Interpolation Error Estimates for Harmonic Coordinates On Polytopes

TL;DR

This paper establishes geometric quality-based interpolation error estimates for harmonic coordinates on polytopes, highlighting the impact of triangulation quality on interpolation accuracy in 2D and 3D.

Contribution

It provides sharp error estimates for convex polygons and polyhedra, and reveals fundamental differences in error behavior for non-convex shapes.

Findings

Poor triangulation quality leads to large interpolation errors.

Convex polygons with bad triangulations produce significant errors.

Non-convex polyhedra can have bounded errors despite close vertices.

Abstract

Interpolation error estimates in terms of geometric quality measures are established for harmonic coordinates on polytopes in two and three dimensions. First we derive interpolation error estimates over convex polygons that depend on the geometric quality of the triangles in the constrained Delaunay triangulation of the polygon. This characterization is sharp in the sense that families of polygons with poor quality triangles in their constrained Delaunay triangulations are shown to produce large error when interpolating a basic quadratic function. Non-convex polygons exhibit a similar limitation: large constrained Delaunay triangles caused by vertices approaching a non-adjacent edge also lead to large interpolation error. While this relationship is generalized to convex polyhedra in three dimensions, the possibility of sliver tetrahedra in the constrained Delaunay triangulation prevent the analogous estimate from sharply reflecting the actual interpolation error. Non-convex polyhedra are shown to be fundamentally different through an example of a family of polyhedra containing vertices which are arbitrarily close to non-adjacent faces yet the interpolation error remains bounded.