C ^k and C ⁰ hp-finite elements on d-dimensional meshes with arbitrary hanging nodes

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 C ⁰ 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 C ⁰ 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.