Related papers: Duality in finite element exterior calculus

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary…

In 2006, Arnold, Falk, and Winther developed finite element exterior calculus, using the language of differential forms to generalize the Lagrange, Raviart--Thomas, Brezzi--Douglas--Marini, and N\'ed\'elec finite element spaces for…

We show how the high order finite element spaces of differential forms due to Raviart-Thomas-N\'edelec-Hiptmair fit into the framework of finite element systems, in an elaboration of the finite element exterior calculus of…

Arnold, Falk, & Winther, in "Finite element exterior calculus, homological techniques, and applications" (2006), show how to geometrically decompose the full and trimmed polynomial spaces on simplicial elements into direct sums of…

The tensor product of two differential forms of degree $p$ and $q$ is a multilinear form that is alternating in its first $p$ arguments and alternating in its last $q$ arguments. These forms, which are known as double forms or…

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…

This paper constructs a unified family of nonconforming finite element spaces for $H\Lambda^k$ in $\mathbb{R}^n$ ($0\leqslant k\leqslant n$, $n\geqslant 1$). The spaces employ piecewise Whitney forms as shape functions, and include the…

We give a systematic and self-contained account of the construction of geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the…

The principal aim of this work is to provide a family of unisolvent and minimal physical degrees of freedom, called weights, for N\'ed\'elec second family of finite elements. Such elements are thought of as differential forms $…

We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields,…

We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. First we develop a framework in which we show how tools from algebraic topology can be applied to the study of their cohomological…

We study dualities between classes of relational topological structures, given by Hom-functors. We show that there exists a 2-element structure with infinitely many relations, which reconstructs all other structures generated by a 2-element…

This paper introduces a novel tangential-normal ($t$-$n$) decomposition for finite element differential forms, presenting a new framework for constructing bases in finite element exterior calculus. The main contribution is the development…

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 show Poincar\'e Duality for $\mathbf{F}_p$-\'etale cohomology of a smooth proper rigid-analytic space over a non-archimedean field $K$ of mixed characteristic $(0, p)$. It positively answers the question raised by P. Scholze in [Sch13a].…

We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic…

In this paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces $ H^p\ (0<p\leqslant 1) $ and functions in their dual spaces. Our decompositions are based on…

We develop finite element spaces of symmetric tensor products of two-forms with polynomial coefficients. In three dimensions, these give higher order finite element spaces of matrix fields with normal-normal continuity, which have…

We construct and study a morphism of spectra implementing the Anderson duality of topological modular forms ($\mathrm{TMF}$). Its differential version will then be introduced, allowing us to pair elements of $\pi_d\mathrm{TMF}$ with spin…

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce…