The author introduces and analyzes a new modified V-cycle multigrid algorithm for cell-centered finite difference (CCFD) approximation of the model problem \(-\nabla p\nabla \widetilde{u}=f\) in \(\Omega,\) \(\widetilde{u}=0\) on \(\partial \Omega\) where \(\Omega\) is the unit square. In this algorithm instead of the natural injection operator for prolongation a new weighted prolongation operator is used. It is proved that its energy norm is bounded by 1 in the case of constant function \(p\) and by \(1+Ch\) in the nonconstant case. The author shows that the V-cycle with a fixed number of smoothing steps is convergent in the case of constant function \(p\) and is a fairly good preconditioned in the nonconstant case. The numerical experiments for both cases demonstrate that the algorithm converges much faster than the conventional schemes.