% formatted on 970304 by morris
\documentstyle{article}
\setcounter{page}{43}
\textheight 9.5in
\textwidth 5.75in
\voffset -1in
\hoffset -0.55in
%\documentstyle{llncs}
\begin{document}
\title{Permutations for preconditioning of discretizations of
convection-diffusion equations}
\author{Daniel B. Szyld \\
Department of Mathematics, Temple University \\ 
1805, N Broad Street, Philadelphia, PA 19122-6094, USA. \\
E-mail : szyld@math.temple.edu}

\date{}

\maketitle
Consider a two-dimensional convection-diffusion equation of the form 
$$
-\epsilon \Delta u + p(x,y) u_{x} + q(x,y) u_{y} = f 
$$
on $ \Omega $, $\epsilon > 0$, with some appropriate boundary conditions.
Once discretized, the corresponding linear system may be very difficult
to solve. By difficult to solve we mean that the modern Krylov-type methods,
such as GMRES(k), QMR or BiCGStab, fail to converge when preconditioned
by ILU(0), ILU(1), Threshold ILU, and with approximate factored inverse
preconditioners. 
It is well known that the ordering of the variables may have
a considerable effect on the convergence of Krylov-type methods; see, e.g.
Duff and Meurant, BIT, 1989].
Ir this paper, we review the algorithm TPABLO (Threshold Parametrized
Block Ordering) which, given a given matrix, produces a permutation
in such a way that the permuted matrix has dense blocks along the
diagonal. Furthermore, the entries in the off diagonal blocks,
all lie below the prescribed threshold.
We present several numerical experiments, which show how the same
Krylov-type methods, with the same preconditioners, converge rapidly,
when applied to the permuted matrix reordered by TPABLO.
\end{document}

---------------------------------
Daniel B. Szyld                          szyld@math.temple.edu
Department of Mathematics (038-16)       http://www.math.temple.edu/~szyld
Temple University, 1805 N Broad Street   phone + 1 215 204 7288
Philadelphia, PA 19122-6094, USA         fax + 1 215 204 6433         

