  We develop a cut finite element method (CutFEM) for the convection problem in
a so called fractured domain which is a union of manifolds of different
dimensions such that a $d$ dimensional component always resides on the boundary
of a $d+1$ dimensional component. This type of domain can for instance be used
to model porous media with embedded fractures that may intersect. The
convection problem can be formulated in a compact form suitable for analysis
using natural abstract directional derivative and divergence operators. The cut
finite element method is based on using a fixed background mesh that covers the
domain and the manifolds are allowed to cut through a fixed background mesh in
an arbitrary way. We consider a simple method based on continuous piecewise
linear elements together with weak enforcement of the coupling conditions and
stabilization. We prove a priori error estimates and present illustrating
numerical examples.
