  Development of particle in cell methods using finite element based methods
(FEMs) have been a topic of renewed interest; this has largely been driven by
(a) the ability of finite element methods to better model geometry, (b) better
understanding of function spaces that are necessary to represent all Maxwell
quantities, and (c) more recently, the fundamental rubrics that should be
obeyed in space and time so as to satisfy Gauss' laws and the equation of
continuity. In that vein, methods have been developed recently that satisfy
these equations and are agnostic to time stepping methods. While is development
is indeed a significant advance, it should be noted that implicit FEM transient
solvers support an underlying null space that corresponds to a gradient of a
scalar potential $\nabla \Phi(\textbf{r})$ (or $t \nabla \Phi (\textbf{r})$ in
the case of wave equation solvers). While explicit schemes do not suffer from
this drawback, they are only conditionally stable, time step sizes are mesh
dependent, and very small. A way to overcome this bottleneck, and indeed,
satisfy all four Maxwell's equation is to use a quasi-Helmholtz formulation on
a tessellation. In the re-formulation presented, we strictly satisfy the
equation of continuity and Gauss' laws for both the electric and magnetic flux
densities. Results demonstrating the efficacy of this scheme will be presented.
