Frequency-domain and time-domain finite-element solution of Maxwell's equations using spectral Lanczos decomposition method

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

“…Therefore, with the growing need for solving geometrically complex electromagnetic problems, the development of flexible time-domain Maxwell equation solvers on advanced grids has received considerable interest in the literature. Guided by the consideration of numerical grids, numerous finite volume time-domain methods [28][29][30] and finite element time-domain methods [31][32][33][34] have been put forward to systematically handle geometrically complex problems in CEM, since the most flexible grid, the unstructured grid, is best adapted to the finite volume and finite element approaches. Many vector elements, such as Nedelec elements [31], Whitney forms [35,36], and curl/div conforming vector elements [37] are constructed to provide a discrete analog to the continuous vector algebra and to enforce only minimal continuity across element boundaries.…”

Alternating-Direction Implicit Formulation of the Finite-Element Time-Domain Method

“…In general, (15) and (16) can be summarized as the following matrix form: (23) (for advancement from the th to th time step)…”

“…They lead to a fully implicit system. Several lumping techniques have been proposed in order to obtain explicit schemes without solving a linear system at each time step [14], [23]- [25]. Moreover, a recently developed approach avoids lumping altogether by constructing a set of orthogonal vector basis functions that yield a diagonal mass matrix [26], [27].…”

Three-dimensional orthogonal vector basis functions for time-domain finite element solution of vector wave equations

“…Obviously, the above expansion preserves the tangential continuity of the fields across element interfaces (at the central points), whereas allows for the correct normal discontinuity. By substituting (12) into (11), we obtain the ordinary differential equation (13) where , , and are square matrices, assembled from their corresponding element matrices given by (14) and (15) at the bottom of the next page and (16) in which denotes , , and , and denote volume and surface integration, respectively. In (13), is the unknown vector given by , and is a vector contributed by the excitation on , which can be assembled from .…”

“…One directly discretizes Maxwell's equations [6]- [9], [18], which typically results in an explicit, finite difference-like leap-frog scheme that does not leverage our extensive knowledge of frequency-domain FE solvers. The other discretizes the second-order vector wave equation, also known as the curl-curl equation, obtained by eliminating one of the field variables from Maxwell's equations [10]- [16], [19]. Despite its ability to handle unstructured meshes and its capacity to impose continuity conditions across material interfaces, the TDFEM does not enjoy widespread popularity compared to the FDTD method.…”