  In this paper we introduce an additive Schwarz method for a Crouzeix-Raviart
Finite Volume Element (CRFVE) discretization of a second order elliptic problem
with discontinuous coefficients, where the discontinuities are both inside the
subdomains and across and along the subdomain boundaries. We show that,
depending on the distribution of the coefficient in the model problem, the
parameters describing the GMRES convergence rate of the proposed method depend
linearly or quadratically on the mesh parameters $H/h$. Also, under certain
restrictions on the distribution of the coefficient, the convergence of the
GMRES method is independent of jumps in the coefficient.
