New Mixed SETD and FETD Methods to Overcome the Low-Frequency Breakdown Problems by Tree-Cotree Splitting

Traditional time-domain finite-element and spectral-element methods are known to suffer from the so-called low-frequency breakdown problem, where the system matrix can become ill-conditioned, and results can be unstable for electrically fine structures. In this article, a new mixed spectral-element time-domain (SETD) and finite-element time-domain (FETD) methods are proposed to overcome this low-frequency breakdown problem by constraint equations and tree-cotree splitting. Two forms of gradient matrices or the null space of the stiffness matrix of high-order basis functions are calculated to construct the constraint equations. Then, the constraint equations are directly applied in the time-stepping matrix equation by tree-cotree splitting to improve the properties of system matrices when the implicit Newmark-beta algorithm is adopted in two different forms. In the calculation of the first form of gradient matrix, the electric field is expanded by high-order edge basis functions; in the second form of gradient matrix, high-order edge basis functions associated with cotree edges and nodal basis functions defined on the free nodes are both employed to expand the electric field. Several numerical examples demonstrate that the new mixed SETD and mixed FETD methods can effectively overcome the low-frequency breakdown with high accuracy and without increasing the number of degrees of freedom over the conventional SETD and FETD methods.