The Study Of Discontinuous Galerkin Method For Maxwell’s Equations

This thesis consists of two parts. In the first part, we introduce the discontin-uous Galerkin method for Maxwell’s equations in both double negative media and Cole-Cole dispersive media. In the second part, we focus on both the superconver-gence analysis of Averaging Discontinuous Galerkin method (ADG) for two point boundary value problem and convergence analysis of ADG method for Maxwell’s equations in free space. There are six chapters which are described as follows.In Chapter 1, the background of our research and the main results of the re-search are introduced.In Chapter 2, we turn to discuss the lossy Drude model meta-materials. Actu-ally, it is depicted mathematically by a system of integro-differential equations and solve it by an implicit DG method. Both the theoretical analysis and the numerical examples show that this scheme is unconditionally stable and has a convergent rate of O(τ2+h1+1/2)in L2-norm.In Chapter 3, we mainly discuss three kinds of important meta-materials i.e., the lossy Drude model, the Plasma-Lorentz model and the Lorentz model. First-ly, the corresponding mathematical model is proposed to generalize some physical models which described the electromagnetic wave propagation in each of them. Meanwhile, the similarly of those models is mentioned. Then the CG-DG method (Continuous Galerkin method in time and discontinuous Galerkin method in space) is introduced to solve those models. Unconditionally stability and a convergent rate of O(τr+1+hk+1/2) in L2-norm is strictly proven. Finally, numerical results in both 2D and 3D are given to validate the theoretical results.In Chapter 4, a more complicated dispersive media, i.e., the Cole-Cole dis-persive media is studied. It is hard to solve this model due to the existence of the fractional derivative term. For this model, an implicit DG scheme is introduced to solve it. Numerical results indicate that this scheme is unconditional stable and has a convergent rate of O(τ2-α+hk+1/2).In Chapter 5, the superconvergence of ADG method for two point boundary value problem is studied. Firstly, based on the Local discontinuous Galerkin method (LDG), the ADG method is introduced and used to solve the two point boundary value problem. The convergent rate of O(hk+1) for k-th order ADG method with k even is proved. Moreover, the superconvergence rate of O(h2k+2) for the numerical flux is proved. To our knowledge, this is the highest superconvergence order in the discontinuous Galerkin method up to now. Finally, numerical results are provided to validate the theoretical results.In Chapter 6, an implicit ADG method for time-domain Maxwell’s equations in free space is studied. The convergent rate of O(hk+1) when k-th order ADG method with k even and of O(hk) when k-th order ADG method with k odd are proved. Finally, numerical results are provided to validate the theoretical results.