  We calculate electrostatic potential landscapes for an external probe charge
in the presence of a set of metallic islands. Our numerical calculation in
three dimensions (3D)uses an efficient grid relaxation technique. The
well-known relaxation algorithm for solving the Poisson equation in two
dimensions is generalized to 3D. In addition,all charges on the system, free as
well as induced charges,are determined accurately and self-consistently to
satisfy the desired boundary conditions. This allows the straightforward
calculation of the potential on the outer boundary using the free space
electrostatic Green's function,as well as the calculation of the entire
capacitance matrix of the system. Physically interesting examples of nanoscale
systems are presented and analyzed.
