In this article, we discuss a V-cycle multigrid algorithm based on the cell-centered finite difference scheme for solving a second-order elliptic problem with discontinuous coefficients. Due to the poor approximation property of piecewise constant spaces and the non-variational nature of our scheme, one step of symmetric linear smoothing in our V-cycle multigrid scheme fails to be a contraction. However, we have very promising computational results obtained by taking slightly different pre-smoothing and post-smoothing strategies. Our resulting multigrid scheme is still symmetric in the elliptic operator related discrete inner products and is a good contraction operator according to our computational experiments. Again, because of the simple structure of the piecewise constant spaces, prolongation and restriction are trivial; we save significant computation time.