Error estimates for generalized barycentric interpolation

Authors:
Andrew Gillette;Alexander Rand;Chandrajit Bajaj
Affiliations:
Department of Mathematics, University of Texas at Austin, Austin, USA;Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, USA;Department of Computer Science, Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, USA
Venue:
Advances in Computational Mathematics
Year:
2012

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Abstract

We prove the optimal convergence estimate for first-order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.