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{\bf Applications of Domain Decomposition Methods to Flows in Porous
Media} \\
~~\\
R.E.~Ewing \\
Institute for Scientific Computation, Texas A\&M University, College
Station, Texas
~~\\
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We consider various models of flows in porous media including single
phase
flows, two phase saturated/unsaturated flows and multi-phase
multicomponent
flows. These models play an essential role in the ground-water
bioremediation, the transport and decay of radionuclides in aquifers,
oil
production etc.  These models consist of coupled nonlinear systems of
elliptic, parabolic, and hyperbolic equations in complex domains with
highly varying and discontinuous coefficients.

The numerical methods for solving these equations have to preserve the
main
properties of the physical problem: monotonicity, local mass
conservation,
and global energy balance. Such methods are the mixed finite element
method
and the finite volume method. We give a short overview of the
approximation
strategies of the flow problems by these two discretization
techniques,
which lead to very large systems of algebraic equations.

It is widely accepted that the domain decomposition method is the only

practical way to efficiently solve these large systems.
This class of  method is based on parallel
processing of the subdomain problems and utilizes in full the
capabilities and the resources of the existing multiprocessor computer

architectures.

In this talk we shall present some recent results in the domain
decomposition
methods obtained by the Numerical Analysis Group at Texas A\&M
University.
Namely, we shall discuss some domain decomposition algorithms for
mixed
finite element approximations of the pressure equation for 3-D
tetrahedral
partitions, including the case of non-matching grids along the
interface of
the subdomains. Next, we shall present some mortar finite element
approximations
and domain decomposition algorithms with optimal multilevel
preconditioning of the system obtained on the mortar space.
Finally, we shall discuss some domain decomposition algorithms
using mortar technique for finite volume discretizations
on non-matching grids.

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