Preconditioners For The System Of Linear Equations Arising From The Time-Harmonic Eddy Current Problems

The time-harmonic eddy current model is often used to simulate the electro-magnetic phenomena with respect to alternating currents at low frequencies. In order to find the numerical solution of such problems, we can utilize the finite element method to discrete the original problem, then we can get the approximate solution via solving a large scale system of linear equations. As direct methods scale poorly with problem size in terms of operation counts and memory require-ments, especially on problems arising from the discretization of PDEs in three space dimensions, iterative methods are the only option available. Because of the indefiniteness and poor spectral properties of the problem, the convergence rate of iterative methods, especially for the Krylov subspace methods, is usually very slow, and sometimes these methods even fail to converge. In addition, the sys-tem is complex, which makes it more difficult to solve. In order to improve the performance and reliability of the iterative methods, preconditioning is necessary. It is widely recognized that preconditioning is the most critical ingredient in the development of efficient solvers for challenging problems in scientific computation.In this thesis, we consider the preconditioned iterative methods for the large scale complex linear systems arising from the time-harmonic eddy current problem. Our work is as follows:(1) We establish a class of positive definite and positive semi-definite splitting methods for the generalized saddle-point problem, and we prove that the methods converge under certain conditions. With special choices of the splitting matrices, a sequence of positive definite and positive semi-definite splitting methods can be generated.(2) The preconditioners base on positive defi-nite and positive semi-definite splitting are established for the time-harmonic eddy current problem, and the eigenvalue distribution of the preconditioned matrix are discussed in details.(3) The preconditioner based on the relaxed dimensional factorization is given. It can see that the preconditioned matrix may have more clustered spectrum.(4) The implementation of the proposed preconditioners are discussed in details, and the numerical tests are carried out to illustrate the effi-ciency and robustness of the proposed preconditioners.