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 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 finite elements, and the basis functions are more canonical and easier to construct. Our finite elements for H(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 finite element systems.
Bibliographical noteFunding Information:
Snorre H. Christiansen is supported by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, Project 278011 STUCCOFIELDS. Jun Hu is supported by the NSFC (Projects 11625101, 91430213, 11421101). Kaibo Hu is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement 339643.