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PreprintCC BY 4.0arXiv

A recovery based linear finite element method for the Monge--Ampère equation

  • Tianyang Chu
  • Hailong Guo
  • Zhimin Zhang
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Abstract

In this paper, we study finite element approximations of a fourth-order perturbed semilinear partial differential equation that arises naturally in the computation of viscosity solutions to the fully nonlinear Monge--Ampère equation. We develop a novel recovery-based C0 linear finite element method for discretizing this model problem. The key analytical tool is a Miranda--Talenti-type estimate for the recovery operator, which guides the formulation of the discrete scheme and guarantees its well-posedness. We rigorously establish convergence in the H2 norm by employing a fixed-point argument. Furthermore, optimal H1-error estimates are derived using a duality argument. Numerical experiments confirm the efficiency of the proposed method and validate the theoretical convergence results.

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Cite

BibTeXRIS
BibTeX
@misc{chu2026recovery,
  title = {{A recovery based linear finite element method for the Monge--Ampère equation}},
  author = {Tianyang Chu and Hailong Guo and Zhimin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29536},
  eprint = {2610.29536},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29536},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A recovery based linear finite element method for the Monge--Ampère equation
AU  - Tianyang Chu
AU  - Hailong Guo
AU  - Zhimin Zhang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29536
UR  - https://arxiv.org/abs/2610.29536
AB  - In this paper, we study finite element approximations of a fourth-order perturbed semilinear partial differential equation that arises naturally in the computation of viscosity solutions to the fully nonlinear Monge--Ampère equation. We develop a novel recovery-based C^0 linear finite element method for discretizing this model problem. The key analytical tool is a Miranda--Talenti-type estimate for the recovery operator, which guides the formulation of the discrete scheme and guarantees its well-posedness. We rigorously establish convergence in the H^2 norm by employing a fixed-point argument. Furthermore, optimal H^1-error estimates are derived using a duality argument. Numerical experiments confirm the efficiency of the proposed method and validate the theoretical convergence results.
ER  -