Iterative Methods And Preconditioning Techniques For Solving Electromagnetic Field Problems

With the development of the computer science and techniques, effective numerical methods become more and more important. in many engineering fields. The finite element method and the method of moment are two important and widely used numerical methods to analyze the electromagnetic problems. However, the application of these two methods usually yield large complex and symmetric linear systems, and the time for solving this kind of linear system is the main part of which for whole numerical simulation process. Therefore the researches on the algorithms of high efficiency and low memory needs are necessary. In this dissertation, the work is mainly based on the effective algorithm for solving the large complex symmetric and highly indefinite linear systems arising from the FEM analyzing electromagnetic field Helmholtz equations, and the MOM analyzing EFIE equations. And the work is devoted to iterative and preconditioning algorithms.We first apply the symmetrical iterative methods such as COCG, SQMR, and the simplified linear BCG(LBCG) for solving complex symmetric dense and sparse linear systems, and detailedly investigate the convergence behaviour by compared with several conventional iterative methods. We then proposed a co-iterative method which can avoid the isotropic breakdowns and outperform the conventional iterative methods for solving several electromagnetic problems.We apply several conventional preconditioners to solve complex symmetric dense and sparse linear systems, and detailedly investigate their performance when combined with different iterative methods. For highly indefinite FEM linear systems, we propose a modified AINV(MAINV) preconditioner. It is designed by adding pivots compensation strategy to the basic AINV process, thus the preconditioner can be more stable and effective. Based on this algorithm, we employed a complex shifted Laplace operater scheme to reduce the computational cost and get a more robust preconditioner. We also propose a modified relaxed IC (RIC) algorithm and investigate its convergence beheavior when applied to different iterative methods. It uses the small elements to compensate the correspond diagonal pivots to get a more stable preconditioner. Conventional preconditioner is usually constructed based on the coefficient matrix A, in this dissertation we develop an novel preconditioner, which is based on the real part of matrix A, for solving specific electromagnetic problems. Compared with the conventional ones, this preconditioner is not only more efficient but also can reduce the memory needs.Besides, we investigate the preprocessing techniques, including scaling and reordering, which is very useful for the incomplete factorization and the factorized sparse approximate inverse algorithms. We use a scaling technique to decrease the condition number of the coefficient matrix. We also use the reordering techniques to achieve the efficient factorization and low memory needs. And the preconditioner is more effective when compared with which resulting from the factorization without preprocessing.