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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) betterunderstanding of function spaces that are necessary to represent all Maxwellquantities, and (c) more recently, the fundamental rubrics that should beobeyed in space and time so as to satisfy Gauss' laws and the equation ofcontinuity. In that vein, methods have been developed recently that satisfythese equations and are agnostic to time stepping methods. While is developmentis indeed a significant advance, it should be noted that implicit FEM transientsolvers support an underlying null space that corresponds to a gradient of ascalar potential $\nabla \Phi(\textbf{r})$ (or $t \nabla \Phi (\textbf{r})$ inthe case of wave equation solvers). While explicit schemes do not suffer fromthis drawback, they are only conditionally stable, time step sizes are meshdependent, and very small. A way to overcome this bottleneck, and indeed,satisfy all four Maxwell's equation is to use a quasi-Helmholtz formulation ona tessellation. In the re-formulation presented, we strictly satisfy theequation of continuity and Gauss' laws for both the electric and magnetic fluxdensities. Results demonstrating the efficacy of this scheme will be presented.

Quasi-Helmholtz Decomposition, Gauss' Laws and Charge Conservation for Finite Element Particle-in-Cell