Two New Finite Element Schemes and Their Analysis for Modeling of Wave Propagation in Graphene

© 2020 The Author(s) In this paper, we investigate a system of governing equations for modeling wave propagation in graphene. Compared to our previous work (Yang et al., 2020), here we re-investigate the governing equations by eliminating two auxiliary unknowns from the original model. A totally new stability for the model is established for the first time. Since the finite element scheme proposed in Yang et al. (2020) is only first order in time, here we propose two new schemes with second order convergence in time for the simplified modeling equations. Discrete stabilities inheriting exactly the same form as the continuous stability are proved for both schemes. Convergence error estimates are also established for both schemes. Numerical results are presented to justify our theoretical analysis

Arbitrary High Order Finite Difference Methods with Applications to Wave Propagation Modeled by Maxwell\u27s Equations

This dissertation investigates two different mathematical models based on the time-domain Maxwell\u27s equations: the Drude model for metamaterials and an equivalent Berenger\u27s perfectly matched layer (PML) model. We develop both an explicit high order finite difference scheme and a compact implicit scheme to solve both models. We develop a systematic technique to prove stability and error estimate for both schemes. Extensive numerical results supporting our analysis are presented. To our best knowledge, our convergence theory and stability results are novel and provide the first error estimate for the high-order finite difference methods for Maxwell\u27s equations

Electromagnetic Wave Scattering at Biperiodic Surfaces: Variational Formulation, Boundary Integral Equations and High Order Solvers

In this thesis we consider time-harmonic electromagnetic wave scattering at impenetrable biperiodic surfaces in a homogeneous medium. Besides their rigorous analysis in biperiodic Sobolev spaces, which aims at answering the questions about existence and uniqueness of solutions, we will derive a high order solver for its numerical approximation - a collocation method based on trigonometric polynomials