WWRC 92-12
Well-Conditioned Iterative Schemes for Mixed Finite-Element Models of Porous-Media Flows
Abstract
Mixed finite-element methods are attractive for modeling flows in porous media
since they can yield pressures and velocities having comparable accuracy. In solving the resulting
discrete equations, however, poor matrix conditioning can arise both from spatial heterogeneity in
the medium and from the fine grids needed to resolve that heterogeneity. This paper presents two
iterative schemes that overcome these sources of poor conditioning. The first scheme overcomes poor
conditioning resulting from the use of fine grids. The idea behind the scheme is to use spectral
information about the matrix associated with the discrete version of Darcy's law to precondition the
velocity equations, employing a multigrid method to solve mass-balance equations for pressure or
head. This scheme still exhibits slow convergence when the permeability or hydraulic conductivity is
highly variable in space. The second scheme, based on the first, uses mass lumping to precondition
the Darcy equations, thus requiring more work per iteration and minor modifications to the multigrid
algorithm. However, the scheme is insensitive to heterogeneities. The overall approach should also
be useful in such applications as electric field simulation and heat transfer modeling when the media
in question have spatially variable material properties.
Key words, mixed finite elements, iterative solution schemes, heterogeneous porous media
AMS(MOS) subject classification. 65
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