Adaptively refined multilevel spline spaces from generating systems

The truncated basis of adaptively refined multilevel spline spaces was introduced by Giannelli et al. (2012, 2013). It possesses a number of advantages, including the partition of unity property, decreased support of the basis functions, preservation of coefficients and strong stability, that may make it highly useful for geometric modeling and numerical simulation. We generalize this construction to hierarchies of spaces that are spanned by generating systems that potentially possess linear dependencies. This generalization requires a modified framework, since the existing approach relied on the linear independence of the functions generating the spaces in the hierarchy. Many results, such as the preservation of coefficients, can be extended to the more general setting. As applications of the modified framework, we introduce a hierarchy of hierarchical B-splines, which enables us to perform local refinement in the presence of features, and we also extend the adaptive multilevel framework to spaces spanned by Zwart-Powell (ZP) elements, which are special box splines defined on the criss-cross grid. In the latter case we show how to identify the linear dependencies that are present in the truncated hierarchical generating system and use this result to perform adaptive surface fitting with multilevel ZP elements.