Gradient bounds for Wachspress coordinates on polytopes

Michael Floater, Andrew Gillette, N. Sukumar

Research output: Contribution to journalArticle

61 Scopus citations

Abstract

We derive upper and lower bounds on the g radients 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 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 H1-norm of the error in the method converges at a linear rate with respect to the size of the mesh elements.

Original languageEnglish (US)
Pages (from-to)515-532
Number of pages18
JournalSIAM Journal on Numerical Analysis
Volume52
Issue number1
DOIs
StatePublished - Mar 28 2014
Externally publishedYes

Keywords

  • Generalized barycentric coordinates
  • Interpolation estimate
  • Polyhedral finite element method
  • Wachspress coordinates

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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