A family of symmetric, optimized quadrature rules for pentatopes. (1st September 2020) - Williams, David M.; Frontin, Cory V.; Miller, Edward A.; Darmofal, David L.
Abstract: This article introduces a new family of quadrature rules for integrating smooth functions on 4-dimensional simplex elements (i.e. pentatopes). These quadrature rules have 1, 5, 15, 35, 70, and 126 points, and are capable of exactly integrating polynomials of degrees 1, 2, 3, 5, 6, and 8, respectively. One main advantage of these rules, is that they have a 'pentatopic number' of points, which means that they can exactly interpolate 4-dimensional polynomials of degrees 0 through 5. As a result, the proposed rules can be used for both quadrature and interpolation purposes. Furthermore, these rules are fully symmetric, as they remain invariant under affine transformations (rotations and reflections) of the pentatope back to itself. In addition, these rules are optimal, in the sense that the truncation error associated with each rule has been minimized via a rigorous optimization procedure. Finally, they have positive weights, and all quadrature points reside strictly within the interior of the pentatope.