Nodal Finite Element de Rham Complexes
For computational scientists using finite element methods, this provides more efficient and canonical basis functions for H(div) and H(curl) spaces, though the improvement is incremental over existing higher-order elements.
This paper constructs 2D and 3D finite element de Rham sequences of arbitrary polynomial degrees with extra smoothness, using nodal degrees of freedom that reduce global degrees of freedom compared to classical Nédélec and BDM elements. The new families generalize scalar Hermite and Lagrange elements and branch into known complexes as regularity decreases.
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 Nédélec 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.