  The elliptic Monge-Amp\`ere equation is a fully nonlinear partial
differential equation which has been the focus of increasing attention from the
scientific computing community. Fast three dimensional solvers are needed, for
example in medical image registration but are not yet available. We build fast
solvers for smooth solutions in three dimensions using a nonlinear
full-approximation storage multigrid method. Starting from a second-order
accurate centered finite difference approximation, we present a nonlinear
Gauss-Seidel iterative method which has a mechanism for selecting the convex
solution of the equation. The iterative method is used as an effective
smoother, combined with the full-approximation storage multigrid method.
Numerical experiments are provided to validate the accuracy of the finite
difference scheme and illustrate the computational efficiency of the proposed
multigrid solver.
