New $C^0$ interior penalty method for Monge-Amp\`ere equations

TL;DR

This paper introduces a novel $C^0$ interior penalty method inspired by the Miranda-Talenti estimate for approximating viscosity solutions of the Monge-Ampère equation, a key nonlinear PDE.

Contribution

The paper develops a new $C^0$ interior penalty scheme for Monge-Ampère equations using the vanishing moment approach and discrete Miranda-Talenti estimates, with proven well-posedness and error bounds.

Findings

The method effectively approximates viscosity solutions.

The scheme is well-posed and stable.

Error estimates are rigorously derived.

Abstract

Monge-Amp\`{e}re equation is a prototype second-order fully nonlinear partial differential equation. In this paper, we propose a new idea to design and analyze the $C^0$ interior penalty method to approximation the viscosity solution of the Monge-Amp\`{e}re equation. The new methods is inspired from the discrete Miranda-Talenti estimate. Based on the vanishing moment representation, we approximate the Monge-Amp\`{e}re equation by the fourth order semi-linear equation with some additional boundary conditions. We will use the discrete Miranda-Talenti estimates to ensure the well-posedness of the numerical scheme and derive the error estimates.