This paper should be considered as an addendum to [A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci. 26 (2016) 1–25] and [A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates, Math. Models Methods Appl. Sci. 27 (2017) 2781–2802] where Poincaré and approximation estimates are used as theoretical tools to study properties of adaptive numerical methods based on hierarchical B-splines. After noting that the support of truncated hierarchical B-splines may be disconnected (and thus no Poincaré estimate can hold), we study minimal extensions of their support on suitable mesh configurations such that (i) Poincaré estimates can be established on them and (ii) their overlaps stay independent of the number of levels. The Poincaré estimates proposed in this note should replace the ones used in the proofs of Theorem 11 and Lemma 7 in [A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci. 26 (2016) 1–25] and [A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates, Math. Models Methods Appl. Sci. 27 (2017) 2781–2802], respectively, in order to include the most general meshes, i.e. the cases when the support of truncated basis functions can be disconnected.

中文翻译：

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关于截断层次 B 样条的 Poincaré 和插值估计的备注

本文应被视为 [A. Buffa 和 C. Gianneli，具有层次样条的自适应等几何方法：误差估计器和收敛，数学。模型方法应用程序。科学。 26(2016) 1-25] 和 [A. Buffa 和 C. Gianneli，具有层次样条的自适应等几何方法：最优性和收敛速度，数学。模型方法应用程序。科学。 27(2017) 2781–2802] 其中庞加莱和近似估计被用作理论工具来研究基于分层 B 样条的自适应数值方法的特性。在注意到截断的分层 B 样条的支持可能是断开的（因此庞加莱估计不能成立）之后，我们研究了它们对合适网格配置的支持的最小扩展，以便（i）可以在它们上建立庞加莱估计和（ii ) 它们的重叠与层数无关。本说明中提出的 Poincaré 估计应该取代 [A. Buffa 和 C. Gianneli，具有层次样条的自适应等几何方法：误差估计器和收敛，数学。模型方法应用程序。科学。 26(2016) 1-25] 和 [A. Buffa 和 C. Gianneli，具有层次样条的自适应等几何方法：最优性和收敛速度，数学。模型方法应用程序。科学。 27(2017) 2781–2802] 分别是为了包括最一般的网格，即截断基函数的支持可以断开的情况。