  The effectiveness of the hyperbolic relaxation method for solving the
Einstein constraint equations numerically is studied here on a variety of
compact orientable three-manifolds. Convergent numerical solutions are found
using this method on manifolds admitting negative Ricci scalar curvature
metrics, i.e. those from the $H^3$ and the $H^2\times S^1$ geometrization
classes. The method fails to produce solutions, however, on all the manifolds
examined here admitting non-negative Ricci scalar curvatures, i.e. those from
the $S^3$, $S^2\times S^1$, and the $E^3$ classes. This study also finds that
the accuracy of the convergent solutions produced by hyperbolic relaxation can
be increased significantly by performing fairly low-cost standard elliptic
solves using the hyperbolic relaxation solutions as initial guesses.
