Abstract: The general and short title might be better specified and then stands for the mathematical foundation of piecewise quadratic non-conforming and fifth-order conforming finite element methods for nonlinear smooth low-order perturbations of the biharmonic equation. There is in fact a competition of low-budget nonconforming methodologies like the popular Morley, WOPSIP, two discontinuous Galerkin,as well as the C0 interior penalty for piecewise quadratic polynomials and the more involved fifth-order Argyris finite element methods. The role of a smoother in the source as well in the nonlinearities is discussed for the streamline-vorticity formulation of the of incompressible 2D Navier-Stokes equations and the von-Karman plate problems as well as for the biharmonic eigenvalue problem.
The adaptive variants of the Morley and Argyris approximations are compared and result in the rehabilitation of the high-order techniques.
The presentation is based on joint work with B. Gr\"a\ss le (University of Zurich) and N. Nataraj (IITB in Mumbai) partly reflected in the references below.
C. Carstensen, N. Nataraj, G.C. Remesan, D. Shylaja: Lowest-order FEM for fourth-order semi-linear problems with trilinear nonlinearity. Num Math 154 (2023) 323–368
Carstensen, C. and Gr\"a\ss le B.: Rate-optimal higher-order adaptive conforming FEM for biharmonic eigenvalue problems on polygonal domains. CMAME 425 (2024) 116931
Carstensen, C. and Gr\"a\ss le, B.: Adaptive Morley FEM for 2D stationary Navier-Stokes. Math Comp (2015) //doi.org/10.1090/mcom/4069
