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Corrigendum to “Convergence of adaptive, discontinuous Galerkin methods”
Mathematics of Computation ( IF 2.2 ) Pub Date : 2020-12-21 , DOI: 10.1090/mcom/3611 Christian Kreuzer , Emmanuil H. Georgoulis
Mathematics of Computation ( IF 2.2 ) Pub Date : 2020-12-21 , DOI: 10.1090/mcom/3611 Christian Kreuzer , Emmanuil H. Georgoulis
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 of nested admissible partitions produced by the adaptive discontinuous Galerkin method (ADGM) we have , and . In the first line of the proof of [KG18, Lemma 11 on p. 2620], we used that
where denotes the Lebesgue measure. This, however, is not true in general, since there are counter examples where is dense in and 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.
中文翻译:
“自适应,不连续Galerkin方法的收敛”更正
摘要:我们最近的论文[KG18](Math。Comp。87(2018),no.314,2611-2640)中的引理11的第一个陈述是不正确的:对于由自适应不连续Galerkin方法产生的嵌套可允许分区的序列(ADGM)我们有,和。在[KG18]证明的第一行中,引号11 p。2620],我们使用了
其中表示勒贝格测度。但是,总的来说,这是不正确的,因为在一些反例中, 下面,我们提出所需的较小修改,以完成对证明[KG18] ADGM收敛的主要结果的证明,并解决有关dG范式破损的一些错字。可从arXiv:1909.12665v2获得本文的完整版本的更正。
更新日期:2021-01-05
where denotes the Lebesgue measure. This, however, is not true in general, since there are counter examples where is dense in and
- [DGK19] A. Dominicus, F. Gaspoz, and C. Kreuzer,
Convergence of an adaptive -interior penalty galerkin method for the biharmonic problem,
1909.12665v2, 2020. Tech.report, Fakultät für Mathematik, TU Dortmund, January 2019, Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 593. - [KG18]
中文翻译:
“自适应,不连续Galerkin方法的收敛”更正
摘要:我们最近的论文[KG18](Math。Comp。87(2018),no.314,2611-2640)中的引理11的第一个陈述是不正确的:对于由自适应不连续Galerkin方法产生的嵌套可允许分区的序列(ADGM)我们有,和。在[KG18]证明的第一行中,引号11 p。2620],我们使用了
其中表示勒贝格测度。但是,总的来说,这是不正确的,因为在一些反例中,
- [DGK19] A. Dominicus,F。Gaspoz和C. Kreuzer,针对双谐波问题
的自适应内部惩罚伽勒金方法的收敛性,
1909.12665v2,2020年。Tech.report,FakultätfürMathematik,多特蒙德大学,2019年1月,Ergebnisberichte德昂大学数学研究所(Nummer 593)。 - [KG18]