The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long time to realise the necessity of an extension of the classical finite element spaces to make them hierarchical. This paper establishes the novel adaptive schemes for the biharmonic eigenvalue problems and provides a mathematical proof of optimal convergence rates towards a simple eigenvalue and numerical evidence thereof. This makes the suggested algorithm highly competitive and clearly justifies the higher computational and implementational costs compared to low-order nonconforming schemes. The numerical experiments provide overwhelming evidence that higher polynomial degrees pay off with higher convergence rates and underline that adaptive mesh-refining is mandatory. Five computational benchmarks display accurate reference eigenvalues up to 30 digits.

翻译：经典Argyris有限元方法的后验误差分析可追溯至1996年，而相关自适应有限元格式的最优收敛速率直至2021年才得以建立。历经漫长探索，研究者意识到必须扩展经典有限元空间使其具备层次化结构。本文针对双调和特征值问题建立了新型自适应格式，从数学上证明了该格式对于单特征值的最优收敛速率，并给出了数值验证。这使得所提算法具有显著竞争力，充分证明了相较于低阶非协调格式在计算与实现成本上的合理性。数值实验充分表明，高阶多项式次数能够换取更优的收敛速率，同时凸显了自适应网格加密的必要性。五项基准测试获得了精确至30位有效数字的参考特征值。