The Second Boundary Value Problem for a Discrete Monge–Ampère Equation

In this work we propose a discretization of the second boundary condition for the Monge–Ampère equation arising in geometric optics and optimal transport. The discretization we propose is the natural generalization of the popular Oliker–Prussner method proposed in

1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.