Adaptive Finite Element Method And Application In Frequency Domain Electromagnetic Problems

The adaptive finite element method is an efficient numerical method for solving partial differential equations in modern computational science and engineering,and has gradually become an important research field,the a posterior error estimator provides important feedback for the design of the adaptive algorithm.Recovery type a posterior error estimator is usually constructed based on superconvergence recovery technique.It is favored by many scholars due to its simple structure and easy programming implementation.In this article,based on recovery type a posterior error estimator,we study an adaptive finite element method for frequency domain electromagnetic problems.Next,we extend the adaptive finite element method to the Helmholtz equation and the Allen-Cahn equation,the main content of this article mainly includes the following four parts:The first part:we study the adaptive finite edge element method for the time-harmonic Maxwell's equations in metamaterials.A postwriori error estimators based on the recovery type and residual type are proposed,respectively.Based on our a posteriori error estimators,the adaptive finite edge element method is designed and applied to simulate the backward wave propagation,electromagnetic splitter,rotator,concentrator.Numerical examples are presented to illustrate the reliability and efficiency of the proposed a posteriori error estimations for the adaptive method.The second part:based on the transforming optical theory as well as the idea of complementary medium,we propose a reliable and effective hierarchical design method for the information-open cloaking device.First,we consider the information-open cloaking device based on linear and nonlinear coordinate transformation.Next,we separately adopt equidistant layering and hierarchical layering design for these two devices to simplify the material structure.Finally,through a large number of numerical simulations,we conclude that information-open cloaking device based on nonlinear coordinate transformation,the hierarchical layering design method can be used to obtain the information-open cloking device with good cloaking performance in a small number of layers.The third part:because B is the basic quantity concerned in physics,and its position is equivalent to electric field E.At present,there are few mathematical theories and numerical algorithms about magnetic flux density B.Therefore,this article establishes the following time-harmonic magnetic flux density model equation:Furthermore,we derive the model equation coupling the time-harmonic magnetic flux density equation and the perfectly matched layer equation.Based on the Hodge decomposition method,they are converted into two scalar elliptic boundary value problems,and the corresponding error analysis are given.Finally,numerical examples are used to verify our theoretical analysis and solve the approximate solutions of the two fundamental fields B and E in physics.The fourth part:we extend the adaptive finite element algorithm to the acoustic wave problem as well as the Allen-Cahn equation in the phase field.On the one hand,we consider the acoustic propagation problem in linear and nonlinear acoustic media.Based on the residual and recovery type a posteriori error estimators,the effectiveness of the adaptive algorithm is verified by a large number of numerical simulations for the Helmholtz equation in different media.On the other hand,we propose an adaptive operator splitting method for the Allen-Cahn equation in the phase field.Firstly,the Allen-Cahn equation is split into linear equation and nonlinear equation.For linear equation,we use the CrankNicolson format to discretize it and solve it by finite element method.For nonlinear equation,we can solve it accurately,reduce the amount of calculation and improve the efficiency of the algorithm.Based on the superconvergent cluster recovery technology and the relative error of the two meshes before and after,the space and time direction error estimators of the Allen-Cahn equation are constructed separately.According to the proposed a posteriori error estimators,we design an adaptive algorithm for the Allen-Cahn equation.Finally,numerical examples verify the effectiveness of the adaptive method.