On the linear independence of truncated hierarchical generating systems

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• 作者：Ur&scaron ; ka Zore^a ; ^{urska.zore@jku.at" class="auth_mail" title="E-mail the corresponding author} ; Bert Jü ; ttler^a ; ^{bert.juettler@jku.at" class="auth_mail" title="E-mail the corresponding author} ; Jiří ; Kosinka^b ; ^{j.kosinka@rug.nl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Multi-level spline space ; Hierarchical generating system ; Truncation ; Doubly hierarchical B-splines ; Zwart-Powell elements ; Hierarchical subdivision splines
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：306
• 期：Complete
• 页码：200-216
• 全文大小：559 K

文摘

Motivated by the necessity to perform adaptive refinement in geometric design and numerical simulation, the construction of hierarchical splines from generating systems spanning nested spaces has been recently studied in several publications. Linear independence can be guaranteed with the help of the local linear independence of the spline basis at each level. The present paper extends this framework in several ways. Firstly, we consider spline functions that are defined on domain manifolds, while the existing constructions are limited to domains that are open subsets of R^d${\mathbb{R}}^d$. Secondly, we generalize the approach to generating systems containing functions which are not necessarily non-negative. Thirdly, we present a more general approach to guarantee linear independence and present a refinement algorithm that maintains this property. The three extensions of the framework are then used in several relevant applications: doubly hierarchical B-splines, hierarchical Zwart-Powell elements, and three different types of hierarchical subdivision splines.