Back to Basix: Construction of arbitrary order finite element DOF maps on polygonal and polyhedral cell meshes

We develop an approach to generating degree-of-freedom (DOF) 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 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, divergence-conforming Raviart–Thomas and curl-conforming Nédélec 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. This method has been implemented in Basix, FEniCSx's element tabulator.

References

• [1] Matthew W Scroggs, Jørgen S Dokken, Chris N Richardson and Garth N Wells. Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes, arΧiv: 2102.11901, 2021.