We briefly discuss why GEM is interesting We state the computational problem for GEM and analyze parallel computing issues for key steps We map into a particle dynamics analogy using Green's function formalism and consider O(N2) cut-off O(N) and fast multipole methods We go in detail through the fast multipole method We expect object oriented approaches to be very important booth at language level and in using technologies like CORBA to integrate world-wide collaboration, simulations and experiments.