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Error Estimates for Generalized Barycentric Interpolation.

Andrew Gillette1, Alexander Rand, Chandrajit Bajaj

  • 1Department of Mathematics, University of Texas at Austin, agillette@math.utexas.edu.

Advances in Computational Mathematics
|January 23, 2013
PubMed
Summary
This summary is machine-generated.

This study analyzes three finite element methods for polygonal domains, finding optimal convergence estimates are achievable under specific geometric conditions for Wachspress, Sibson, and Harmonic interpolants.

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Area of Science:

  • Numerical Analysis
  • Computational Geometry
  • Finite Element Methods

Background:

  • Finite element methods (FEM) rely on accurate interpolants for convergence.
  • Generalizing interpolants beyond triangles to polygonal domains is a key challenge.
  • Barycentric interpolation functions offer a foundation for such generalizations.

Purpose of the Study:

  • To establish optimal convergence estimates for first-order interpolants in FEM for convex planar polygonal domains.
  • To compare three distinct approaches: Wachspress, Sibson, and Harmonic.
  • To identify geometric conditions influencing the convergence of these interpolants.

Main Methods:

  • Investigated Wachspress rational functions for interpolation.
  • Analyzed Sibson interpolants utilizing Voronoi diagrams.
  • Examined Harmonic interpolants derived from partial differential equations (PDEs).
  • Evaluated convergence properties based on polygon geometric characteristics.

Main Results:

  • All three methods (Wachspress, Sibson, Harmonic) can achieve optimal convergence estimates.
  • The optimal convergence depends on specific geometric conditions of the polygon.
  • The maximum interior angle condition, crucial for triangles, is necessary for Wachspress functions but not for Sibson functions.

Conclusions:

  • The choice of interpolation method impacts convergence requirements for polygonal domains.
  • Sibson interpolants offer greater flexibility regarding polygon geometry compared to Wachspress.
  • These findings advance the development of robust finite element methods for complex geometries.