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arXiv · 2409.00434

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

Tianyang Chu·Hailong Guo·Zhimin Zhang
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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.

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Tianyang Chu, Hailong Guo, Zhimin Zhang. 2024-08-31. New $C^0$ interior penalty method for Monge-Amp\`ere equations. https://arxiv.org/abs/2409.00434

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