Nonconforming virtual element method for the Monge-Amp\`ere equation
Numerical Analysis
2026-04-27 v1 Numerical Analysis
Abstract
In this article, we develop the -nonconforming -conforming virtual element method (VEM) for the vanishing moment approximation of the second-order fully nonlinear Monge-Amp\`ere equation in two dimensions. In the vanishing moment equation an artificial biharmonic term is introduced which produces a quasilinear fourth order problem. We derive optimal a priori error estimates in the -, - and -norms for the virtual element method, and show the existence and uniqueness of the virtual element solution. We perform several numerical experiments to validate the convergence rate of the error with respect to the mesh size.
Cite
@article{arxiv.2604.22589,
title = {Nonconforming virtual element method for the Monge-Amp\`ere equation},
author = {Scott Congreve and Alice Hodson and Anwesh Pradhan},
journal= {arXiv preprint arXiv:2604.22589},
year = {2026}
}