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

TL;DR

This paper introduces a new $h$-adaptive finite element algorithm for the Poisson equation that guarantees error reduction and optimal convergence rates with constants that are robust against polynomial degree variations.

Contribution

It proposes a $p$-robust $h$-adaptive algorithm driven by equilibrated-flux estimators with proven error contraction and optimal algebraic convergence rates, independent of polynomial degree.

Findings

Error contraction at each step with degree-independent constants.

Optimal algebraic convergence rate achieved under certain marking parameters.

Numerical experiments confirm theoretical robustness and effectiveness.

Abstract

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 {\sl a posteriori} verifiable criterion is satisfied. We further show that this algorithm converges at optimal algebraic rate $s$ if the D\"orfler 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 {\sl a posteriori} criterion is always satisfied for one or a few local mesh refinement steps by newest-vertex bisection.