  We construct and analyze a finite volume scheme for numerical solution of a
three-dimensional Poisson equation. This is an extension of a two-dimensional
approach by Suli 1991. Here we derive optimal convergence rates in the discrete
H^1 norm and sub-optimal convergence in the maximum norm, where we use the
maximal available regularity of the exact solution and minimal smoothness
requirement on the source term. We also find a gap in the proof of a key
estimate in a reference in Suli 1991 for which we present a modified and
completed proof. Finally, the theoretical results derived in the paper are
justified through implementing some canonical examples in 3D.
