# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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by Christian Kreuzer and Emmanuil H. Georgoulis
Math. Comp. 90 (2021), 637-640 Request permission

## Abstract:

The first statement of Lemma 11 in our recent paper [KG18] (Math. Comp. 87 (2018), no. 314, 2611–2640) is incorrect: For the sequence $\{\mathcal {G}_{k}\}_k$ of nested admissible partitions produced by the adaptive discontinuous Galerkin method (ADGM) we have $\mathcal {G}^+\coloneq \bigcup _{k\ge 0}\bigcap _{j\ge k}\mathcal {G}_{j}$, and $\Omega ^+\coloneq \operatorname {interior}\left (\bigcup \{ E: E\in \mathcal {G}^+\}\right )$. In the first line of the proof of [KG18, Lemma 11 on p. 2620], we used that \begin{align*} |\Omega |=|\operatorname {interior}(\Omega \setminus \Omega ^+)|+|\Omega ^+|, \end{align*} where $|\cdot |$ denotes the Lebesgue measure. This, however, is not true in general, since there are counter examples where $\Omega ^+$ is dense in $\Omega$ and $0=|\operatorname {interior}(\Omega \setminus \Omega ^+)|<|\Omega \setminus \Omega ^+|.$

Below, we present the required minor modifications to complete the proof of the main result stating convergence of the ADGM of [KG18] and address some typos regarding the broken dG-norm. A corrected full version of the article is available at arXiv:1909.12665v2.

References
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• Christian Kreuzer
• Affiliation: Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44227 Dortmund, Germany
• MR Author ID: 833122
• ORCID: 0000-0003-2923-4428
• Email: christian.kreuzer@tu-dortmund.de
• Emmanuil H. Georgoulis
• Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom; and Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou 157 80, Greece
• MR Author ID: 750860
• Email: Emmanuil.Georgoulis@le.ac.uk
• Received by editor(s): October 1, 2019
• Received by editor(s) in revised form: September 8, 2020
• Published electronically: December 21, 2020
• Additional Notes: The research of the first author was supported by DFG research grant KR 3984/5-1.