  We introduce two hybridizable discontinuous Galerkin (HDG) methods for
numerically solving the Monge-Ampere equation. The first HDG method is devised
to solve the nonlinear elliptic Monge-Ampere equation by using Newton's method.
The second HDG method is devised to solve a sequence of the Poisson equation
until convergence to a fixed-point solution of the Monge-Ampere equation is
reached. Numerical examples are presented to demonstrate the convergence and
accuracy of the HDG methods. Furthermore, the HDG methods are applied to
r-adaptive mesh generation by redistributing a given scalar density function
via the optimal transport theory. This r-adaptivity methodology leads to the
Monge-Ampere equation with a nonlinear Neumann boundary condition arising from
the optimal transport of the density function to conform the resulting
high-order mesh to the boundary. Hence, we extend the HDG methods to treat the
nonlinear Neumann boundary condition. Numerical experiments are presented to
illustrate the generation of r-adaptive high-order meshes on planar and curved
domains.
