Conforming Finite Element Function Spaces in Four Dimensions, Part II: The Pentatope and Tetrahedral Prism

TL;DR

This paper develops explicit conforming finite element function spaces for the pentatope and tetrahedral prism in four dimensions, extending previous work and focusing on elements used in space-time finite element methods.

Contribution

It introduces new conforming finite element spaces for complex 4D elements without full tensor-product structure, using finite element exterior calculus techniques.

Findings

Explicit basis functions and degrees of freedom are provided.

The constructed spaces conform with the de Rham complex in 4D.

Results are translated into linear algebra for practical implementation.

Abstract

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 element function spaces that maintain conformity with infinite-dimensional spaces of a carefully chosen de Rham complex. This paper is a natural extension of the companion paper entitled "Conforming Finite Element Function Spaces in Four Dimensions, Part I: Foundational Principles and the Tesseract" by Nigam and Williams, (2023). In contrast to Part I, in this paper we focus on two of the most popular elements which do not possess a full tensor-product structure in all four coordinate directions. We note that these elements appear frequently in existing space-time finite element methods. In order to build our finite element spaces, we utilize powerful techniques from the recently developed 'Finite Element Exterior Calculus'. Subsequently, we translate our results into the well-known language of linear algebra (vectors and matrices) in order to facilitate implementation by scientists and engineers.