Apparent horizon finder for a special family of spacetimes in 3D numerical relativity

We present a method to find the apparent horizon (AH) on a special family of three-dimensional (3D) spacelike hypersurfaces which has $\pi$-rotation symmetry around the $z$ axis as well as the reflection one with respect to the equatorial plane. In a nonaxisymmetric 3D hypersurface, the AH, if it exists, is determined by solving a 2D elliptic-type equation. In the present method, we solve the elliptic-type equation as a boundary value problem. To test this method, we apply it to a variety of nonaxisymmetric 3D hypersurfaces which can be obtained by solving the constraint equations in general relativity. We find that the present method works well in all cases.