  The main drawback for the application of the conforming Argyris FEM is the
labourious implementation on the one hand and the low convergence rates on the
other. If no appropriate adaptive meshes are utilised, only the convergence
rate caused by corner singularities [Blum and Rannacher, 1980], far below the
approximation order for smooth functions, can be achieved. The fine
approximation with the Argyris FEM produces high-dimensional linear systems and
for a long time an optimal preconditioned scheme was not available for
unstructured grids. This paper presents numerical benchmarks to confirm that
the adaptive multilevel solver for the hierarchical Argyris FEM from
[Carstensen and Hu, 2021] is in fact highly efficient and of linear time
complexity. Moreover, the very first display of optimal convergence rates in
practically relevant benchmarks with corner singularities and general boundary
conditions leads to the rehabilitation of the Argyris finite element from the
computational perspective.
