A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general source allows for quasi-best approximation with lowest-order finite element methods. This paper establishes the stability and a priori error control in the piecewise energy and weaker Sobolev norms under minimal hypotheses. Applications include the stream function vorticity formulation of the incompressible 2D Navier-Stokes equations and the von K\'{a}rm\'{a}n equations with Morley, discontinuous Galerkin, $C^0$ interior penalty, and weakly over-penalized symmetric interior penalty schemes. The proposed new discretizations consider quasi-optimal smoothers for the source term and smoother-type modifications inside the nonlinear terms.

翻译：针对具有三线性非线性和一般源项的四阶半线性问题，本文建立了一个统一框架，实现了最低阶有限元方法下的拟最优逼近。在最小化假设条件下，本文建立了分段能量范数和较弱Sobolev范数下的稳定性与先验误差控制。应用场景包括不可压缩二维纳维-斯托克斯方程的流函数涡度公式，以及采用Morley元、间断伽辽金法、$C^0$内罚法和弱过罚对称内罚格式的冯·卡门方程。所提出的新离散方法考虑了源项的拟最优光滑化以及非线性项内部的光滑型修正。