  We introduce a generalized finite difference method for solving a large range
of fully nonlinear elliptic partial differential equations in three dimensions.
Methods are based on Cartesian grids, augmented by additional points carefully
placed along the boundary at high resolution. We introduce and analyze a
least-squares approach to building consistent, monotone approximations of
second directional derivatives on these grids. We then show how to efficiently
approximate functions of the eigenvalues of the Hessian through a multi-level
discretization of orthogonal coordinate frames in $\mathbb{R}^3$. The resulting
schemes are monotone and fit within many recently developed convergence
frameworks for fully nonlinear elliptic equations including non-classical
Dirichlet problems that admit discontinuous solutions, Monge-Amp\`ere type
equations in optimal transport, and eigenvalue problems involving nonlinear
elliptic operators. Computational examples demonstrate the success of this
method on a wide range of challenging examples.
