Discontinuous Galerkin Finite Element Method (DG-FEM) has been further developed in this dissertation. We give a complete proof of stability and error estimate for the DG-FEM combined with Runge Kutta which is commonly used in different fields. The proved error estimate matches those numerical results seen in technical papers. Numerical simulations of metamaterials play a very important role in the design of invisibility cloak, and sub-wavelength imaging. We propose a leap-frog discontinuous Galerkin Finite Element Method to solve the time-dependent Maxwell\u27s equations in metamaterials. The stability and error estimate are proved for this scheme. The proposed algorithm is implemented and numerical results supporting the analysis are provided. The wave propagation simulation in the double negative index metamaterials supplemented with perfectly matched layer (PML) boundary is given with one discontinuous Galerkin time difference method (DGTD), of which the stability and error estimate are proved as well in this dissertation. To illustrate the effectiveness of this DGTD, we present some numerical result tables which show the consistent convergence rate and the simulation of PML in metamaterials is tested in this dissertation as well. Also the wave propagation simulation in metamaterals by this DGTD scheme is consistent with those seen in other papers. Several techniques have appeared for solving the time-dependent Maxwell\u27s equations with periodically varying coefficients. For the first time, I apply the discontinuous Galerkin (DG) method to this homogenization problem in dispersive media. For simplicity, my focus is on obtaining a solution in two-dimensions (2D) using 2D corrector equations. my numerical results show the DG method to be both convergent and efficient. Furthermore, the solution is consistent with previous treatments and theoretical expectations
