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\title{\vspace*{-0.5cm}
'$E\xi$\hspace{-1mm}
$\begin{array}[b]{c}
\mbox{\tiny c}\\[-4mm]\varepsilon\end{array}\!\!\nu\acute{o}s$ 
$\pi\acute{\alpha}\nu\tau\alpha$: 
from the unitary Korn's type inequality 
in subspaces to 
a unified way for developing iterative 
methods\\ for 3$D$ thin elastic structures with uniform convergence
}
\author{
\Large Leonidas S. Xanthis  
}
\date{}

\begin{document}

\maketitle

\vspace*{-8mm}

\begin{center}
Centre for Techno-Mathematics {\em \&} Scientific Computing Laboratory \\
University of Westminster, London HA1 3TP, UK ({\tt lsx@wmin.ac.uk})
\end{center}

\vspace*{4mm}

\begin{center}
{\LARGE Abstract}
\end{center}

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``An accurate, fully three-dimensional simulation of a very thin body 
is beyond the power of even the most powerful computers and
computational techniques \ldots The development and mathematical validation of
accurate numerical methods for shells which are reliable for the full range
of shell geometries \ldots  is one of the most challenging issues in numerical 
computation''. This quotation is from the rationale given by the
Mathematical Sciences Research Institute (MSRI) of the USA
in announcing a workshop for the year-2000  on
`Elastic Shells: Modeling, Analysis and 
Numerics' 
organized by the eminent
D.N. Arnold (Penn State), 
I. Babu\v{s}ka (Austin), F. Brezzi (Pavia),
Ph.G. Ciarlet (Paris) and J. Pitk\"{a}ranta (Helsinki).\\ 

In this paper we address the fundamental {\em fin-de-mill\'{e}naire}
computational issue
of how to solve efficiently  large-scale 3$D$ elasticity problems for
{\em thin} structures of arbitrary geometry, for example, shells. 
We show that by invoking the {\em Ovtchinnikov-Xanthis's (OX's)
\footnote{Not to be confused with Ox's as in 
{\em Doctor Ox's Experiment}, the novella by Jules Verne and new opera 
by Gavin Bryars to have its
world premi\`{e}re in June 1998 at the English National Opera.}
Inequality} 
[1-5] -- the {\em only} Korn's type inequality  
providing pivotal information for the radical improvement of
the performance of iterative methods for thin 
structures -- we can establish an effective and unified framework  for 
developing robust iterative methods for thin elastic structures 
with convergence rate independent of both the thickness and the 
discretization parameters. Thus \\[-3mm]
we recall and commend the all-pervasive and unifying Logos of
Heraclitus:
$\begin{array}[b]{c}
\mbox{,}\\[-2mm] \varepsilon\end{array}\!\!\xi$\hspace{-1mm} 
$\begin{array}[b]{c}
\mbox{\tiny c}\\[-2.5mm]\varepsilon\end{array}\!\!\nu\acute{o}s$ 
$\pi\acute{\alpha}\nu\tau\alpha$.

%
%\end{abstract}

{\small
\begin{thebibliography}{1}

\bibitem{CMAME1}
E.~E.~Ovtchinnikov and L.~S.~Xanthis.
A new {K}orn's type inequality for thin domains and its application
  to iterative methods.
Comput. Methods Appl. Mech. Engrg. 138 (1996) 299--315.

\bibitem{CRAS}
E.~E.~Ovtchinnikov and L.~S.~Xanthis.
A new {K}orn's type inequality for thin elastic structures.
C. R. Acad. Sci. Paris, S\'erie I 324 (1997) 577--583.

\bibitem{RS1}
E.~E.~Ovtchinnikov and L.~S.~Xanthis.
The {K}orn's type inequality in subspaces 
  and thin elastic structures.
Proc. R. Soc. Lond. A 453 (1997) 2003--2016.

\bibitem{CMAME2}
E.~E.~Ovtchinnikov and L.~S.~Xanthis.
The discrete {K}orn's type inequality for thin domains and its application
  to iterative methods.
Comput. Methods Appl. Mech. Engrg. 160 (1998) 23-37.

\bibitem{RS2}
E.~E.~Ovtchinnikov and L.~S.~Xanthis.
Iterative subspace correction methods for thin
elastic structures and {K}orn's type inequality in subspaces. 
Proc. R. Soc. Lond. A 454 (1998) 2023-2039.
\end{thebibliography}
}
\end{document}