Solving the Einstein constraints numerically on compact three-manifolds using hyperbolic relaxation

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 H3 and the H2 x S1 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 S3, S2 x S1, and the E3 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.