On the threshold condition for Dörfler marking

It is an open question if the threshold condition $θ< θ_\star$ 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) and which is potentially converging with exponential rates. However, for Dörfler marking $θ> θ_\star$ the algebraic converges rate can be made arbitrarily small.

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