Foundations of space-time finite element methods: polytopes, interpolation, and integration

Abstract

The main purpose of this article is to facilitate the implementation of spacetime finite element methods in four-dimensional space. Generally speaking, researchers and engineers have previously relied on their intuition and ability to visualize two-and three-dimensional spaces. Unfortunately these attributes do not naturally extend to four-dimensional space, and therefore the development of numerical methods in this context is a unique and difficult challenge. In order to construct a finite element method in four-dimensional space, 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

Enabling four-dimensional conformal hybrid meshing with cubic pyramids

“…A similar approach for generating four-dimensional simplex meshes has been employed by Neumüller and Karabelas in [25]. A detailed survey of four-dimensional structured and unstructured simplicial meshing techniques is provided by Caplan [8] and Frontin et al [18]. Finally, examples of refinement strategies for high-dimensional simplicial meshes have been developed in the work of Brandts et al [6], and Korotov and Křížek [20].…”

“…Finally, we conclude our work by developing numerical integration procedures for the cubic pyramid elements. We note that, while such procedures are already well-established for the tesseract and pentatope elements [18,39], they have yet to be developed for cubic pyramid elements.…”