Nodal Finite Element de Rham Complexes

TL;DR

This paper develops new finite element de Rham complexes with higher smoothness and nodal degrees of freedom, simplifying basis construction and reducing degrees of freedom compared to classical elements.

Contribution

It introduces novel finite element de Rham sequences with arbitrary polynomial degrees, higher regularity, and nodal DoFs, generalizing Hermite and Lagrange elements.

Findings

Reduced global degrees of freedom compared to classical elements

Basis functions are more canonical and easier to construct

Finite elements for H(div) with regularity 2 match known nonstandard elements

Abstract

We construct 2D and 3D finite element de Rham sequences of arbitrary polynomial degrees with extra smoothness. Some of these elements have nodal degrees of freedom (DoFs) and can be considered as generalisations of scalar Hermite and Lagrange elements. Using the nodal values, the number of global degrees of freedom is reduced compared with the classical N\'{e}d\'{e}lec and Brezzi-Douglas-Marini (BDM) finite elements, and the basis functions are more canonical and easier to construct. Our finite elements for ${H}(\mathrm{div})$ with regularity $r=2$ coincide with the nonstandard elements given by Stenberg (Numer Math 115(1): 131-139, 2010). We show how regularity decreases in the finite element complexes, so that they branch into known complexes. The standard de Rham complexes of Whitney forms and their higher order version can be regarded as the family with the lowest regularity. The construction of the new families is motivated by the finite element systems.%, and we also establish local exact sequences (geometric decomposition) for the new elements.