Error estimates for the fast multipole method

Error estimates for algorithms based on truncations for evaluating electrostatic interactions in molecular dynamics applications are very important for several reasons. For example, the estimates are necessary to establish the validity of the simulations and can be used to estimate various simulation parameters. Very precise estimates have been found for the Ewald method and the related particle mesh Ewald method. However, for the very popular fast multipole method such a precise estimate is not available. In this paper, we illustrate the rather complicated error behavior of the fast multipole method and we use statistical methods to derive an estimate for the root mean square error on the forces. Furthermore, the expected maximum error on the force acting on a single particle is studied. The estimates are tested against errors obtained from simulations and are found to be very precise.