ABSTRACT

This talk explores recent advances [5, 1, 2, 3] on rate-optimal adaptive finite element methods (AFEM) for conforming and non-conforming discretisations of fourth-order problems. Nonconforming methods, such as the Morley element, are particularly attractive for their low polynomial degrees. The optimality of the Morley AFEM for the biharmonic source problem is known for over a decade. Since then, these results have been extended to semilinear problems, including eigenvalue problems and the Von K´arm´an equations. More recently, the first result [2] on optimal convergence towards a regular root of the 2D stationary Navier–Stokes equations was established based on novel error estimators and a careful discretisation of the nonlinearity. The analysis of the adaptive algorithm for stabilised nonconforming schemes remains challenging and is unresolved in most cases. However, recent insights [3] into the stabilised WOPSIP AFEM reveal a supercloseness to Morley and lead to optimal convergence rates in terms of a non-standard error estimator.

The final part of this talk discusses conforming discretisations based on the Argyris element [5, 1]. While its high polynomial degree mandates adaptive mesh refinement, optimal convergence rates were established only very recently [4], in 2021. It took a long time, since the first a posteriori error analysis of the classical Argyris FEM dating back to 1996, to realize the necessity of a hierarchical extension. Numerical experiments provide striking evidence that higher polynomial degrees pay off with optimal higher-order convergence rates and an efficient computation of eigenvalues up to more than 30 digits.