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

• Published in: Applied numerical mathematics 2021-08, Vol.166, p.92-113
• Main Authors: Frontin, Cory V., Walters, Gage S., Witherden, Freddie D., Lee, Carl W., Williams, David M., Darmofal, David L.
• Format: Article
• Language: English
• Subjects: Finite element methods; Pentatope; Quadrature; Space-time; Tesseract; Tetrahedral prism
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2021.03.019

•We develop new sequences of 4D polytopes for uniform and hybrid space-time meshing.•We illustrate the sequences with appropriate 4D to 2D projections.•We recover the standard tesseract, tetrahedral prism, and pentatope elements.•We review the polynomial bases and symmetry groups for these standard elements.•We develop new quadrature/cubature rules for these standard elements. 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.