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

TL;DR

This paper develops explicit high-order conforming finite element spaces on four-dimensional tesseracts, utilizing finite element exterior calculus to ensure stability, exactness, and practical implementation for advanced space-time finite element methods.

Contribution

It introduces a comprehensive construction of conforming finite element spaces on 4-cubes, including explicit degrees of freedom and theoretical properties, advancing high-dimensional finite element theory.

Findings

Constructed explicit high-order conforming finite elements on 4-cubes

Established exactness of finite element sequences

Provided detailed implementation tools and degrees of freedom

Abstract

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.