A Finite Element Based P$^3$M Method for $N$-Body Problems

Abstract

In this paper we introduce a new mesh-based method for $N$-body calculations. The method is founded on a particle-particle--particle-mesh (P$^3$M) 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 accurately with a finite element method, leading ultimately to a sparse matrix problem that is solved efficiently with standard multigrid methods, though the short-range calculation is now more involved than P$^3$M particle-mesh-Ewald (PME) methods. We introduce the method, analyze its key properties, and demonstrate the accuracy of the algorithm.