Electromagnetic Problems Fast Solution Of Finite Element Methods

In many practical engineering areas, the application of numerical simulation methods can not only decrease the costs of productions, but also guarantee the reliability of productions. The superiority of an excellent numerical simulation method depends not only on the accuracy of simulating practical physical problems, but also on the efficiency of solving physical problems. In other words, both the CPU time and the memory requirement should be as few as possible. Based on the solution of boundary value problems for the vector Helmholtz equations, this thesis mainly focuses on the efficient analysis for the electromagnetic problems with the vector finite element method.At the beginning of the thesis, the theory of the finite element method is presented and two preconditioning methods for the vector finite element method are discussed. At the same time, a new type of preconditioning technology is introduced as the auxiliary space preconditioning. A Krylov method is preconditioned by a technique that approximately solves equivalent problems in two auxiliary spaces:a space of scalar functions and a space of piecewise linear,"nodal", vector functions. In this thesis, a two-level spectral preconditioning technology based on the auxiliary space preconditioning is proposed. Numerical results are presented to demonstrate the efficiency of the proposed method.Full-wave solutions of Maxwell’s equations break down at low frequencies. Firstly, a rigorous method is introduced to solve the low-frequency breakdown problem. The additional computation will be brought in, although the accuracy can be guaranteed. Therefore, a fast full-wave finite-element-based solution is developed to eliminate the low frequency breakdown problem in this thesis. It is applicable to general3-D problems involving ideal conductors as well as nonideal conductors immersed in inhomogeneous, lossless, and lossy materials. The proposed method retains the rigor of a theoretically rigorous full-wave solution and significantly speeds up the low-frequency computation.