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We propose an efficient algorithm for the evaluation of the potential and itsgradient of gravitational/electrostatic $N$-body systems, which we callparticle mesh multipole method (PMMM or PM$^3$). PMMM can be understood both asan extension of the particle mesh (PM) method and as an optimization of thefast multipole method (FMM).In the former viewpoint, the scalar density andpotential held by a grid point are extended to multipole moments and localexpansions in $(p+1)^2$ real numbers, where $p$ is the order of expansion. Inthe latter viewpoint, a hierarchical octree structure which brings its$\mathcal O(N)$ nature, is replaced with a uniform mesh structure, and weexploit the convolution theorem with fast Fourier transform (FFT) to speed upthe calculations. Hence, independent $(p+1)^2$ FFTs with the size equal to thenumber of grid points are performed.  The fundamental idea is common to PPPM/MPE by Shimada et al. (1993) and FFTMby Ong et al. (2003). PMMM differs from them in supporting both the open andperiodic boundary conditions, and employing an irreducible form where both themultipole moments and local expansions are expressed in $(p+1)^2$ real numbersand the transformation matrices in $(2p+1)^2$ real numbers.  The computational complexity is the larger of $\mathcal O(p^2 N)$ and$\mathcal O(N \log (N/p^2))$, and the memory demand is $\mathcal O(N)$ when thenumber of grid points is $\propto N/p^2$.

Particle mesh multipole method: An efficient solver for  gravitational/electrostatic forces based on multipole method and fast  convolution over a uniform mesh