This Paper deals with the initial value problems of the Modified KdV(MKdV) equation. First, the analytic solutions of solitary wave and interaction of two solitons are obtained by the Bäcklund Transformation. Then, a numerical scheme of a kind of Petrov-Galerkin finite element methods is derived and used to solve the initial value problems of MKdV equation numerically. Finally, the comparison of numerical and analytic solutions is made and shows the excellent accurancy and stability of the numerical scheme used in this paper.