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\title{Adaptive Refinement Procedure for Singularly Perturbed
Convection-Diffusion Problems.}
\author{ Mariana Nikolova and Owe Axelsson \\
Faculty of Mathematics and Informatics\\
University of Nijmegen\\
The Netherlands\\
e-mail: nikolova@sci.kun.nl, axelsson@sci.kun.nl}
\date{}

\begin{document}
\maketitle

\begin{abstract}
\noindent
A finite difference method is presented for singularly perturbed
convection-diffusion problems with discretization error estimates of nearly
second order, which hold uniformly in the singular perturbation parameter
$\vep$. The method is based on a defect-correction technique and Shishkin
type meshes in the layer region(s). These a priori adapted meshes give good
results for exponential and parabolic layers.\\
Nevertheless, the use of Shishkin meshes to solve problems with interior
layers and more complicated layers, the location of which is unknown a
priori, is still questionable. In this case we use the black box tool - a
posteriori adapted meshes obtained by some adaptive refinement procedure.
The advantages and disadvantages of {\em a priori-}  and {\em a posteriori-}
adaptive refinement techniques are illustrated by numerical experiments in
2D.\\



\noindent
{\bf \it AMS Subject Classifications:} 35B25, 65N06, 65N15, 65N50.

\noindent
\keywords{singularly perturbed convection-diffusion problems,
defect-correction technique, order of discretization error, a priori adapted
mesh - Shishkin mesh, a posteriori adapted mesh}
\end{abstract}

\end{document}