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

摘要: The main purpose of this article is to facilitate the implementation space-time finite element methods in four-dimensional space. In order develop a method setting, it necessary create numerical foundation, or equivalently infrastructure. This foundation should include collection suitable elements (usually hypercubes, simplices, closely related polytopes), interpolation procedures orthonormal polynomial bases), and integration quadrature rules). It well known that each these areas has yet be fully explored, present article, we attempt directly address issue. We begin by developing concrete, sequential procedure for constructing generic (4-polytopes). Thereafter, review key properties several canonical elements: tesseract, tetrahedral prism, pentatope. Here, provide explicit expressions bases on elements. Next, construct symmetric rules with positive weights are capable exactly integrating high-degree polynomials, e.g. up degree 17 tesseract. Finally, successfully tested using set experiments transcendental functions.