Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 52, Issue 1, Pages 515-532Publisher
SIAM PUBLICATIONS
DOI: 10.1137/130925712
Keywords
Wachspress coordinates; interpolation estimate; generalized barycentric coordinates; polyhedral finite element method
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Funding
- NSF [DMS-0715146]
- NBCR at UC San Diego
- NSF at UC Davis [CMMI-1334783]
- Directorate For Engineering [1334783] Funding Source: National Science Foundation
- Div Of Civil, Mechanical, & Manufact Inn [1334783] Funding Source: National Science Foundation
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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 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.
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