Three families of grad-div-conforming finite elements

TL;DR

This paper introduces three new families of grad-div conforming finite elements in 3D, with simple elements and proven effectiveness for quad-div problems through numerical validation.

Contribution

It constructs three families of grad-div conforming finite elements in 3D, including the simplest elements with minimal degrees of freedom, and demonstrates their efficiency for quad-div problems.

Findings

Simple elements with 8 and 14 DOFs for tetrahedron and cuboid

Numerical experiments validate correctness and efficiency

Conforming approximations for quad-div problems

Abstract

Several smooth finite element de Rham complexes are constructed in three-dimensional space, which yield three families of grad-div conforming finite elements. The simplest element has only 8 degrees of freedom (DOFs) for a tetrahedron and 14 DOFs for a cuboid. These elements naturally lead to conforming approximations to quad-div problems. Numerical experiments for each family validate the correctness and efficiency of the elements for solving the quad-div problem.