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A unified framework for fourth-order semilinear problems with trilinearnonlinearity and general source allows for quasi-best approximation withlowest-order finite element methods. This paper establishes the stability and apriori error control in the piecewise energy and weaker Sobolev norms underminimal hypotheses. Applications include the stream function vorticityformulation of the incompressible 2D Navier-Stokes equations and the vonK\'{a}rm\'{a}n equations with Morley, discontinuous Galerkin, $C^0$ interiorpenalty, and weakly over-penalized symmetric interior penalty schemes. Theproposed new discretizations consider quasi-optimal smoothers for the sourceterm and smoother-type modifications inside the nonlinear terms.

Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems