2021 UNC Greensboro PDE Conference.
Electron. J. Diff. Eqns., Conference 26 (2022), pp. 13-32.

Title: Discrete Aleksandrov solutions of the Monge-Ampere equation

Author: Gerard Awanou (Univ. of Illinois, Chicago, USA)
 
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.

Published August 25, 2022.
Math Subject Classifications: 39A12, 35J60, 65N12, 65M06.
Key Words: Discrete Monge-Ampere; Aleksandrov solution; weak convergence of measures.