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Composite basis functions for pyramidal elements on the spaces $H^1(\Omega)$, $H(\mathrm{curl},\Omega)$, $H(\mathrm{div},\Omega)$ and $L^2(\Omega)$ are presented. In particular, we construct the lowest-order composite pyramidal elements and…
In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these…
In this paper, the first family of conforming finite element divdiv complexes on cuboid grids in three dimensions is constructed. Besides, a new family of conforming finite element divdiv complexes with enhanced smoothness on tetrahedral…
We present degrees of freedom to accompany the approximation spaces already presented in a companion paper and thus complete the definition of families of high-order conforming finite elements on pyramids for the spaces of the de Rham…
We discuss the construction of finite element spaces of differential forms which satisfy the crucial assumptions of the finite element exterior calculus, namely that they can be assembled into subcomplexes of the de Rham complex which admit…
We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the…
Finite element de Rham complexes and finite element Stokes complexes with various smoothness in three dimensions are systematically constructed. First smooth scalar finite elements in three dimensions are derived through a non-overlapping…
This paper presents the first family of conforming finite element divdiv complexes on tetrahedral grids in three dimensions. In these complexes, finite element spaces of $H(\text{divdiv},\Omega;\mathbb{S})$ are from a current preprint [Chen…
We construct new families of direct serendipity and direct mixed finite elements on general planer convex polygons that are $H^1$ and $H(div)$ conforming, respectively, and possess optimal order of accuracy for any order. They have a…
The aim of this article is to derive discontinuous finite elements vector spaces which can be put in a discrete de-Rham complex for which an harmonic gap property may be proven. First, discontinuous finite element spaces inspired by…
In this study, two-dimensional finite element complexes with various levels of smoothness, including the de Rham complex, the curldiv complex, the elasticity complex, and the divdiv complex, are systematically constructed. Smooth scalar…
We construct smooth finite element de Rham complexes in two space dimensions. This leads to three families of curl-curl conforming finite elements, two of which contain two existing families. The simplest triangular and rectangular finite…
In this paper, we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some…
In this paper, we present explicit expressions for conforming finite element function spaces, basis functions, and degrees of freedom on the pentatope and tetrahedral prism elements. More generally, our objective is to construct finite…
Using the language of finite element exterior calculus, we define two families of $H^1$-conforming finite element spaces over pyramids with a parallelogram base. The first family has matching polynomial traces with tensor product elements…
We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously…
We construct finite element Stokes complexes on tetrahedral meshes in three-dimensional space. In the lowest order case, the finite elements in the complex have 4, 18, 16, and 1 degrees of freedom, respectively. As a consequence, we obtain…
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…
In this paper we present the notion of de Rham cohomology with compact support for diffeological spaces. Moreover we shall discuss the existence of three long exact sequences. As a concrete example, we show that long exact sequences exist…
A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A…