New $C^0$ interior penalty method for Monge-Amp\`ere equations
Numerical Analysis
2024-09-04 v1 Numerical Analysis
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 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.
Keywords
Cite
@article{arxiv.2409.00434,
title = {New $C^0$ interior penalty method for Monge-Amp\`ere equations},
author = {Tianyang Chu and Hailong Guo and Zhimin Zhang},
journal= {arXiv preprint arXiv:2409.00434},
year = {2024}
}