We introduce a framework for spline spaces of hierarchical type, based on a parent–children relation, which is very convenient for the analysis as well as the implementation of adaptive isogeometric methods. This framework exploits the innate refinement by functions in the B-splines context, rather than by elements or cells, which is more natural in the finite element context. Furthermore, it entails a new language to handle hierarchical spline spaces, which allows to tackle fundamental questions in a very simple manner. For example, it makes it simple to create hierarchical bases with several desired properties with a refinement procedure which has linear complexity, i.e., the resulting bases have cardinality bounded by the number of initially marked functions.

我们基于父子关系引入了用于分层类型的样条空间的框架，这对于分析和实现自适应等几何方法非常方便。此框架通过B样条曲线上下文中的函数而不是元素或单元来利用先天的细化，这在有限元素上下文中更为自然。此外，它需要一种新的语言来处理层次样条空间，从而可以以非常简单的方式解决基本问题。例如，通过具有线性复杂度的细化过程，可以轻松创建具有多个所需属性的层次基础，即所得基础的基数受初始标记函数的数量限制。