This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Kármán equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.

本文研究了具有三线性非线性的四阶半线性椭圆方程的Hessian离散化方法（HDM）。HDM为几种数值方法的收敛性分析提供了通用框架，例如合格和不合格的有限元方法（ncFEM）以及基于梯度恢复（GR）运算符的方法。首次分析了Adini ncFEM和GR方法，该方法基于廉价的，基于分段线性函数的高阶导数的局部重建，并且针对具有三线性非线性的四阶半线性椭圆方程进行了分析。四个属性，即矫顽力，一致性，极限一致性和紧凑性，使HDM框架中的收敛分析成为可能，不需要任何精确解的规则性。讨论了应用中的两个重要问题，即流函数涡度公式中的Navier-Stokes方程和板弯曲的vonKármán方程。给出了Morley和Adini ncFEM和GR方法的数值实验结果。