In this paper, the construction of C^k basis functions is proposed for paraxial d-dimensional rectangular meshes with arbitrary hanging nodes and arbitrary polynomial degree distributions. The construction is based on the large-support approach introduced in [1] for C⁰ basis functions in 2D and uses hierarchical tensor-product shape functions which combine Hermite shape functions with Gegenbauer polynomials enabling the support of the basis functions to be independent of k (in contrast to basis functions based on B-spline approaches). Moreover, these shape functions allow for an efficient recursive computation of the constraints coefficients in the application of constrained approximation for hanging nodes without the need for collocation. An appropriate indexing of the shape functions is introduced in order to prove the differentiability properties of the basis functions. The construction is also suitable for the extension to C⁰ finite elements on meshes which are not necessarily rectangular. In particular, the orientation problem resulting from differently oriented edges or faces can be appropriately treated within this extension. Numerical examples illustrate the feasibility of the proposed approach. Moreover, some aspects concerning the condition number of the system matrix resulting from the discretization of Poisson's problem are discussed.

本文提出了具有任意悬挂节点和任意多项式分布的近轴d维矩形网格的C ^k基函数的构造。该构造基于[1]中针对2D中的C ⁰基函数的大支持方法，并使用分层的张量积形状函数，该函数将Hermite形状函数与Gegenbauer多项式结合在一起，从而使基函数的支持独立于k（与基于B的基本函数相反-样条方法）。而且，这些形状函数允许在对悬挂节点进行约束近似的应用中对约束系数进行有效的递归计算，而无需搭配使用。为了证明基本函数的可微性，引入了形状函数的适当索引。该结构也适用于扩展到C ⁰网格上的有限元不一定是矩形的。特别地，可以在该扩展范围内适当地处理由不同取向的边缘或面引起的取向问题。数值算例说明了该方法的可行性。此外，还讨论了有关由泊松问题离散化导致的系统矩阵条件数的某些方面。