  In this paper, we present two variants of the Additive Schwarz Method for a
Crouzeix-Raviart finite volume element (CRFVE) discretization of second order
elliptic problems with discontinuous coefficients where the discontinuities are
only across subdomain boundaries. One preconditioner is symmetric while the
other is nonsymmetric. The proposed methods are almost optimal, in the sense
that the residual error estimates for the GMRES iteration in the both cases
depend only polylogarithmically on the mesh parameters.
