New C⁰ interior penalty method for Monge-Ampère equations

2024/08/31 by Chu, Tianyang, Guo, Hailong, Zhang, Zhimin
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2409.00434

Abstract

Monge-Ampère equation is a prototype second-order fully nonlinear partial differential equation. In this paper, we propose a new idea to design and analyze the C⁰ interior penalty method to approximation the viscosity solution of the Monge-Ampère equation. The new methods is inspired from the discrete Miranda-Talenti estimate. Based on the vanishing moment representation, we approximate the Monge-Ampère 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.

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