Decay rates of adaptive finite elements with Dorfler marking

We investigate the decay rate for an adaptive finite element discretization of a second order linear, symmetric, elliptic PDE. We allow for any kind of estimator that is locally equivalent to the standard residual estimator. This includes in particular hierarchical estimators, estimators based on the solution of local problems, estimators based on local averaging, equilibrated residual estimators, the ZZ-estimator, etc. The adaptive method selects elements for refinement with Dorfler marking and performs a minimal refinement in that no interior node property is needed. Based on the local equivalence to the residual estimator we prove an error reduction property. In combination with minimal Dorfler marking this yields an optimal decay rate in terms of degrees of freedom.