In view of the calculation of elliptic partial differential equations, compact difference schemes with fourth and sixth order precision are firstly established, followed by an adoption of the Richardson extrapolation method to obtain the extrapolation schemes with sixth and eighth order precision on this basis. The established difference scheme can be verified by two Poisson equation examples. The numerical example results show that the adoption of the Richardson extrapolation method based on the compact difference scheme is able to obtain effective and robust high-precision numerical solutions.