Application of the tree-cotree splitting technique to the transient full-wave analysis based on the time-domain finite element method

The time-domain finite element method (TDFEM) [1, 2] models electromagnetic phenomena by solving the second-order vector wave equation for electric fields expanded with vector edge basis functions. Since the time-step size has a significant effect on the property of the TDFEM system matrix, the actual choice of the time-step size is not only limited by the maximum frequency of interest, but also constrained by the speed of convergence at each time step when an iterative solver is applied to solve the TDFEM system. It is observed that as the time-step size increases, a typical TDFEM system matrix becomes more ill-conditioned, and thus the number of iterations needed to solve such a system at each time step increases drastically. The situation becomes more severe when the geometry of the problem imposes further challenge in the simulation, for example, when very small elements are required in the finite-element discretization or when the disparity in the geometrical size of individual components leads to an extremely nonuniform mesh. In practical applications, these situations are quite common and cannot be avoided. As a result, in the TDFEM analysis the time-step size usually has to be much smaller than required by the temporal sampling rate for achieving certain temporal discretization accuracy. Moreover, when the input signal contains relatively slow-varying components, such a small time-step size will lead to an unacceptably long simulation time for the solutions to respond to the slow-varying signal and finally reach a steady state. In this paper, we apply the tree-cotree splitting (TCS) algorithm [3, 4] to the TDFEM to alleviate the constraint on the time-step size and to improve the convergence of iterative solutions at each time step. Compared with the conventional TDFEM, application of the TCS algorithm maintains the accuracy of the TDFEM solution but significantly reduces the number of iterations per time step for a preconditioned iterative solver to converge when the time-step size becomes relatively large. This desirable feature allows us to adopt a larger time-step size within the requirement of the temporal sampling rate to achieve a faster time-marching with a marginal additional cost. In addition, the proposed formulation effectively suppresses the late-time linear drift or instability, a problem associated with the conventional TDFEM.