| Summary: | We present a variational multiscale mixed
finite element method for the solution of Darcy flow
in porous media, in which both the permeability field
and the source term display a multiscale character.
The formulation is based on a multiscale split of the
solution into coarse and subgrid scales. This decomposition
is invoked in a variational setting that leads
to a rigorous definition of a (global) coarse problem
and a set of (local) subgrid problems. One of the key
issues for the success of the method is the proper
definition of the boundary conditions for the localization
of the subgrid problems. We identify a weak
compatibility condition that allows for subgrid communication
across element interfaces, a feature that
turns out to be essential for obtaining high-quality
solutions. We also remove the singularities due to
concentrated sources from the coarse-scale problem
by introducing additional multiscale basis functions,
based on a decomposition of fine-scale source terms
into coarse and deviatoric components. The method
is locally conservative and employs a low-order approximation
of pressure and velocity at both scales.
We illustrate the performance of the method on several
synthetic cases and conclude that the method
is able to capture the global and local flow patterns
accurately.
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