Fast Algorithms Of The Finite-element Method, Boundary Integral Method And Their Hybrid Methods For Complicated Electromagnetic Problems

Computational electromagnetics have been seen a considerable surge in research on efficient accurate numerical methods, which have been stimulated by the demands for electromagnetic targets simulations. The research of this dissertation is focus on the finite element method (FEM), the boundary integral method (BI) and the finite element-boundary integral method (FE-BI), specially majored on their applications to three-dimensional electromagnetic scattering.At the beginning of the dissertation, the curvilinear elements and the high order hierarchical vector basis functions are studied to improve the accuracy of the numerical results. Two preconditioning techniques have been developed to accelerate the convergence of Krylov iterative method, which are used to solve the large system of linear equations resulted from high order FEM. One is derived based on the shifted Laplace operator and the diagonally perturbed incomplete Cholesky factorization is used to approximate the inverse of this operator. Another is a p-version multigrid method combined with flexible GMRES solver.The application of high order FEM combined with the perfectly matched layers (PML) to the analysis of electromagnetic scattering is studied. A locally conformal PML is extended to the high order FEM combined with curvilinear elements and high order basis functions, both of which can bring high accuracy and efficiency. This combination is able to handle challenging geometrics with arbitrary curvature with few unknown numbers, especially those with curvature discontinuities.The set of the numerical truncation boundary in FEM usually sacrifice the accuracy for its approximation of open domain. Therefore, the boundary integral methods have been adopted extensively. However, the large dense complex linear system matrices that arise from integral equations usually have bad conditions which lead to inefficient iteration convergence. Two preconditioning techniques have been studied to accelerate the iteration convergence. One is the spectral multiresolution (MR) preconditioner, which uses a spectral preconditioner in a two-step manner to update an existing MR preconditioner so as to recover the global information. Another is an efficient Calderon multiplicative preconditioner (CMP) based on the curvilinear elements, which exploits the self-regularizing property of the electric filed integral equation (EFIE), i.e., the fact that the square of the EFIE operator does not have eigenvalues accumulating at zero or infinity, and thus can give rise to a well-conditioned EFIE MoM system independent of the discretization density.FE-BI is investigated in the last part of the dissertation. First is the application of curl-conforming and div-conforming high order hierarchical vector basis functions to the FE-BI with the improvement of the numerical accuracy. For the difference of mesh density requirements between FEM and BI and the hierarchical characteristic of the basis functions, a mixed order FE-BI has been developed, i.e., the high order basis functions are used in the FEM part and the low order basis functions in the BI part. The mixed order FE-BI can effectively reduce the number of the unknowns but keep the high accuracy as the high order FE-BI. The second is application of FE-BI for the analysis of the scattering from the complex media targets, such as the anisotropic media, bi-isotropic media, and bi-anisotropic media. Besides that, FE-BI is also applied to analyze the electromagnetic scattering from arbitrary objects in the half-space environment. Combined with the real image method to approximate the half-space Green's function, the half-space FE-BI can be extended for the electrically large problems by the fast multipole method (FMM).An efficient preconditioner is also constructed by the finite-element part and the near-field value terms of the boundary integral, both of which can be easily obtained from the matrix of FE-BI with no additional computing cost and memory requirement. During the iterative solution of the preconditioned FE-BI, an inner solver is needed for the solution of the preconditioning matrix equation. For the limited RAM, some preconditioning techniques, such as SSOR, ILU (0), ILU(1) and ILUT are used to accelerate the convergence of iterative method. Convergence properties are compared and analyzed. For the adequate RAM, an H-matrix-based fast direct solver is introduced for the inner solution. The memory requirement and computational complexity are analyzed in this dissertation. Compared with the other direct solver for the inverse, it has advantages in the memory requirement and computational complexity.