An explicit construction of optimized interpolation points on the 4-simplex

In this work, a family of symmetric interpolation points are generated on the four-dimensional simplex (i.e., the pentatope). Interpolation points are crucial toward the furtherance of space–time finite element methods, which themselves find applications in shock fitting, fluid structure interactions, and even rotating detonation engines. The points generated herein are optimized in order to minimize the Lebesgue constant. The process of generating these points closely follows that outlined by Warburton (J Eng Math 56:247–262, 2006). Here, Warburton generated optimal interpolation points on the triangle and tetrahedron by formulating explicit geometric warping and blending functions and applying these functions to equidistant nodal distributions. The locations of the resulting points were Lebesgue-optimized. In our work, we extend this procedure to four dimensions and construct interpolation points on the pentatope up to order 10. The Lebesgue constants of our nodal sets are calculated and are shown to outperform those of equidistant nodal distributions, as well as many of the point distributions of Isaac (SIAM J Sci Comput 42(6):4046–4062, 2020).