  In this paper, we present a multigrid $V$-cycle preconditioner for the linear
system arising from piecewise linear nonconforming Crouzeix-Raviart
discretization of second order elliptic problems with jump coefficients. The
preconditioner uses standard conforming subspaces as coarse spaces. We showed
that the convergence rate of the multigrid $V$-cycle algorithm will deteriorate
rapidly due to large jumps in coefficient. However, the preconditioned system
has only a fixed number of small eigenvalues, which are deteriorated due to the
large jump in coefficient, and the effective condition number is bounded
logarithmically with respect to the mesh size. As a result, the multigrid
$V$-cycle preconditioned conjugate gradient algorithm converges nearly
uniformly. Numerical tests show both robustness with respect to jumps in the
coefficient and the mesh size.
