It is an open question if the threshold condition θ < θ ⋆ \theta<\theta_{\star} for the Dörfler marking parameter is necessary to obtain optimal algebraic rates of adaptive finite element methods. We present a (non-PDE) example fitting into the common abstract convergence framework (axioms of adaptivity) which allows for convergence with exponential rates. However, for Dörfler marking θ > θ ⋆ \theta>\theta_{\star} , the algebraic convergence rate can be made arbitrarily small.

中文翻译：

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关于 Dörfler 标记的阈值条件

Dörfler 标记参数的阈值条件 θ < θ ⋆ \theta<\theta_{\star} 是否对于获得自适应有限元方法的最佳代数率是必要的，这是一个悬而未决的问题。我们提出了一个（非 PDE）示例，该示例适合通用抽象收敛框架（自适应公理），该框架允许以指数速率收敛。然而，对于 Dörfler 标记 θ > θ ⋆ \theta>\theta_{\star} ，代数收敛速度可以任意小。