A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semi-linear problems with trilinear non-linearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space, and more importantly, modifies the trilinear term in the stream-function vorticity formulation of the incompressible 2D Navier-Stokes and the von K\'{a}rm\'{a}n equations. This enables the first efficient and reliable a posteriori error estimates for the 2D Navier-Stokes equations in the stream-function vorticity formulation for Morley, two discontinuous Galerkin, $C^0$ interior penalty, and WOPSIP discretizations with piecewise quadratic polynomials.

翻译：本文针对两类具有三线性非线性和一般源项的四阶半线性问题，对五种最低阶有限元方法进行了通用后验误差分析。一种拟最优平滑器将源项延拓至离散试探空间，更重要的是，修正了不可压缩二维Navier-Stokes方程流函数涡度公式及von Kármán方程中的三线性项。这使得对于Morley元、两种间断Galerkin方法、$C^0$内罚方法和WOPSIP离散格式（采用分片二次多项式），首次实现了二维Navier-Stokes方程流函数涡度公式的高效可靠后验误差估计。