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\begin{center}
\textbf{\Large INCREMENTAL UNKNOWNS} \\ 
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\textbf{\Large AND} \\	
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\textbf{\Large GRAPH TECHNIQUES}	
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\

\begin{center}
Salvador Garcia \\
Instituto de Matem\'aticas y F\'{\i}sica \\
Universidad de Talca \\
Casilla 721, Talca, Chile \\
E-mail: sgarcia@pehuenche.utalca.cl
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\begin{center}
\textbf{\Large Summary}
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\

\qquad The condition
number of the incremental unknowns matrix associated to the Laplace
operator is $ O(1/h_{0}^{2}) O((\log h)^{2}) $ where $ h_{0} $ is
the mesh size of the coarsest grid and where $ h $ is the mesh size of the
finest grid. Furthermore, if block diagonal scaling is used then the
condition number of the preconditioned incremental unknowns matrix
associated to the Laplace operator comes out to be
$ O((\log h)^{2}) $.
 
\

\qquad Incremental unknowns were introduced by Temam (1990) in the dynamical system
theory where the
objective is to study the long-term dynamic behavior of the solutions of
dissipative evolutionary equations when finite-difference approximations
in a variational framework of such equations are used and when
several levels of discretization are considered.
On the whole, a general
elliptic linear differential equation has to be solved at each time step;
incremental unknowns are efficient in the numerical solution
of such equations but no rigorous theoretical justification
was available.
Hereafter, using graph techniques, we prove that the condition
number of the incremental unknowns matrix associated to the Laplace
operator is $ O(1/h_{0}^{2}) O((\log h)^{2}) $ where $ h_{0} $ is
the mesh size of the coarsest grid and where $ h $ is the mesh size of the
finest grid; furthermore, if block diagonal scaling is used then the
condition number of the preconditioned incremental unknowns matrix
associated to the Laplace operator comes out to be
$ O((\log h)^{2}) $. 

\

\qquad The results are obtained in essence by deriving
appropriate bounds on the generalized Rayleigh quotient 
\begin{equation*}
\ttfrac{(v,h^{2}(-\Delta_{h})v)}{(v,(S\mathcal{K}^{-1}S^{T})^{-1}v)}, 
\end{equation*}
as stated by Elman and Zhang (1995); here $ \Delta_{h} $ is the
finite-difference Laplace operator and 
$ \mathcal{K} $ is either the identity or a block diagonal part of
the incremental unknowns matrix $ S^{T} (-\Delta_{h}) S $, where $ S $
stands for the transfer matrix from the incremental unknowns $ \hat{x} $
to the nodal unknowns $ x $, i.e., $ x = S \hat{x} $;
the coefficients of this incremental unknowns matrix  
are computed (bounded) using the finite-difference variational approach.
It follows promptly from linear algebra 
lemmas and from the
Ger\v{s}gorin theorem that the
maximum eigenvalue of the incremental unknowns matrix $ S^{T}
(-\Delta_{h}) S $ is
bounded by an absolute constant independent of $ i $ (the number of
levels); then we infer the result stated before. 



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