On the threshold condition for D\"orfler marking

Abstract

It is an open question if the threshold condition $\theta < \theta_\star$ for the D\"orfler 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 D\"orfler marking $\theta > \theta_\star$ the algebraic converges rate can be made arbitrarily small.