Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes

TL;DR

This paper introduces a local, permutation-based method for constructing degree-of-freedom maps for arbitrary order finite element spaces on various cell shapes, including complex polytopes, without relying on mesh orientation.

Contribution

It presents a novel, local approach for generating degree-of-freedom maps applicable to any cell shape and type, improving flexibility over existing methods.

Findings

Applicable to various element types including Lagrange, divergence-, and curl-conforming elements

Works on general polytopal meshes with mixed cell types

Implemented in an open-source finite element library

Abstract

We develop a method for generating degree-of-freedom maps for arbitrary order 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 Lagrange, 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 an open-source finite element library.