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\begin{center}
\large{\bf
Incremental Unknowns for solving convection-diffusion
problems in an open channel
        } \\[.5cm]
Pascal POULLET and Fran\c cois BADE  \\[.2cm]
\footnotesize{
D\'epartements de Maths et de Physique,
Universit\'e des Antilles et de la Guyane,\\ 97159
Pointe-\`a-Pitre Cedex, Guadeloupe F.W.I \\
\hfill\\
e-mail: Pascal.Poullet@univ-ag.fr
}
%\\[.5cm]
\end{center}
\begin{abstract}
Incremental Unknowns (IU) have been introduced 
in order to compute solutions on approximate inertial manifolds (dynamical 
systems theory),
in finite differences. Later, this tool has been
often used to compute approximate solutions of elliptic partial 
differential equations with multilevel discretization. 
It has been proved recently that the method is also useful as a 
preconditioner for other linear and nonlinear problems 
However, the expansion of Nonlinear Galerkin concept in finite difference 
discretization has been 
hindered because of the non-small size of {\it increments}. Typically, in 
fluid Mechanics, the IU, which represent the small scale components, are 
proportional to the gradient norm of the solution, and effectively can 
vary substantially. 
One way to overcome this problem is to increase the order of magnitude
for the definition of the increments. 
Nevertheless, 
the necessary number of interpolation points increases with the order of 
interpolation. 
Following that, the corresponding transfer matrix from the hierarchical 
basis of higher order to the nodal basis is much more filled, therefore 
the solver which performs iteratively this operator is penalized.
Another basic idea may consist in adapting the spatial mesh to reduce
the size of the increments. While doing it with a uniform mesh, the class 
of large size numerical problems is easily reached.
Yet, the reduction of spatial mesh can be achieved with adaptive 
meshes or nonuniform meshes. 

Our aim is to focus onto a linear equation, the convection-diffusion 
equation in a rectangular domain : the open duct.
Although, the problem may seem to be simple, it is known
that with some boundary conditions, we have the development of a boundary 
layer perpendicular to the flow. 
Then, we choose a discretization by nonuniform finite differences in the 
perpendicular direction and a uniform mesh in the remaining direction. 
To catch the boundary layer, we decide to use a distribution of 
narrowed points by the tanh mapping.
A comparative study of several multilevel preconditioners including 
Incremental Unknowns has been performed, for different values of parameters. 
Some partial results show that in some cases, the IU on nonuniform meshes 
(which have been defined with various weigthed average)% \cite{CHEHAB97}) 
improves the speed to achieve the solution.
\end{abstract}
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\end{document}