Ck and C0 hp-finite elements on d-dimensional meshes with arbitrary hanging nodes

https://doi.org/10.1016/j.finel.2021.103529Get rights and content
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Highlights

  • Ck basis functions on paraxial d-dimensional rectangular meshes with arbitrary hanging nodes are constructed.

  • Arbitrary polynomial degree distributions are supported.

  • Global DOFs are constructed from shape functions on appropriate unions of mesh elements.

  • The differentiability properties are rigorously proven.

  • Exponential convergence is achieved for nonsmooth problems in 2D and 3D.

Abstract

In this paper, the construction of Ck 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.

Keywords

hp-FEM
Arbitrary hanging nodes
Differentiable basis functions

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The authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis” under the project SCHR 1244/4-2.