\documentstyle[12pt,a4]{article}
% Abstract for the conference NMA'98 in Sofia
\setcounter{page}{1}
\textwidth 6in
\textheight 9in
\oddsidemargin 0pt
\evensidemargin 0pt
\topmargin 0pt
\newcommand{\eps}{\varepsilon}

\begin{document}
\sloppy

\begin{center}
\bf GRID APPROXIMATION OF SINGULARLY PERTURBED ELLIPTIC
EQUATIONS WITH CONVECTIVE TERMS IN UNBOUNDED DOMAINS
\end{center}

\begin{center}
Grigorii I. Shishkin\\[1ex]
{\it Institute of Mathematics and Mechanics,
Russian Academy of Sciences, Ekaterinburg 620219,  Russia}
\end{center}

\bigskip
On a half-plane ~$D=\{x:\ 0<x_1<\infty, \ x_2\in R\}$,
we consider the construction of difference schemes for
singularly perturbed elliptic equations with convective terms,
where the flow is directed to the boundary. We restrict ourselves
to the case if either (1) the limit equations (i.e., equations for
$\eps=0$,
where $\eps$ is a small parameter multiplying the highest derivatives)
include the reaction terms which describe a decay of the substance,
or (2) the sources have a bounded support of width $\delta$. In the
last
case the power of the sources is inversely proportional to the value of
$\delta$.
The perturbation parameter $\eps$ and the parameter $\delta$ can take
arbitrary values in the half-intervals $(0,1]$ and $(0,M]$,
respectively, where $M<\infty$. As $\eps\to 0$, a boundary layer
appears in a neighbourhood of the set ~$\{x_1=0, \ x_2\in R\}$.

\medskip
Difficulties arising in the numerical solution of singular perturbation
problems on bounded subdomains are sufficiently well-known. However,
the situation is worse in the case of unbounded domains because the
number of mesh points in the discretizations is infinite.
Moreover, the problem may have a singular right-hand side of the
equation. If $\delta\to 0$, even for $\eps=1$, the singularity of the
solution appears caused by the "almost" concentrated source. Therefore,
there is a need to develop special numerical methods to handle
boundary layers and other singularities mentioned above.


\medskip
For these two problems with "reaction terms"
and with "source of the finite width" we construct, respectively,
$\eps$-uniformly and ($\eps,\delta$)--uniformly convergent schemes.
The each class of the above schemes constructed includes:

(a) formal schemes, i.e., schemes on meshes with an infinite number of
the nodes, that converge on the whole half-plane $D$; ~
and

(b) constructive schemes, i.e., schemes on meshes with the finite
number of
the nodes, that converge on the bounded subdomains from this
half-plane.\\
\noindent
Note that parameter-uniformly convergent schemes were not considered
previously in the case of unbounded (with respect to both $x_1$ and
$x_2$) domains.

\medskip
To construct schemes, we use classical difference approximations of the
differential equations on piecewise uniform meshes which
condense in the neighbourhood of the boundary layer. It is shown that,
for the problem with "source of the finite width", there exist no
constructive
schemes (b) that converge $(\eps,\delta)$--uniformly in the case
$\delta\in (0,\infty)$. We discuss also the validity of the obtained
results for non-stationary problems.\\


This work was supported in part by the Russian Foundation for
Basic Research under Grant N 98-01-00362.
\end{document}