Volume 57, Number 1, January-February 2023
|Page(s)||329 - 366|
|Published online||21 February 2023|
Guaranteed contraction of adaptive inexact hp-refinement strategies with realistic stopping criteria
Inria, 2 rue Simone Iff, 75589 Paris, France
2 CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée, France
* Corresponding author: firstname.lastname@example.org
Accepted: 29 September 2022
The purpose of this contribution it to theoretically analyze the adaptive refinement strategies for conforming hp-finite element approximations of elliptic problems proposed for exact algebraic solvers in (P. Daniel, A. Ern, I. Smears and M. Vohralk, Comput. Math. Appl. 76 (2018) 967–983.) and for inexact algebraic solvers in (P. Daniel, A. Ern, and M. Vohralk, Comput. Methods Appl. Mech. Eng. 359 (2020) 112607.). Both of these strategies are driven by guaranteed equilibrated flux energy error estimators. The employed hp-refinement criterion stems from solving two separate local residual problems posed only on the patches of elements around marked vertices selected by a bulk-chasing criterion. In the above references, we have derived a fully computable guaranteed bound on the ratio of the error on two successive steps of the hp-adaptive loop. Here, our focus is to prove that this ratio is uniformly smaller than one, and thus the convergence of the adaptive and adaptive inexact hp-refinement strategies. To be able to achieve this goal, we have to introduce some additional assumptions on the h- and p-refinements, namely an extension of the marked region, as well as a sufficient h- or p-refinement of each marked patch. We investigate two such strategies, where one ensures a polynomial-degree-robust guaranteed contraction. In the inexact case, a sufficiently precise stopping criterion for the algebraic solver is requested, but this criterion remains fully computable and also realistic in the sense that in our numerical experiments, it does not request the algebraic error to be excessively small in comparison with the total error.
Mathematics Subject Classification: 65N30 / 65N15 / 65N50
Key words: Elliptic problem / finite element method / a posteriori error estimate / equilibrated flux / hp-adaptivity / inexact solver / error reduction / convergence analysis
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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