On p-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators
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 of degree p−1, 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.
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BibTeX
@misc{chaumontfrelet2026probust,
title = {{On $p$-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators}},
author = {Théophile Chaumont-Frelet and Zhaonan Dong and Gregor Gantner and Martin Vohralík},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://inria.hal.science/hal-05544956v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On p-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators
AU - Théophile Chaumont-Frelet
AU - Zhaonan Dong
AU - Gregor Gantner
AU - Martin Vohralík
PY - 2026
DA - 2026/09/10
PB - HAL
UR - https://inria.hal.science/hal-05544956v2
AB - 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 of degree p−1, 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.
ER -