An Improved Fast Finite Element Time-Domain Method Based on Compressive Sensing for Cavity Problems

To avoid directly solving linear equations for major time marching in the finite element time-domain (FETD) method, the compressive sensing (CS) theory is introduced to construct an explicit iterative scheme instead of an implicit form. In this method, Whitney elements and a leapfrog model are applied to time-dependent Maxwell curl equations for space and time discretization. After a small number of iterations, steps for linear equations in the initial time, the electric field intensity and magnetic flux density are updated just by the linear combination of results of previous time steps. Furthermore, a restart mechanism is built to control the computational complexity and accuracy over a long simulation time. The effectiveness of the new algorithm is verified by analyzing 3-D resonant cavities.

I. Introduction

The finite element time-domain (FETD) method has attracted considerable attention over the past few years, due to its potential advantages in handling complex geometry and obtaining wideband data [1]–[3]. A variety of FETD methods, which can be grouped into two categories, have been proposed. One directly solves Maxwell’s equations and the other uses the wave equation. With the development of FETD, many technologies have been proposed to improve its computational efficiency, such as domain decomposition, Hierarchical matrix and single-variable fully implicit solver [4]–[7]. In our previous works [8], the compressive sensing (CS) [9]–[11] was introduced to FETD method to accelerate the solution of matrix equation, in which CS is used to construct an underdetermined linear system. However, all of them have to solve a linear system of equations at each time step, which leads to a high computation complexity, especially in the simulation of large scale problems or complex structures.