On p-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators

Building on existing $hp$-adaptive algorithms driven by equilibrated-flux estimators from [ESAIM Math. Model. Numer. Anal. 57 (2023), 329-366] and the references therein, we propose a novel h-adaptive algorithm for a fixed polynomial degree p. We consider a conforming finite element discretization of the Poisson equation in two or three space dimensions. Supposing piecewise polynomial right-hand side, we show that the algorithm yields error contraction at each step, with a contraction factor that is independent of p provided that a certain a posteriori verifiable criterion is satisfied. We further show that this algorithm converges at optimal algebraic rate s if the Dörfler marking parameter is chosen below some specified p-independent upper threshold. The constants involved here are p-robust, although they may depend on the rate s. The theoretical results are supported by numerical experiments, in which the a posteriori criterion is always satisfied for one or a few local mesh refinement steps by newest-vertex bisection.