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

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.Comment: 34 pages, 18 figure

Output error behavior for discretizations of ergodic, chaotic systems of ordinary differential equations

The use of numerical simulation for prediction of characteristics of chaotic dynamical systems inherently involves unpredictable processes. In this work, we develop a model for the expected error in the simulation of ergodic, chaotic ordinary differential equation (ODE) systems, which allows for discretization and statistical effects due to unpredictability. Using this model, we then generate a framework for understanding the relationship between the sampling cost of a simulation and the expected error in the result and explore the implications of the various parameters of simulations. Finally, we generalize the framework to consider the total cost—including unsampled spin-up timesteps—of simulations and consider the implications of parallel computational environments to give a realistic model of the relationship between wall-clock time and the expected error in simulation of a chaotic ODE system. </jats:p