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.

本文的主要目的是促进在三维空间中实现时空有限元方法。为了在这种情况下开发有限元方法，必须创建一个数值基础，或者等效地建立一个数值基础结构。该基础应包括适当元素（通常是超立方体，单纯形或紧密相关的多态）的集合，数值插值过程（通常是正交多项式基）和数值积分过程（通常是正交规则）。众所周知，这些领域中的每个领域都尚未得到充分探索，在本文中，我们尝试直接解决此问题。我们从开发一个具体的，顺序的过程开始，以构建通用的四维元素（4-多面体）。之后，我们回顾了几种规范元素的关键数值特性：金属镶嵌，四面体棱镜和五角形。在这里，我们基于这些元素为正交多项式提供显式表达式。接下来，我们构造具有正权重的完全对称的正交规则，该规则能够精确积分高阶多项式，例如，在tesseract上最高为17级。最后，使用多项式和超越函数的一组规范数值实验成功地测试了正交规则。直至在tesseract上达到17度。最后，使用多项式和超越函数的一组规范数值实验成功地测试了正交规则。直至在tesseract上达到17度。最后，使用多项式和超越函数的一组规范数值实验成功地测试了正交规则。