Abstract:
This PhD dissertation introduces the Discrete Time Vector Finite Element Method (DTVFEM), a novel approach for solving Maxwell's equations on three-dimensional unstructured grids. The primary goal of the research was to develop a method that is provably stable, as well as energy and charge conserving. The DTVFEM achieves this by using specific types of vector finite elements: covariant elements (edge elements) to represent the electric field and contravariant elements (face elements) for the magnetic flux density. This is a key feature, as these elements are complementary and correctly model the natural continuity of the fields across material interfaces—tangential for the electric field and normal for the magnetic flux density. The method employs a Galerkin approximation to discretize Ampere's and Faraday's laws into a system of ordinary differential equations, which are then advanced in time using the leapfrog method. A significant portion of the work addresses the challenge of solving the large, sparse linear system that arises at every time step, showing that for unstructured grids, iterative methods are competitive and the method's complexity is comparable to "explicit" methods. The DTVFEM was implemented in the VFEM3D software and validated through several computational experiments on resonant cavities, waveguides, and antennas, confirming its accuracy and performance on both serial and parallel supercomputers.
