Foundations of space-time finite element methods: polytopes, interpolation, and integration

TL;DR

This paper develops the numerical foundation for four-dimensional space-time finite element methods by constructing elements, polynomial bases, and quadrature rules, enabling more accurate and efficient simulations in 4D settings.

Contribution

It introduces a systematic procedure for constructing 4D elements, explicit orthonormal bases, and symmetric quadrature rules capable of high-degree polynomial integration.

Findings

Explicit expressions for polynomial bases on 4-polytopes

Quadrature rules that integrate degree 17 polynomials exactly

Successful numerical experiments validating the methods

Abstract

The main purpose of this article is to facilitate the implementation of space-time finite element methods in four-dimensional space. In order to develop a finite element method in this setting, it is necessary to create a numerical foundation, or equivalently a numerical infrastructure. This foundation should include a collection of suitable elements (usually hypercubes, simplices, or closely related polytopes), numerical interpolation procedures (usually orthonormal polynomial bases), and numerical integration procedures (usually quadrature rules). It is well known that each of these areas has yet to be fully explored, and in the present article, we attempt to directly address this issue. We begin by developing a concrete, sequential procedure for constructing generic four-dimensional elements (4-polytopes). Thereafter, we review the key numerical properties of several canonical elements: the tesseract, tetrahedral prism, and pentatope. Here, we provide explicit expressions for orthonormal polynomial bases on these elements. Next, we construct fully symmetric quadrature rules with positive weights that are capable of exactly integrating high-degree polynomials, e.g. up to degree 17 on the tesseract. Finally, the quadrature rules are successfully tested using a set of canonical numerical experiments on polynomial and transcendental functions.