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、精确性及效率关键取决于测试函数与试探函数的逼近空间选择，这对需满足inf-sup条件以确保稳定性的高阶混合有限元方法尤为关键。基于此，本文及其姊妹篇的首要目标是提供四维空间中一系列显式定义的协调有限元空间。本文在四维超立方体上构造了显式高阶协调有限元——该构造运用了近期发展的“有限元外微积分”工具。我们以实际实现为导向，提供了包括Piola型变换在内的详细信息，以及体积、面、边、棱和顶点自由度的显式表达式。此外，我们建立了有限元序列的恰当性与自由度的唯一可解性等重要理论性质。