Abstract
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 Dörfler
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 Dörfler marking this yields an optimal decay rate in terms
of degrees of freedom.
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