Gradient bounds for Wachspress coordinates on polytopes

TL;DR

This paper establishes bounds on the gradients of Wachspress coordinates for convex polytopes, analyzes their sharpness, and demonstrates their practical use in finite element methods for solving PDEs on polyhedral meshes.

Contribution

It provides new gradient bounds for Wachspress coordinates on convex polytopes, including sharp bounds in 2D and analysis for specific shapes, along with a practical implementation for PDE solving.

Findings

Gradient bounds are sharp in 2D.

Error in finite element solution converges linearly.

Implementation demonstrates practical applicability.

Abstract

We derive upper and lower bounds on the gradients of Wachspress coordinates defined over any simple convex d-dimensional polytope P. The bounds are in terms of a single geometric quantity h_*, which denotes the minimum distance between a vertex of P and any hyperplane containing a non-incident face. We prove that the upper bound is sharp for d=2 and analyze the bounds in the special cases of hypercubes and simplices. Additionally, we provide an implementation of the Wachspress coordinates on convex polyhedra using Matlab and employ them in a 3D finite element solution of the Poisson equation on a non-trivial polyhedral mesh. As expected from the upper bound derivation, the H^1-norm of the error in the method converges at a linear rate with respect to the size of the mesh elements.