A Finite Element Based P3M Method for N-body Problems

TL;DR

This paper presents a novel finite element based P3M method for N-body problems that improves accuracy and scalability by using polynomial bases for potential decomposition and finite element methods for long-range interactions.

Contribution

The paper introduces a new P3M approach employing polynomial bases for potential screening and finite element methods for exact long-range calculations, enhancing accuracy and efficiency.

Findings

Method achieves scalable N-body computations with high accuracy.

Finite element approach leads to sparse matrix problems solved efficiently.

The approach outperforms traditional Gaussian screen-based methods.

Abstract

We introduce a fast mesh-based method for computing N-body interactions that is both scalable and accurate. The method is founded on a particle-particle--particle-mesh P3M approach, which decomposes a potential into rapidly decaying short-range interactions and smooth, mesh-resolvable long-range interactions. However, in contrast to the traditional approach of using Gaussian screen functions to accomplish this decomposition, our method employs specially designed polynomial bases to construct the screened potentials. Because of this form of the screen, the long-range component of the potential is then solved exactly with a finite element method, leading ultimately to a sparse matrix problem that is solved efficiently with standard multigrid methods. Moreover, since this system represents an exact discretization, the optimal resolution properties of the FFT are unnecessary, though the short-range calculation is now more involved than P3M/PME methods. We introduce the method, analyze its key properties, and demonstrate the accuracy of the algorithm.