_i\times(0,T],\ i=2,3$, where
$\Gamma_2$ and $\Gamma_3$ are, respectively, the left and right
boundaries of the set $D$ so that $\Gamma=\Gamma_2 \cup \Gamma_3$; \
$S_{0i}=S_0\cup S_i$, $ S^{(0i)}=S_0\cap\ov{S}_i$, $i=1,2,3$.
By $G^{(j)}$ and $G^{(0i)}$ we denote the extensions of $G$ beyond the sets
$S_j$ and $S_{0i}$. These extensions are assumed to be tube domains
that contain the corresponding boundary sets including their
$m$-neighbourhoods. We denote by $L^{(j)}_{(2.1)}$, $L^{(0I)}_{(2.1)}$ \ and \
$v^{(j)}(x,t)$, $v^{(0i)}(x,t)$ the extensions of the
operator $L_{(2.1)}$ and the function $v(x,t),\;(x,t)\in \ov{G}$
onto $\ov{G}^{(j)}$ and $\ov{G}^{(0i)}$, respectively, which retain the
properties of the operator $L_{(2.1)}$ and the smoothness of the
function $v(x,t)$.
{\bf 5.1.}
In the case of condition (2.6) we divide the range of the
parameters into the two additional subsets
$$
\te_2^{\,2}\te_1^{\,-1}\le M,
\eqno (5.1) $$
$$
\te_2^{\,2}\te_1^{\,-1}\ge m,
\eqno (5.2) $$
where $\te_i=\te_{i(2.5)}(\eps)$.
The solution of the boundary value problem under conditions
(2.6), (5.1) can be represented as a sum of the functions\\[1ex]
\centerline{$
u(x,t)=U(x,t)+V_1(x,t), \quad (x,t)\in \ov{G}$,}\\[1ex]
where $U(x,t),\ V_1(x,t)$ are the regular and singular parts of
the solution.
The function $U(x,t)$ is a restriction on $\ov{G}$ of the function
$U^{(1)}(x,t),\; (x,t)\in \ov{G}^{(1)}$, where
$U^{(1)}(x,t)$ is the solution of the problem
\beqv
L^{(1)}_{(2.1)} U^{(1)}(x,t)&=&F^{(1)}(x,t,\eps), \quad (x,t)\in G^{(1)},\\[0.2ex]
U^{(1)}(x,t)&=&\f^{(1)}(x,t),\quad (x,t)\in S^{(1)}.\no
\eeqv
Here $S^{(1)}=S(G^{(1)}),\; \f^{(1)}(x,t)$ is the smooth extension of the
function $\f(x,t)$, defined on $S_0$, i.e.,
$\f^{(1)}(x,t)=\f(x,t),\;(x,t)\in S_0$.
The functions $F^{(1)}(x,t,\eps) \ii \f^{(1)}(x,t)$ are assumed
to vanish outside of sufficiently small neighbourhoods of the sets
$\ov{G} \ii S_0$, respectively.
The function $V_1(x,t)$, i.e., the boundary layer in
the neighbourhood of the set $S_1=S^L$,
is the solution of the problem \vspace{-2mm}
\beqv
& L_{(2.1)} V_1(x,t)=0,\quad (x,t)\in G, &\\[0.3ex]
& V_1(x,t)=\f(x,t)-U(x,t),\quad (x,t)\in S. & \no
\eeqv
Thus, the singular component $V_1(x,t)$ is described by
the parabolic equation; this function $V_1(x,t)$ is the parabolic
boundary layer.
In the case of conditions (2.6), (5.2) provided that \
$
\te_1,\; \te_2=o(1)
$
the solution of the problem has such a representation\\[1ex]
\centerline{$
u(x,t)=U(x,t)+V_2(x,t)+V_3^{\ast}(x,t),\quad (x,t)\in\ov{G}$,}\\[1ex]
while for \ $\te_2\geq m$ \ it take the form\\[1ex]
\centerline{$
u(x,t)=U(x,t)+V_2(x,t),\quad (x,t)\in \ov{G}$.}\\[1ex]
Here the functions $V_2(x,t)$ and $V_3^{\ast}(x,t)$ are, respectively,
the regular and hyperbolic boundary layers which satisfy the equations
$$
L_{(5.3)}V_2(x,t)\equiv \left\{ \ti{\eps}_1^{\,2} a(x,t)
\frac{\pa^2}{\pa x^2}+ \ti{\eps}_2^{\,2} b(x,t)
\right\} V_2(x,t)=0,\ \ (x,t)\in G,
\eqno(5.3)
$$
$$
L_{(5.4)}V_3^{\ast}(x,t)\!\equiv\! \left\{\!
\ti{\eps}_2^{\,2} b(x,t)\frac{\pa}{\pa x} -
\ti{\eps}_3^{\,2} p(x,t)\frac{\pa}{\pa t} -
\ti{\eps}_4^{\,2} c(x,t)\right\}\! V_3^{\ast}(x,t)\!=\!0,\;
(x,t)\!\in\! G. ~~
\eqno(5.4)
$$
Here and below the boundary conditions for the singular components
can be written out in terms of the function $U(x,t)$, the regular
part of the solution.
\renewcommand{\theequation}{5.5}
The singularly perturbed differential equation
\beq
&& L_{(5.5)}u(x,t)\equiv \left\{\ov{\eps}^2_1 a(x,t)\frac{\pa^2}{\pa x^2} +
\ov{\eps}^2_2 b(x,t)\frac{\pa}{\pa x}-p(x,t)\frac{\pa}{\pa t}-
\right. \\[1ex]
&& \mbox{\hspace{2cm}} \left. ~~~
\phantom{\frac{\pa^2}{\pa x^2}}-\,c(x,t)\right\}u(x,t)=f(x,t),
\quad (x,t)\in G,\quad \ov{\eps}_1,\, \ov{\eps}_2\in (0,1], \no
\eeq
where $\ov{\eps}_1,\, \ov{\eps}_2$ are the perturbation parameters,
is Eq. (2.1a), in which the parameters obey condition (2.6). Thus,
in the case of a Dirichlet problem for Eq. (5.5) both regular and parabolic or
hyperbolic boundary layers can appear depending on the ratio of the
parameters $\ov{\eps}_1$ and $\ov{\eps}_2$. No initial layer appears under
condition (2.6).
{\bf 5.2.} In the case of condition (2.7) we consider the solution in such a
representation\\
\centerline{$
u(x,t)=U(x,t)+V_0(x,t),\quad (x,t)\in \ov{G}$,}\\[1ex]
where $U(x,t),\;V_0(x,t)$ are the regular and singular parts of the solution.
The regular part $U(x,t)$ can be found by restriction of the function
$U^{(0)}(x,t)$, $(x,t)\in\ov{G}^{\,(0)}$ on the set $\ov{G}$.
The function $V_0(x,t)$, i.e., the initial layer,
is the solution of the parabolic equation \
$L_{(2.1)}V_0(x,t)=0,\quad (x,t)\in G$. \
Under condition (2.7) no boundary layer appears.
\renewcommand{\theequation}{5.6}
The singularly perturbed differential equation with the small parameter
$\ov{\eps}$
\beq
L_{(5.6)}u(x,t)\!\equiv\!\left\{a(x,t)\frac{\pa^2}{\pa x^2}+
b(x,t)\frac{\pa}{\pa x}-\ov{\eps}^{\,2} p(x,t)\frac{\pa}{\pa t}
-c(x,t)\right\}\!u(x,t) \!=\!f(x,t),~~ &&\\[0.5ex]
(x,t)\in G,\quad \ov{\eps}\in(0,1]\no \mbox{\hspace{50mm}}&&
\end{eqnarray}
is Eq. (2.1a), in which the parameters obey condition (2.7).
Thus, in the case of a Dirichlet problem for Eq. (5.6) the initial layer
observed is a parabolic layer.
{\bf 5.3.} If condition (2.8) is fulfilled and also $\eps_1,\ \eps_3=o(1)$,
we represent the solution in the form\\[0.3ex]
\centerline{$
u(x,t)=U(x,t)+V_0^{\ast}(x,t)+V_2(x,t)+V_{02}(x,t),\;(x,t)\in \ov{G}$.}\\[1ex]
Here $V_0^{\ast}(x,t)$, $V_2(x,t)$ and $V_{02}(x,t)$ are, respectively,
the hyperbolic initial layer, the regular boundary layer and
the parabolic corner layer (in a neighbourhood of the set
$S^{(02)}$).The function $U(x,t)$, $(x,t)\in\ov{G}$ is a restriction on
$\ov{G}$ of the function $U^{(02)}(x,t)$, $(x,t)\in \ov{G}^{\,(02)}$.
The function $V_{02}(x,t)$ is the solution of the parabolic equation\\[1ex]
\centerline{$
L_{(2.1)}V_{02}(x,t)=0,\quad (x,t)\in G$}\\[1ex]
with the boundary conditions which are rapidly decreasing (in an exponential
way) when we recede from the set $S^{(02)}$.
For $\te_1\!\geq\! m$ and/or $\te_3\!\geq\! m$ condition (2.8) turns
into one of conditions (2.6),\,(2.7).
Thus, in the case of a Dirichlet problem for the singularly perturbed
equation
$$
L_{(5.7)}u(x,t)\equiv \left\{\ov{\eps}_1^2 a(x,t)\frac{\pa ^2}{\pa x^2}+
b(x,t)\frac{\pa}{\pa x}-
\ov{\eps}_3^2 p(x,t)\frac{\pa}{\pa t}-c(x,t)\right\}u(x,t)=f(x,t),
\eqno(5.7)
$$
\vspace{-6mm}
\centerline{$(x,t)\in G,\quad \ov{\eps}_1,\ \ov{\eps}_3 \in (0,1]$}\\[0.5ex]
regular boundary, initial hyperbolic and corner parabolic layers appear
as $\ov{\eps}_1,\ \ov{\eps}_3\to 0$.
{\bf 5.4.} Under condition (2.9) the range of the parameters
is divided into two subsets (5.1) and (5.2). Under condition (5.1)
we represent the solution of the problem in the form
$$
u(x,t)=U(x,t)+\sum_{i=0,2,3}V_i(x,t)+\sum_{i=2,3}V_{0i}(x,t),\quad
(x,t)\in \ov{G}.
$$
Here $V_0(x,t)$ and $V_2(x,t),\ V_3(x,t)$ are the regular initial and
boundary layers, $V_{02}(x,t)$ and $V_{03}(x,t)$ are the corner
parabolic layers in the neighbourhoods of the sets
$S^{(02)}$ and $S^{(03)}$, respectively.
In the case of condition (5.2) we assume
$$
u(x,t)=U(x,t)+\sum_{i=0,2,3}V_i(x,t)+V_{02}(x,t)+V_{03}^{\ast}(x,t),\;
(x,t)\in \ov{G}.
$$
Here, in contrast to $V_{03}(x,t)$ from the previous expansion,
$V_{03}^{\ast}(x,t)$ is the hyperbolic corner layer in the neighbourhood
of the set $S^{(03)}$. The regular part $U(x,t)$ in these expansions is
obtained by restriction on $\ov{G}$ of the function
$U^{(01)}(x,t)$, $(x,t)\in \ov{G}^{\,(01)}$.
Under condition (2.9) and for \ $\ti{\eps}_1,\ \ti{\eps}_2,\ \ti{\eps}_3=o(1)$
layers appear in a neighbourhood of each side of the boundary $S$.
If one of the components $\ti{\eps}_1,\ \ti{\eps}_2,\ \ti{\eps}_3$ is
a quantity of the order of 1, then condition (2.9) turns into
one of conditions (2.6)--(2.8).
Thus, in the case of a Dirichlet problem for the singularly perturbed equation
$$
L_{(5.8)}u(x,t)\equiv \left\{\ov{\eps}_1^2 a(x,t)\frac{\pa ^2}{\pa x^2}+
\ov{\eps}^2_2 b(x,t)\frac{\pa}{\pa x}-
\ov{\eps}_3^2 p(x,t)\frac{\pa}{\pa t}-c(x,t)\right\}u(x,t)=f(x,t),
\eqno(5.8)
$$
\vspace{-6mm}
\centerline{$(x,t)\in G,\quad \ov{\eps}_1,\ \ov{\eps}_2,\
\ov{\eps}_3 \in (0,1]$}\\[0.5ex]
regular initial and boundary layers, and also parabolic and hyperbolic
corner layers appear as $\ov{\eps}_1,\ \ov{\eps}_2,\ \ov{\eps}_3\to 0$.
{\bf 5.5.} The corner parabolic layers are defined only by the local
(near the "corners" $S^{(02)}$ and $S^{(03)}$) boundary conditions and,
therefore, in virtue of their simplicity they admit to construct
$\eps$-uniformly convergent schemes based on the fitted operator
methods.
The parabolic boundary layers in problem (2.1) appear (in a neighbourhood
of the lateral boundary of the domain) provided that\\[1ex]
\centerline{$
\eps_1=o(1),\quad \eps_2=O(\eps_1^{1/2}),\quad
\eps_3\geq m,\quad \eps_4=O(1)$,}\\[1ex]
while ~the \,parabolic ~initial ~layers
~(\,in\, a \,neighbourhood\, of \,the ~lower ~boundary\,) \\
arise in the case of the conditions \\[1ex]
\centerline{$
\eps_1\geq m,\quad \eps_2=O(1),\quad
\eps_3=o(1),\quad \eps_4=O(1)$.}\\[1ex]
Thus, in the case of boundary value problems for singularly
perturbed Eqs. (5.5)--(5.8) and (2.1a) there do not exist fitted
operator schemes convergent $\eps$-uniformly in the discrete maximum norm.
This motivates the necessity of using the techniques of special
condensing meshes for the construction of $\eps$-uniformly
convergent schemes for the above-stated problem.
Note that the author previously examined the construction
of $\eps$-uniformly convergent schemes using special refined meshes,
for the case of problem (2.1), only for $\eps_3\geq m$, $\eps_4=O(1)$
and either $\eps_2=0$ or $\eps_2=1$ (see, for example, Shishkin [3], [6],
[16]--[18] and the references therein).
\bc
ACKNOWLEDGEMENTS
\ec
This work was supported in part by the Russian Foundation for
Basic Research under Grant N 95-01-00039.
\bc
\bf REFERENCES
\ec
\small
\baselineskip 14pt
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\end{enumerate}
\end{document}