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.
翻译:本条的主要目的是促进四维空间中实施时空限定元素方法。 为了在这一设置中开发一个有限元素方法, 有必要在此设置一个数字基础, 或等效的数字基础设施 。 此基础应该包括一系列合适的元素( 通常是超立方、 implices 或密切相关的多面体)、 数字内插程序( 通常的 orphory 多元基 ) 和数字集成程序( 通常的四维规则 ) 。 众所周知, 这些区域中的每一区域还有待充分探索, 而在目前的条款中, 我们试图直接解决这一问题。 我们首先为构建通用的四维元素( 4- polytopes ) 开发一个具体、 顺序程序 。 之后, 我们审查几种恒星元素的关键数字属性( 通常是超立方体、 四面镜棱柱和五面体 ) 。 这里, 我们对这些元素的正对称二次矩阵规则的表达方式。 接下来, 我们构建了完全对称的二次等量规则, 其正重能够完全整合高度的四维系的四维元素规则 。 。 。 将最终测试到最后一级 。 的磁度 。