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\title{Coarse scale models for diffusion in heterogeneous media
and for multi-level iterative solvers}

\abstract{
For many applications in environmental modelling,
the coefficients of the mathematical model
incorporate strong variations on several length scales.
An accurate numerical treatment of such problems utilizes
homogenization and upscaling techniques to define an approximate
coarse scale model, capturing the influence of the unresolved
fine scales, and thus preserving the gross features of the flow.\\
Multi-level iterative solvers for problems incorporating
different length scales rely either on these homogenization
or upscaling techniques, or use energy-dependent averaging
procedures to define the coarse grid operators needed.
Conversely these operator- or matrix-dependent averaging
schemes define discrete homogenization procedures and
can be used to compute approximations to the exact
homogenized diffusion coefficient of heterogeneous media.\\
We give a discrete homogenization method arising from
these matrix-dependent averaging methods and compare it
with other schemes such as renormalization.
We discuss advantages and disadvantages of different
approaches both for homogenization and for the definition
of coarse grid operators in multi-level iterative solvers
and clarify the connections between both worlds.
We report on numerical experiments for the
resulting homogenization procedures and
the corresponding multilevel methods.
Specifically we report on simulations of
fluid flow in reservoirs working with homogenized
diffusion coefficients.
}

\author{Stephan Knapek\\
Abteilung f\"ur Wissenschaftliches Rechnen und Numerische Simulation\\
Institut f\"ur Angewandte Mathematik\\
Universit\"at Bonn\\
Wegelerstr. 6, 53115 Bonn, Germany\\
e-mail: knapek@iam.uni-bonn.de
}

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