Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of $C^1$ isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We introduce a new abstract framework for the definition of hierarchical splines, which replaces the hypothesis of local linear independence for the basis of each level by a weaker assumption. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by $C^1$ splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems.

翻译：等几何分析是一种强大范式，利用样条的高光滑性求解高阶偏微分方程的数值解。然而，标准多元B样条模型的张量积结构难以表示复杂几何形状，且为在一般域上保持高连续性，必须在多片几何结构上采用特殊构造。本文聚焦于带有层次样条的自适应等几何方法，将多片平面域上 $C^1$ 等几何样条空间的构造推广至层次化框架。我们提出了一种新的抽象框架用于定义层次样条，该框架将每一层级基的局部线性独立假设替换为更弱的条件。我们还开发了一种细化算法，确保该条件在适当分级的多片层次网格配置下对 $C^1$ 样条成立，并证明该算法具有线性复杂度。通过求解泊松问题和双调和问题，测试了自适应方法的性能。