Gradient Bounds for Wachspress Coordinates on Polytopes
Michael Floater, Andrew Gillette, N. Sukumar · SIAM Journal on Numerical Analysis · 2014
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 $h_\ast$, which denotes the minimum distance between a vertex of $P$ and any hyperplane containing a nonincident 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 three-dimensional finite element solution of the Poisson equation on a nontrivial 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.