A sixth-order compact difference scheme and multigrid V-cycle algorithm are employed to solve the two-dimensional Poisson equation with Dirichlet boundary conditions. This scheme, along with several different relaxation operators, is compared with the fourth-order formula to show the dramatic improvement in both computed accuracy and convergence rate without distinctly increasing computational cost. Further more, Zebra Line Gauss-Seidel (ZLGS) relaxation is a more efficient smoother than other relaxation operators for no matter fourth-order or sixth-orer formula with multigrid method.