arXiv:2610.29536v1 Announce Type: new 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--Amp\`{e}re 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.