Efficient Assembly of $H(\mathrm{div})$ and $H(\mathrm{curl})$ Conforming Finite Elements

In this paper, we discuss how to efficiently evaluate and assemble general finite element variational forms on $H(\mathrm{div})$ and $H(\mathrm{curl})$. The proposed strategy relies on a decomposition of the element tensor into a precomputable reference tensor and a mesh-dependent geometry tensor. Two key points must then be considered: the appropriate mapping of basis functions from a reference element, and the orientation of geometrical entities. To address these issues, we extend here a previously presented representation theorem for affinely mapped elements to Piola-mapped elements. We also discuss a simple numbering strategy that removes the need to contend with directions of facet normals and tangents. The result is an automated, efficient, and easy-to-use implementation that allows a user to specify finite element variational forms on $H(\mathrm{div})$ and $H(\mathrm{curl})$ in close to mathematical notation.