  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.
