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Interpolation error estimates for mean value coordinates over convex polygons

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Abstract

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in Gillette et al. (Adv Comput Math 37(3), 417–439, 2012), we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach π.

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Author information

Authors and Affiliations

  1. CD-adapco, Austin, TX, USA

    Alexander Rand

  2. Department of Mathematics, University of California, San Diego, San Diego, CA, USA

    Andrew Gillette

  3. Department of Computer Science, Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX, USA

    Chandrajit Bajaj

Authors
  1. Alexander Rand
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  2. Andrew Gillette
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  3. Chandrajit Bajaj
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Corresponding author

Correspondence to Chandrajit Bajaj.

Additional information

Communicated by Douglas Arnold.

This research was supported in part by NIH contracts R01-EB00487, R01-GM074258, and a grant from the UT-Portugal CoLab project. This work was performed while the first author was at the Institute for Computational Engineering and Sciences at the University of Texas at Austin.

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Rand, A., Gillette, A. & Bajaj, C. Interpolation error estimates for mean value coordinates over convex polygons. Adv Comput Math 39, 327–347 (2013). https://doi.org/10.1007/s10444-012-9282-z

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  • DOI: https://doi.org/10.1007/s10444-012-9282-z

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