Finite element differential forms on cubical meshes

TL;DR

This paper introduces a comprehensive family of finite element differential forms on cubical meshes across dimensions, unifying and extending existing elements for stable numerical discretizations.

Contribution

It develops a unified framework for finite element differential forms on cubical meshes, including new spaces and generalizations of existing elements, applicable in multiple dimensions.

Findings

Includes all polynomial degrees and form degrees in the family.

Constructs finite element subcomplexes satisfying finite element exterior calculus.

Provides stable discretizations for various differential equations.

Abstract

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 serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the serendipity spaces, and new H(curl) and H(div) finite element spaces. Spaces in the family can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, and hence can be used for stable discretization of a variety of problems. The construction and properties of the spaces are established in a uniform manner using finite element exterior calculus.