In this paper, we mainly study the conforming and nonconforming finite element methods for the Klein-Gordon equation. Firstly, the bi-p-degree finite element over the rectangular meshes is applied to solve this equation, and the superclose property and the superconvergence are derived. Secondly, a kind of Crouzeix-Raviart type anisotropic nonconforming element is applied to the above equation. Based on the special properties of the element and the interpolation technique directly, the error estimates are obtained for semi-discrete and fully-discrete schemes. At last, we discuss the moving grid method for this equation with the above nonconforming finite element, and deduce the corresponding error estimates. |