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

Discrete Aleksandrov solutions of the Monge-Ampere equation

Gerard Awanou

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.

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Gerard Awanou
Department of Mathematics, Statistics, and Computer Science
M/C 249. University of Illinois at Chicago
Chicago, IL 60607-7045, USA
email: awanou@uic.edu

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