In this talk, I will focus on a new family of fourth-order and sixth-order compact difference schemes for the 2D/3D linear elliptic equations. By using finite volume method for derivation, the highest-order compact schemes based on different types of dual partitions are obtained. Moreover, a new fourth-order compact scheme is gained and numerical experiments show the new scheme is much better than other known fourth-order schemes. The outlines for the nonlinear elliptic problems are also given. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference schemes.