2021 UNC Greensboro PDE Conference.
Electron. J. Diff. Eqns., Conference 26 (2022), pp. 1332.
Discrete Aleksandrov solutions of the MongeAmpere equation
Gerard Awanou
Abstract:
We make two relaxations of the OlikerPrussner method for the Dirichlet problem for
the MongeAmpere 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
righthand 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 MongeAmpere; 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 606077045, USA
email: awanou@uic.edu

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