  We develop a framework for constructing mixed multiscale finite volume
methods for elliptic equations with multiple scales arising from flows in
porous media. Some of the methods developed using the framework are already
known \cite{jennylt03}; others are new. New insight is gained for the known
methods and extra flexibility is provided by the new methods. We give as an
example a mixed MsFV on uniform mesh in 2-D. This method uses novel multiscale
velocity basis functions that are suited for using global information, which is
often needed to improve the accuracy of the multiscale simulations in the case
of continuum scales with strong non-local features. The method efficiently
captures the small effects on a coarse grid. We analyze the new mixed MsFV and
apply it to solve two-phase flow equations in heterogeneous porous media.
Numerical examples demonstrate the accuracy and efficiency of the proposed
method for modeling the flows in porous media with non-separable and separable
scales.
