The FMM-BEM Method for the 3D Particulate Stokes Flow

Abstract

This work introduces new functions based on the spherical harmonics and the solid harmonics which have been used to construct a multipole development for the 3D Stokes problem in order to reduce the operations costs in the BEM method. We show that the major properties of those functions are inherited from the solid harmonics. The contribution of this paper is the introduction of new formulas that serve to calculate the multipole moments and the transfer functions that are necessary for the schemes of order $O(N\text{log}N)$. Moreover, new translation formulas are introduced to obtain an $O(N)$ scheme. The error truncation of the resulting scheme is discussed. In comparison to the BEM that attains a limit storage at $O(10^4)$, we present here a method based on FMM-BEM that attains a storage at a limit of $O(10^6)$. The implementation of the method achieves a high accuracy level at a reasonable cost.