A high order compact finite difference scheme is employed in conjunction with the multigrid algorithm to solve the convection-diffusion equations with variable coefficients. Special treatments, such as restriction on the coarsest grid and residual injection scaling factor for accelerating the convergence for both small and large Reynolds number problems, are discussed. A heuristic residual analysis is given to obtain a cost-effective residual injection operator for the diffusion-dominated problems. The multigrid method requires neither a preconditioner nor added dissipation terms for high-Reynolds problems. Numerical experiments are employed to test the stability and efficiency of the proposed method.