Gradient bounds for Wachspress coordinates on polytopes

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.Comment: 18 pages, to appear in SINU