A Posteriori Error Control for Fourth-Order Semilinear Problems with Quadratic Nonlinearity

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 919-945, April 2024.
Abstract. A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semilinear problems with trilinear nonlinearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space and, more important, modifies the trilinear term in the stream-function vorticity formulation of the incompressible two-dimensional Navier–Stokes equations and the von Kármán equations. This enables the first efficient and reliable a posteriori error estimates for the two-dimensional Navier–Stokes equations in the stream-function vorticity formulation for Morley, two discontinuous Galerkin, [math] interior penalty, and weakly overpenalized symmetric interior penalty discretizations with piecewise quadratic polynomials.