Nonconforming virtual element method for the Monge-Amp\`ere equation

TL;DR

This paper introduces a novel virtual element method for solving the Monge-Ampère equation, providing optimal error estimates and validating convergence through numerical experiments.

Contribution

It develops a $C^1$-nonconforming virtual element method for the Monge-Ampère equation with proven error bounds and solution existence and uniqueness.

Findings

Optimal error estimates in multiple norms are derived.

The method demonstrates convergence in numerical experiments.

Existence and uniqueness of the virtual element solution are established.

Abstract

In this article, we develop the $C^1$-nonconforming $C^0$-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 $H^2$-, $H^1$- and $L^2$-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.