On Basis Constructions in Finite Element Exterior Calculus

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 families of finite element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between finite element spaces. These structural results show, for example, how to transfer linear dependencies between canonical spanning sets, or give a new derivation of the degrees of freedom

Symmetry and Invariant Bases in Finite Element Exterior Calculus

We study symmetries of bases and spanning sets in finite element exterior calculus using representation theory. The group of affine symmetries of a simplex is isomorphic to a permutation group and represented on simplicial finite element spaces by the pullback action. We want to know which vector-valued finite element spaces have bases that are invariant under permutation of vertex indices. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djokovic and Malzan's classification of monomial irreducible representations of the symmetric group and use symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Invariant bases are constructed in dimensions two and three for different spaces of finite element differential forms.Comment: 27 pages. Submitte

Conforming Finite Element Function Spaces in Four Dimensions, Part 1: Foundational Principles and the Tesseract

The stability, robustness, accuracy, and efficiency of space-time finite element methods crucially depend on the choice of approximation spaces for test and trial functions. This is especially true for high-order, mixed finite element methods which often must satisfy an inf-sup condition in order to ensure stability. With this in mind, the primary objective of this paper and a companion paper is to provide a wide range of explicitly stated, conforming, finite element spaces in four-dimensions. In this paper, we construct explicit high-order conforming finite elements on 4-cubes (tesseracts); our construction uses tools from the recently developed `Finite Element Exterior Calculus'. With a focus on practical implementation, we provide details including Piola-type transformations, and explicit expressions for the volumetric, facet, face, edge, and vertex degrees of freedom. In addition, we establish important theoretical properties, such as the exactness of the finite element sequences, and the unisolvence of the degrees of freedom.Comment: 35 pages, 1 figure, 1 tabl