Book Description

Abstract: This dissertation is concerned with the development of numerical techniques for solving Maxwell equations in the time-domain. Two of the main challenges to obtain such solution are, first, how to construct explicit (that is, matrix-free) time-updating formulas without relinquishing the advantage of using irregular unstructured meshes in complex geometries, and second, how to best parallelize the algorithm to solve large-scale problems. The finite element time-domain (FETD) and the finite-difference time-Domain (FDTD) are presently the two most popular methods for solving Maxwell equations in the time-domain. FDTD employs a staggered-grid spatial discretization together with leap-frog style time update scheme to produce a method with many desirable properties such as: conservation of charge and energy, absence of spurious mode, and a simple easy-to-code algorithm. Nevertheless, FDTD (in its conventional form) relies on orthogonal grids, which is a disadvantage when modeling complex geometries. On the other hand, FETD is based upon unstructured grids and hence naturally tailored to handle complex geometries. However, in time-domain simulation (as opposed to frequency-domain simulations), FETD requires a matrix solver at every time step. Since the total number of time steps to produce the overall time-domain solution can be quite large, this requirement demands excessive computational resources. To overcome this problem, we develop a FETD algorithm with "FDTD-like" explicit characteristics. Usually, the system matrices generated after discretizing Maxwell equations in irregular grids are very large and sparse matrices, while their inverses are very large and dense matrices. To construct an explicit algorithm, ideally one would need to somehow obtain and use such inverses. However, these dense matrices are of course not useful in a update scheme because they are not only very costly to compute but also very costly to store for most practical problems. For this reason, we investigate the use of approximate sparse inverses to build update schemes for FETD. We show that the most direct choice, which is to use the approximate inverse of the system matrix itself, is not really an adequate choice because of the nature of the corresponding (continuum) operator, with long-range interactions. We therefore consider instead the use of the approximate inverse of the Hodge (or mass) matrix, which a symmetric positive definite matrix representing a strictly local operator in the continuum limit whose inverse is also local, to compute explicit update schemes. This entails the discretization of Maxwell's equations based on discrete differential forms and the use of a "mixed" set of basis functions for the FETD: Whitney one forms for the electric field intensity and Whitney two forms for the magnetic flux density. This choice of basis functions obeys a discrete version of the de Rham diagram and leads to solutions that are free of spurious modes and numerically stable. We construct a parallel approach to compute the approximate inverse, and provide an error analysis of the resulting solutions versus the density of the approximate inverse and the mesh refinement considered. A higher-order version of the mixed FETD algorithm is also constructed, showing good convergence versus the polynomial order.