Efficient Assembly of H(div) and H(curl) Conforming Finite Elements

TL;DR

This paper presents an efficient, automated approach for assembling finite element variational forms on H(div) and H(curl) spaces, leveraging tensor decomposition, proper basis mapping, and orientation strategies.

Contribution

It introduces a novel decomposition method and orientation handling for Piola-mapped elements, enabling easier and faster assembly of finite element forms on H(div) and H(curl).

Findings

Achieves automated and efficient assembly process.

Extends representation theorem to Piola-mapped elements.

Simplifies handling of facet normal and tangent directions.

Abstract

In this paper, we discuss how to efficiently evaluate and assemble general finite element variational forms on H(div) and H(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(div) and H(curl) in close to mathematical notation.