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Interpolation Error Estimates for Mean Value Coordinates over Convex Polygons.

Alexander Rand1, Andrew Gillette, Chandrajit Bajaj

  • 1Institute for Computational Engineering and Sciences, University of Texas at Austin, arand@ices.utexas.edu.

Advances in Computational Mathematics
|September 13, 2013
PubMed
Summary
This summary is machine-generated.

This study proves interpolation error estimates for mean value coordinates in finite element analysis. Unlike Wachspress coordinates, mean value coordinates maintain stable gradients for convex polygons, enhancing their practical use.

Keywords:
Barycentric coordinatesfinite element methodinterpolation

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Area of Science:

  • Computational mathematics
  • Numerical analysis
  • Finite element methods

Background:

  • Harmonic, Wachspress, and Sibson coordinates are used in finite element analysis.
  • Mean value coordinates offer practical advantages but lack rigorous error estimates.
  • Previous work established estimates for other coordinate systems.

Purpose of the Study:

  • To derive interpolation error estimates for mean value coordinates on convex polygons.
  • To provide a theoretical basis for the practical advantages of mean value coordinates.
  • To ensure suitability for standard finite element analysis.

Main Methods:

  • Analysis based on uniform bounds of the gradient of mean value functions.
  • Consideration of convex polygons with diameter one and specific geometric restrictions.
  • Comparison with existing coordinate systems like Wachspress coordinates.

Main Results:

  • Interpolation error estimates for mean value coordinates are proven.
  • A uniform bound on the gradient of mean value functions is established for relevant polygons.
  • The gradient of mean value coordinates does not become large as interior angles approach π.

Conclusions:

  • Mean value coordinates are rigorously shown to be suitable for finite element analysis.
  • The stability of mean value coordinate gradients is mathematically confirmed.
  • This work validates and explains the observed practical benefits of mean value coordinates.