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

2022/03/25 by Matthew W. Scroggs, Jørgen S. Dokken, Chris Richardson +2 · 221 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #Algorithm #Computational Geometry and Mesh Generation #Computer science #Curl (programming language) #Degree (music) #Degrees of freedom (physics and chemistry) #Finite element method #Geometry #Hexahedron #Mathematics #Polygon mesh #Polytope #Quadrilateral #Rigid transformation #Simplex #Transformation (genetics)

paper · pdf · doi:10.1145/3524456

published in ACM Transactions on Mathematical Software 48(2), 1-23 (Association for Computing Machinery)

openalex publication_date 2022/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

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.

Citations

• Mixed finite elements in ?3
• Firedrake

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