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

TL;DR

This paper extends Warburton's method to construct symmetric, Lebesgue-optimized interpolation points on the four-dimensional simplex (pentatope), demonstrating improved Lebesgue constants up to order ten.

Contribution

It introduces a new explicit geometric construction of optimized interpolation points on the 4-simplex, extending previous 3D methods to higher dimensions.

Findings

Constructed interpolation points up to order ten on the pentatope.

Achieved lower Lebesgue constants compared to equidistant points.

Demonstrated the effectiveness of the method in higher dimensions.

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.