The authors extend Hamfeldt’s work [1] to solve a class of fully nonlinear degenerate elliptic partial differential problems. Hamfeldt previously developed a meshfree finite difference scheme for the weak form of the problem. Here, the authors construct piecewise Cartesian meshes using a quadtree structure. Additional boundary points are included to get a consistent scheme. They show that this set of points guarantees a monotone convergent approximation. The quadtree structure efficiently allows for adaptivity of the mesh in either a priori or a posteriori ways. The inclusion of boundary points helps in solving such partial differential equations (PDEs) on complicated geometries.

Several examples demonstrate the scheme. The first example is the Monge-Ampère equation, which is “only elliptic in the space of convex functions.” They show that the solution is of second-order accuracy; the non-convex, non-smooth domain does not affect the order of convergence. They also demonstrate that the central processing unit (CPU) time does not increase as fast when using a large number of points. The last example shows the benefits of an adaptive scheme: error reduces faster and is smaller for the same number of mesh points.