New $C^0$ interior penalty method for Monge-Amp\`ere equations
<|reference_start|>New $C^0$ interior penalty method for Monge-Amp\`ere equations: 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.<|reference_end|>
@article{chu2024new,
    title={New $C^0$ interior penalty method for Monge-Amp\`ere equations},
    author={Tianyang Chu, Hailong Guo, Zhimin Zhang},
    journal={arXiv preprint arXiv:2409.00434},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.00434},
    primaryClass={math.NA cs.NA}
}