Year: 2026
Type: preprint
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 $C^0$ 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.
Source: arXiv (Cornell University)
Authors: Chu, Tianyang, Guo, Hailong, Zhang, Zhimin
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Award: G7794165260
OpenAlex ID: W7223213432