| In2000, physicists successfully constructed an artificial metamaterial, whichhas both negative permittivity and negative permeability. Due to its exotic proper-ties, the numerical methods of metamaterial have attracted more and more atten-tion since then. This paper is mainly focused on the development and applicationof finite element methods for modeling the backward wave propagation and elec-tromagnetic cloaks in metamaterials.In the first part of this thesis, we study the time-domain finite element methodof backward wave propagation in metamaterials. Since metamaterials are usuallydispersive media, we firstly develop a fully-discrete interior penalty discontinuousGalerkin method for solving the time-dependent Maxwell’s equations in dispersivemedia. The scheme is proved to be unconditionally stable and achieve optimalerror estimates in both L2norm and energy norm. Some numerical results sup-porting our analysis are presented. Then we study the superconvergence of thethree-dimensional time-dependent Maxwell’s equations in metamaterials. Wefind that the convergence rates obtained at element centers are one-order higherthan that predicted by the theoretical analysis. We also present some numericalresults justifying our theoretical analysis. And then, we simplify the time-domainmetamaterial equations to two kinds of simple model equations. We study the ex-istence and uniqueness of the new modeling problems, and propose some efcientfully-discrete finite element methods to solve them. Stability and optimal erroranalysis are obtained. Then the scheme is extended to model wave propagationin heterogeneous media composed of metamaterials and free space, and extensivenumerical results demonstrate the efectiveness of our algorithm for modeling theexotic backward wave propagation phenomenon in metamaterials. At last we dis-cuss the well-posedness of the plasma-Lorentz model in metamaterial, and developtwo fully-discrete finite element methods for solving it. Detailed stability and er-ror analysis are carried out, and3-D numerical results justifying our theoreticalanalysis are presented.In the second part of this thesis, we study electromagnetic cloaks with thefrequency domain and time domain finite element methods. First combining withthe appropriate frequency domain perfectly matched layer method, we constructthe finite element algorithm for modeling of frequency domain electromagnetic cloaks. Numerical results demonstrate that our algorithm is quite efective, andour results are consistent with the results obtained by the commercial softwareComsol. Then we present the time domain model equations and the correspondingfinite element methods of electromagnetic cloaks with arbitrary shapes. The well-posedness for the governing equations and the stability analysis for our proposedscheme are obtained. Finally we obtain the time-domain electromagnetic cloakphenomena with diferent shapes (circular, elliptical, triangular) by using timedomain perfectly matched layer method. The numerical results demonstrate thatthe model equations and the algorithms are correct.In the third part of this thesis, we construct an efcient a posteriori errorestimator by the finite element post-processing technique, and apply it to thefrequency domain electromagnetic cloak problems. We first propose a posteriorierror estimates for the second order elliptic boundary value equations based ona new recovery procedure called the explicit polynomial recovery(EPR). We thenobtain the superconvergence results of EPR on special mesh, and the numericalexperiments show the efectiveness of the EPR indicator. After that, we study aposteriori error estimator by the post-processing recovery technique of edge finiteelement methods for time-harmonic Maxwell equations. Finally, we develop acorresponding adaptive edge finite element method with the adaptive CVDT gridgeneration technique, and the efciency and robustness of the proposed methodare demonstrated by extensive numerical experiments for electromagnetic cloakingproblems. |
PDF Full Text Request