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

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