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.

翻译：时空有限元方法的稳定性、鲁棒性、精度及效率关键取决于试验函数与试探函数的逼近空间选择，这对高阶混合有限元方法尤为重要——此类方法需满足inf-sup条件以确保稳定性。基于此，本文及其姊妹篇的首要目标是为四维空间提供一系列明确表述的协调有限元空间。在本文中，我们在四维超立方体（tesseract）上构造显式高阶协调有限元，该构造借助近期发展的“有限元外微分”工具。面向实际实现，我们提供了Piola型变换的细节，以及体积元、面元、棱元、边元及节点自由度的显式表达式。此外，我们建立了重要理论性质，包括有限元序列的精确性及自由度的唯一可解性。