Preconditioning Method And Its Applications In Numerical Simulation For Electromagnetic

Computational mathematics is involved in many applications of science. It plays avital role in aeromechanics, life sciences, resources exploration and materials designs.By means of modern computers, physical models can be established according to math-ematical theories and will give rise to algebraic systems of linear equations. By solvingthe resulted linear systems, we can obtain useful results which are required in physicalmodel problems. Consuming time and computational work used to solve the linear sys-tems often occupy the whole process, which can be more than 80%. This is due to badspectral property of the coefficient matrix, which lead to the difficulty for solving the lin-ear systems. How to construct an accurate and fast solver of the linear systems tempsto be the bottleneck in solving the practical problems nowadays. So we here focus onstudying of preconditioning techniques which can accelerate the convergence rate of iter-ative methods, and propose a few kinds of target-oriented preconditioners for problems incomputational electromagnetics.M-matrix is a kind of positive definite matrices with many elegant properties and theiterative solvers of such linear systems are always charming for lots of researchers. Howto use these properties to construct a more effective preconditioners is very attractive.Here for a block tridiagonal M-matrix, we study the block ILU factorization method. Weuse some strategies like ILU(k) to keep the sparsity of the ILU factors, and we propose arestarted method to enhance the parallelity theoretically. This will be quite helpful to savematrix-constructing time. Several numerical tests on second order elliptic equations willprove its efficiency.For indefinite linear systems arising from the discretization of Maxwell equation byedge finite element method(FEM), we propose a positive definite preconditioner by useof the stiff matrix and the mass matrix. Such kind of positive definite preconditioner isvery convenient to construct, and it is also easy to solve since it is positive definite. Wenot only show the spectral properties of the preconditioned matrix theoretically, but alsoprovide some numerical tests to show the power of the proposed preconditioner.The vector wave equations are commonly used when we describe the scattering prob-lems. When we discretize it by edge FEM, a large, sparse, indefinite and complex linear system will be eventually got. When the ILU is performed on such kind of matrices, weutilize the perturbation technique to enhance its effectiveness. The modified ILU methodcombined with such perturbation technique will be used to accelerate the convergencerate of the iterative solution, which will be verified by numerical tests.Finite element method and method of moment(MoM) are two important methods incomputational electromagnetics. The hybrid FEM-MoM is more attractive since it takesboth advantages of FEM and MoM. For the linear systems arising from the discretizationof the Maxwell equation by the hybrid method, we propose several different precondi-tioners. For scattering problems of a dielectric body, the SOR preconditioner does notseem to be bad, and several approximations of the coefficient matrices are constructed asthe preconditioners, and they are more efficient and easier to be solved compared to SORpreconditioner. For antenna problems, we study the block ILU and a two level precondi-tioning method, and the appropriate form of the matrix equation for iterative solution isgiven. Finally we give the numerical tests to show the efficiency of the preconditioners.Lots of authors pay much attention to the preconditioned iterative solutions of theHelmholtz equation since it is widely used in computational electromagnetics, acous-tic wave propagation and physical geography problems. The operator preconditioningmethod is different from those which construct preconditioners from the coefficient ma-trix. Instead,the operator preconditioning technique constructs the preconditioner by ap-proximating the original operator by a modified operator, which will be of more physicalmeaning in this way. We here use such preconditioning operator, and utilize the AMGmethod to accelerate the convergence rate of the iterative solution of the Helmholtz equa-tions. From the numerical tests we can find that, by means of such kind of operators basedon AMG preconditioning techniques, the iteration account needed to solve the linear sys-tem nearly does not seem to grow when the dimension of the coefficient matrix becomeslarger.