A Necessarily Incomplete Review of Electromagnetic Finite Element Particle-in-Cell Methods

While finite-difference time-domain methods have long been the foundational basis for particle-in-cell (PIC) codes, there has been increasing momentum in developing a suite of finite element-based PIC methods. The beauty of finite difference-based methods is that it is easily cast within the correct mathematical framework to represent fields, fluxes, currents, and charges. However, more importantly, these methods are cost-effective. In the intervening years, since finite difference methods were developed, the state of the art of field modeling has shifted rather dramatically. Indeed, the most popular and trusted field simulators are based on the finite element method (FEM), thanks, in large part, to the discovery of the correct function spaces for quantities of interest for Maxwell's equations, but also the flexibility that it brings to modeling geometry with the ability to refine in space and numerical order to better capture the underlying physics. Together with time-stepping schemes that are unconditionally stable, these methods provide the framework necessary to correctly capture the nuance of the physical evolution with high fidelity. The intent of this article is to review advances in electromagnetic finite element PIC (EM-FEMPIC). We will address the progress made in fundamental challenges in such a method for charge conservation to more programmatic ones, such as computational complexity.