Award Date
May 2023
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
First Committee Member
Jichun Li
Second Committee Member
Monika Neda
Third Committee Member
Pengtao Sun
Fourth Committee Member
Hongtao Yang
Fifth Committee Member
Yi-Tung Chen
Number of Pages
142
Abstract
The Perfectly Matched Layer (PML) technique is an effective tool introduced by B´erenger [13] to reduce the unbounded wave propagation problem to a bounded domain problem. This dissertation focuses on two different PML models and their applications to wave propagation problems with Maxwell’s equation in complex media. We investigate these models using two popular numerical methods: the Finite Difference Method (FDM) in Chapters 2 and 3, and the Finite Element Method (FEM) in Chapters 4 and 5.In Chapter 2, we focus on analyzing the stability of a PML developed by B’ecache et al. [10] for simulating wave propagation in the Drude metamaterial. While this PML was originally shown to be stable through modal analysis in [11], we establish its stability using the energy method. Additionally, we develop and analyze an FDTD scheme and present numerical simulations that demonstrate the PML’s effectiveness in absorbing outgoing waves in the Drude medium. In Chapter 3, we propose and analyze a fourth-order finite difference scheme for solving Ziolkowski’s PML model [118]. While fourth-order schemes have been commonly used in solving Maxwell’s equations, few papers provide a rigorous convergence analysis. One of the key contributions of our work is to fill this gap by demonstrating the fourth-order pointwise convergence of our proposed scheme. We present numerical results to support our analysis and demonstrate the effectiveness of the Perfectly Matched Layers in absorbing outgoing waves for this PML model. In Chapter 4, our focus is on developing more efficient numerical methods for Ziolkowski’s PML model by reducing the number of unknowns. To achieve this goal, we reformulate the original model into an integro-differential form and propose two new finite element methods for solving this equivalent PML model. We provide stability and convergence analysis for both schemes and present numerical results to support our analysis and demonstrate the effectiveness of the wave absorption for this equivalent PML. In Chapter 5, we present a new variational form for simulating the propagation of surface plasmon polaritons on graphene sheets, where the graphene is treated as a thin sheet of current with effective conductivity. We propose a novel finite element method to solve this graphene model and prove discrete stability and error estimates for our proposed method. We also present numerical results to demonstrate the effectiveness of this graphene model in simulating the surface plasmon polaritons propagating on graphene sheets.
Keywords
Finite Difference Time Domain Method; Finite Element Time Domain Method; Graphene; Numerical Analysis; Perfectly Matched Layer
Disciplines
Applied Mathematics | Electromagnetics and Photonics | Engineering Physics | Other Physics | Physics
Degree Grantor
University of Nevada, Las Vegas
Language
English
Repository Citation
Zhu, Li, "Analysis and Application of Finite Element and High-Order Finite Difference Methods for Maxwell’s Equations in Complex Media" (2023). UNLV Theses, Dissertations, Professional Papers, and Capstones. 4808.
https://digitalscholarship.unlv.edu/thesesdissertations/4808
Rights
IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/
Included in
Applied Mathematics Commons, Electromagnetics and Photonics Commons, Engineering Physics Commons, Other Physics Commons