Rate optimality of AFEM with maximum marking

Abstract

For Poisson's equation, we prove optimal algebraic convergence rates for adaptive conforming finite element methods driven by standard maximum marking in arbitrary space dimension, for every marking parameter in $(0,1]$. Previously, instance optimality of a modified maximum marking was proved in [L. Diening, C. Kreuzer, and R. Stevenson, Instance optimality of the adaptive maximum strategy, Found. Comput. Math., 16 (2016), pp. 33--68]. The basic ingredients of our anlaysis consists of discrete reliability, discrete efficiency, overlay and closure estimates of local mesh refinement. The discrete efficiency requires a facet-interior node property of simplicial mesh refinement. The novel part of the optimality analysis is an elementary integral-sum argument that converts energy decrease into a cumulative cardinality bound of marked elements, yielding rate optimality without a contraction argument.