Recursive, parameter-free, explicitly defined interpolation nodes for simplices (2002.09421v2)

Abstract: A rule for constructing interpolation nodes for $n$th degree polynomials on the simplex is presented. These nodes are simple to define recursively from families of 1D node sets, such as the Lobatto-Gauss-Legendre (LGL) nodes. The resulting nodes have attractive properties: they are fully symmetric, they match the 1D family used in construction on the edges of the simplex, and the nodes constructed for the $(d-1)$-simplex are the boundary traces of the nodes constructed for the $d$-simplex. When compared using the Lebesgue constant to other explicit rules for defining interpolation nodes, the nodes recursively constructed from LGL nodes are nearly as good as the "warp & blend" nodes of Warburton in 2D (which, though defined differently, are very similar), and in 3D are better than other known explicit rules by increasing margins for $n > 6$. By that same measure, these recursively defined nodes are not as good as implicitly defined nodes found by optimizing the Lebesgue constant or related functions, but such optimal node sets have yet to be computed for the tetrahedron. A reference python implementation has been distributed as the recursivenodes package, but the simplicity of the recursive construction makes them easy to implement.