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.

翻译：本文给出了五胞体和四面体棱柱单元上共形有限元函数空间、基函数和自由度的显式表达式。更一般地，我们的目标是构建与精心选择的de Rham上同调无穷维空间保持共形的有限元函数空间。本文是Nigam和Williams（2023）所著姊妹论文《符合四维共形有限元函数空间，第一部分：基本原理与超立方体》的自然延续。与第一部分不同，本文聚焦于两种最常见但并非在所有四个坐标方向都具有完整张量积结构的单元。我们注意到这些单元频繁出现在现有的时空有限元方法中。为了构建有限元空间，我们利用了近期发展的"有限元外微积分"的强有力技术。随后，我们将结果转化为线性代数（向量与矩阵）的通用语言，以方便科学家和工程师实现。