Discrete Aleksandrov solutions of the Monge-Ampere equation

Abstract

We make two relaxations of the Oliker-Prussner method for the Dirichlet problem for the Monge-Ampere equation. First we relax the convexity requirement and consider mesh functions which are only discrete convex. The second relaxation consists in  using a finite stencil. The discrete nonlinear equations are solved with a damped  Newton's method. We give two proofs of convergence of the resulting scheme for right-hand side a density, on domains which are convex and not necessarily strictly convex, under the assumption that the boundary data has a continuous convex extension. The first proof is based on the notion of Aleksandrov solution while the second uses  viscosity solutions. For more information see https://ejde.math.txstate.edu/conf-proc/26/a2/abstr.html