Finite Element Methods For Maxwell's Equations With Variable Coefficients In Metamaterials And Its Application In Transformation Optics

In the past ten years,the development of transformation optics technology provides a powerful tool for the design of functional physical devices.With the development of computer technology and numerical calculation method,numerical method provides an important research method for solving more and more complex practical problems in electromagnetic field engineering.In this thesis,the finite element theory of Maxwell's equations with variable coefficients and its application in several transformation optics devices are studied.Firstly,we establish time domain the mathematical models of wave propagation in several transformation optics devices,including electromagnetic wave concentrator,rotator and splitter,and design the corresponding time domain finite element method to simulate the wave propagation in these transformation optics devices.We implement the proposed algorithm,and the numerical results verify the effectiveness of our mathematical model and finite element method.Then,we design a new coordinate transformation and deduce the anisotropic material parameters of the quadrilateral thermal invisibility cloak according to the principle of transformation thermodynamics.Because the material parameters is anisotropic,it is difficult to make use of the material.The anisotropic material is simplified by the effective medium theory.The thermal stealth device is designed by using the layered structure of metamaterial which is only composed of two kinds of isotropic materials.Numerical simulation results show that the designed layered structure stealth cloak has good long-time thermal stealth performance.Secondly,we study the convergence of the adaptive edge finite element method for the arbitrary order Nedelec element of the time harmonic Maxwell's equations with variable coefficients.It is proved that the energy error and the error estimator is strictly monotonically decreasing when the initial mesh is small enough.We give the variational problem of the time harmonic Maxwell's equations with variable coefficients and the posteriori error estimator of the residual type.We establish the quasi orthogonality,the global upper bound of the error and the compressibility of the error estimator,and prove the convergence results.Numerical results verify the correctness of our theoretical analysis,that is,the effectiveness of the error estimator.At the same time,we study the adaptive staggered discontinuous Galerkin(SDG)method for time harmonic Maxwell's equations with variable coefficients.Two types of posterior error estimators are given,one is recovery estimator,the other is residual estimator.We propose the curl recovery method of the staggered discontinuous Galerkin method(SDGM),and construct an asymptotically accurate recovery type error estimator based on the superconvergence result of the latter understanding.The reliability and effectiveness of the residual type posterior error estimator are proved.The effectiveness of the indicator and the robustness of ASDGM are proved by numerical experiments.Finally,we study a finite element method for solving time domain Maxwell's equations in nonlinear Kerr media.The fully discrete scheme is proved to be conditionally stable and optimally convergent in space and time.Numerical results verify our theoretical results and the propagation of solitons in Kerr medium.We also propose a simple and efficient radial basis function meshless method for solving time domain Maxwell's equations.The main idea is to solve a series of interesting mathematical models derived from transform optics by using the super optimal interpolation property of radial basis function and the explicit property of leap-frog scheme.The stability of the method is proved,and the effectiveness of the meshless method is verified by numerical experiments.