Domain Decomposition Framework for Maxwell Finite Element Solvers and Application to PIC

The most popular method for self-consistent simulation of fields interacting with charged species is using finite difference time domain (FDTD) methods together with Newton's laws of motion to evolve locations and velocities of particles. Despite their popularity, the limitations of FDTD particle-in-cell (EM-FDTDPIC) methods are well-known. To address these, there has been significant interest over the past decade in exploring alternatives. In the past few years, the advances in electromagnetic finite element methods for particle-in-cell (EM-FEMPIC) have advanced by leaps and bounds. The mathematics necessary for implicit FEM methods that are unconditionally stable and charge-conserving are now well understood. Some of these advances are more recent. The next bottleneck necessary to make EM-FEMPIC competitive with the FDTD-based scheme is overcoming computational cost. Our approach to resolving this challenge is to develop two different finite element tearing and integration approaches, and use these to create domain decomposition schemes for EM-FEMPIC. Details of the proposed methodology are presented as well as a number of electrostatic results that demonstrate charge conservation as well as amelioration of costs for a number of problems.