On the Threshold Condition for Dörfler Marking.

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. [ABSTRACT FROM AUTHOR]