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

TL;DR

This paper introduces a quasi-Helmholtz decomposition approach in finite element particle-in-cell methods that strictly enforces Gauss' laws and charge conservation, improving the physical fidelity of electromagnetic simulations.

Contribution

It presents a novel quasi-Helmholtz formulation that satisfies all Maxwell's equations in FEM-based PIC methods, overcoming limitations of traditional implicit and explicit schemes.

Findings

The scheme strictly enforces Gauss' laws for electric and magnetic fields.

It ensures charge conservation in finite element particle-in-cell simulations.

Results demonstrate improved accuracy and stability of the proposed method.

Abstract

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.