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\title{Cascadic Conjugate
Gradient Methods \\ for Elliptic Partial Differential Equations.\\ Algorithm and
Numerical Results}  
\author{Peter Deuflhard
\thanks{Konrad--Zuse--Zentrum Berlin, Heilbronner Str. 10, D--10711
Berlin--Wilmersdorf, Germany}} 
\date{}
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\begin{abstract}
Cascadic conjugate gradient methods for the
numerical solution of elliptic partial differential equations consist of
Galerkin finite element methods as outer iteration and (possibly 
preconditioned) conjugate gradient methods as inner iteration. Both
iterations are known to minimize the energy norm of the arising iteration
errors. The present paper derives a unified frame to study the relative merits
of different preconditioners versus the case of no preconditioning.
Surprisingly, in the performed numerical experiments the cascadic
conjugate gradient method {\it without any preconditioning} (to be called CCG
method) turned out to be not only simplest but also fastest. It appears that
the cascade principle in itself already realizes some kind of preconditioning.
A theoretical explanation of the observed iteration pattern will be given
elsewhere. \end{abstract}

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