Geospace is a vast complex system. A portable quantitative model of the geospace electromagnetic structure is the foundation of many researches in the area of space plasma physics. Work described in this dissertation uses both the analytical and numerical approaches to bring the existing models several steps closer to the ultimate model. An analytical model is developed to simulate the magnetic effect of the magnetopause current by solving the Laplace equation in a complex geometry. The procedure involves the minimization of an integral quantity to determine the coefficients of a harmonic expansion for the magnetic scalar potential of the magnetopause field. This procedure can shield any kind of interior magnetic field, or open the closed magnetosphere up with an arbitrary normal component distribution over the magnetopause. The Toffoletto and Hill 1993 open magnetosphere model is improved by adding a new physical module of the shielded ring current field, and modifying its additional tail field to eliminate the shortcomings that are evident. The new improved version has a better representation of the magnetic field in the inner magnetosphere. A numerical model for the electric potential on closed field lines is developed by extending the polar-cap potential into the low-latitude ionosphere using a finite element method. This significantly enhances the model's capability. By mapping the electric potential into the magnetosphere, the associated magnetospheric plasma convection can be examined to assess the reasonability of the normal component distribution on the magnetopause.