  Apparent horizon plays an important role in numerical relativity as it
provides a tool to characterize the existence and properties of black holes on
three-dimensional spatial slices in 3+1 numerical spacetimes. Apparent horizon
finders based on different techniques have been developed. In this paper, we
revisit the apparent horizon finding problem in numerical relativity using
multigrid-based algorithms. We formulate the nonlinear elliptic apparent
horizon equation as a linear Poisson-type equation with a nonlinear source, and
solve it using a multigrid algorithm with Gauss-Seidel line relaxation. A
fourth order compact finite difference scheme in spherical coordinates is
derived and employed to reduce the complexity of the line relaxation operator
to a tri-diagonal matrix inversion. The multigrid-based apparent horizon finder
developed in this work is capable of locating apparent horizons in generic
spatial hypersurfaces without any symmetries. The finder is tested with both
analytic data, such as Brill-Lindquist multiple black hole data, and numerical
data, including off-centered Kerr-Schild data and dynamical inspiraling binary
black hole data. The obtained results are compared with those generated by the
current fastest finder AHFinderDirect (Thornburg, Class. Quantum Grav. 21, 743,
2003), which is the default finder in the open source code Einstein Toolkit.
Our finder performs comparatively in terms of accuracy, and starts to
outperform AHFinderDirect at high angular resolutions (\sim 1^\circ) in terms
of speed. Our finder is also more flexible to initial guess, as opposed to the
Newton's method used in AHFinderDirect. This suggests that the multigrid
algorithm provides an alternative option for studying apparent horizons,
especially when high resolutions are needed.
