A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general sources 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 Kármán 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.

具有三线性非线性和通用源的四阶半线性问题的统一框架允许使用最低阶有限元方法进行准最佳近似。本文在最小假设下建立了分段能量和较弱 Sobolev 范数的稳定性和先验误差控制。应用包括不可压缩的二维纳维-斯托克斯方程和莫利冯卡门方程的流函数涡度公式、不连续伽辽金、\(C^{0}\)内部惩罚和弱过度惩罚对称内部惩罚方案。所提出的新离散化考虑了源项的准最优平滑器和非线性项内的平滑器类型修改。