  A multi-cube method is developed for solving systems of elliptic and
hyperbolic partial differential equations numerically on manifolds with
arbitrary spatial topologies. It is shown that any three-dimensional manifold
can be represented as a set of non-overlapping cubic regions, plus a set of
maps to identify the faces of adjoining regions. The differential structure on
these manifolds is fixed by specifying a smooth reference metric tensor.
Matching conditions that ensure the appropriate levels of continuity and
differentiability across region boundaries are developed for arbitrary tensor
fields. Standard numerical methods are then used to solve the equations with
the appropriate boundary conditions, which are determined from these
inter-region matching conditions. Numerical examples are presented which use
pseudo-spectral methods to solve simple elliptic equations on multi-cube
representations of manifolds with the topologies T^3, S^2 x S^1 and S^3.
Examples are also presented of numerical solutions of simple hyperbolic
equations on multi-cube manifolds with the topologies R x T^3, R x S^2 x S^1
and R x S^3.
