Remarks on Poincaré and interpolation estimates for Truncated Hierarchical B-splines

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.