On the threshold condition for Dörfler marking

Abstract

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