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× 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× 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.