  We address periodic-image errors arising from the use of periodic boundary
conditions to describe systems that do not exhibit full three-dimensional
periodicity. The difference between the periodic potential, as
straightforwardly obtained from a Fourier transform, and the potential
satisfying any other boundary conditions can be characterized analytically. In
light of this observation, we present an efficient real-space method to correct
periodic-image errors, based on a multigrid solver for the potential
difference, and demonstrate that exponential convergence of the energy with
respect to cell size can be achieved in practical calculations. Additionally,
we derive rapidly convergent expansions for determining the Madelung constants
of point-charge assemblies in one, two, and three dimensions.
