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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