On $p$-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911500758679552 |
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| author | Chaumont-Frelet, Théophile Dong, Zhaonan Gantner, Gregor Vohralík, Martin |
| author_facet | Chaumont-Frelet, Théophile Dong, Zhaonan Gantner, Gregor Vohralík, Martin |
| contents | 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 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 {\sl a posteriori} criterion is always satisfied for one or a few local mesh refinement steps by newest-vertex bisection. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08887 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $p$-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators Chaumont-Frelet, Théophile Dong, Zhaonan Gantner, Gregor Vohralík, Martin Numerical Analysis 65N12, 65N30, 65N50 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 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 {\sl a posteriori} criterion is always satisfied for one or a few local mesh refinement steps by newest-vertex bisection. |
| title | On $p$-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators |
| topic | Numerical Analysis 65N12, 65N30, 65N50 |
| url | https://arxiv.org/abs/2603.08887 |