Construction of Arbitrary Order Finite Element Degree-of-Freedom Maps on Polygonal and Polyhedral Cell Meshes

Author:

Scroggs Matthew W.1ORCID,Dokken Jørgen S.1ORCID,Richardson Chris N.2ORCID,Wells Garth N.1ORCID

Affiliation:

1. Department of Engineering, Universityof Cambridge, Cambridge, United Kingdom

2. BP Institute, University of Cambridge, Cambridge, United Kingdom

Abstract

We develop a method for generating degree-of-freedom maps for arbitrary order Ciarlet-type finite element spaces for any cell shape. The approach is based on the composition of permutations and transformations by cell sub-entity. Current approaches to generating degree-of-freedom maps for arbitrary order problems typically rely on a consistent orientation of cell entities that permits the definition of a common local coordinate system on shared edges and faces. However, while orientation of a mesh is straightforward for simplex cells and is a local operation, it is not a strictly local operation for quadrilateral cells and, in the case of hexahedral cells, not all meshes are orientable. The permutation and transformation approach is developed for a range of element types, including arbitrary degree Lagrange, serendipity, and divergence- and curl-conforming elements, and for a range of cell shapes. The approach is local and can be applied to cells of any shape, including general polytopes and meshes with mixed cell types. A number of examples are presented and the developed approach has been implemented in open-source libraries.

Funder

EPSRC

Publisher

Association for Computing Machinery (ACM)

Subject

Applied Mathematics,Software

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5. Unified form language

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