This paper presents a new approach to the direct numerical simulation of potential problems with many spherical internal boundaries, e.g., many spheres in potential flow. The basic idea is to use a local analytic representation valid near the particle and to match it to an external field calculated by a standard finite-difference (or finite-element) method. In this way the geometric complexity arising from the irregular relation between the particle boundary and the underlying mesh is avoided and fast solvers can be used. The results suggest that the computational effort increases less than proportionally to the number of particles and, additionally, that meshes that would be excessively coarse as measured in terms of particle radius in a conventional calculation can be used without significant loss of accuracy. In separate (if preliminary) work the same approach has been extended to the simulation of viscous flow about spheres and cylinders at finite Reynolds numbers.