A Three-Dimensional Finite Element Poisson Solver for Monte Carlo Particle Simulators

Abstract

A finite element Poisson solver for the calculation of the three-dimensional electric field in self-consistent particle simulations has been implemented in Fortran and integrated into an existing Monte Carlo simulator for particle transport. Robust and efficient algorithms for point location in unstructured two and three-dimensional grids is implemented, used in a new injection routine adapted to unstructured meshes and for assembling the load vector of the finite element system. The linear system arising from the finite element approximation of the Poisson equation is solved using conjugate gradient method preconditioned with an incomplete \(LU\)-factorization, which outperforms two other tested solvers. Special storage schemes are implemented to construct and store the matrices of the linear system and are seamlessly combined with the linear system solvers. The work has resulted in a new program structure called Monte Carlo software with finite element Poisson solver (MCFEM), which has been tested by performing short bias simulations of avalanche photodiodes on different grid refinements with the use of both linear and quadratic polynomial basis functions.