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

TL;DR

The paper addresses constructing Lebesgue-optimized interpolation points on the 4D simplex (the pentatope) by extending Warburton's explicit warping and blending framework to higher dimension. It develops a four-dimensional generalization of the nodal-point construction, starting from equispaced points and applying dimensionally recursive warping and blending controlled by a single optimization parameter $\alpha$ (with $\alpha=\beta=\gamma$ in the experiments). The authors derive a 4D nodal-indexing formula and compute Lebesgue constants $\Lambda$ to compare against equispaced nodes, finding that the optimized nodes consistently yield smaller $\Lambda$ up to order $p=10$, typically by about a factor of four, except for $p=3$. The results demonstrate symmetry and stability benefits for high-dimensional finite element methods and point to clear paths for extending the methodology to even higher dimensions and for validating performance via additional error analyses.

Abstract

In this work, a family of symmetric interpolation points are generated on the four-dimensional simplex (i.e. the pentatope). These points are optimized in order to minimize the Lebesgue constant. The process of generating these points closely follows that outlined by Warburton in "An explicit construction of interpolation nodes on the simplex," Journal of Engineering Mathematics, 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 ten. The Lebesgue constants of our nodal sets are calculated, and are shown to outperform those of equidistant nodal distributions.

Figures (9)

• Figure 1: Effect of warping and blending functions on points on the reference tetrahedron. Note that points located at vertices, midpoints, and barycenters undergo zero warping and/or blending.
• Figure 2: Optimized interpolation points on a projected version of the pentatope for $p = 5$. Black points appear at the vertices, green points appear on the edges, blue points appear on the triangular faces, red points appear on the tetrahedral facets, and yellow points appear in the interior of the pentatope.
• Figure 3: Optimized interpolation points on a projected version of the pentatope for $p = 6$. Black points appear at the vertices, green points appear on the edges, blue points appear on the triangular faces, red points appear on the tetrahedral facets, and yellow points appear in the interior of the pentatope.
• Figure 4: Optimized interpolation points on a projected version of the pentatope for $p = 7$. Black points appear at the vertices, green points appear on the edges, blue points appear on the triangular faces, red points appear on the tetrahedral facets, and yellow points appear in the interior of the pentatope.
• Figure 5: Optimized interpolation points on a projected version of the pentatope for $p = 8$. Black points appear at the vertices, green points appear on the edges, blue points appear on the triangular faces, red points appear on the tetrahedral facets, and yellow points appear in the interior of the pentatope.