Residual-based a posteriori error estimates for a conforming mixed finite element discretization of the Monge-Amp\`ere equation

TL;DR

This paper introduces a new a posteriori error estimate for conforming finite element solutions of the Monge-Ampère equation, ensuring reliable and efficient error control on isotropic meshes in 2D.

Contribution

It develops a novel residual-based a posteriori error analysis for a mixed finite element discretization of the Monge-Ampère equation, extending previous methods.

Findings

The error estimate is proven to be reliable.

The error estimate is proven to be efficient.

Applicable to isotropic meshes in 2D.

Abstract

In this paper we develop a new a posteriori error analysis for the Monge-Amp\`ere equation approximated by conforming finite element method on isotropic meshes in 2D. The approach utilizes a slight variant of the mixed discretization proposed by Gerard Awanou and Hengguang Li in International Journal of Numerical Analysis and Modeling, 11(4):745-761, 2014. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient.