Advances in Computational Mathematics

, Volume 41, Issue 3, pp 489–505

Convergence of Wachspress coordinates: from polygons to curved domains

Article

DOI: 10.1007/s10444-014-9370-3

Cite this article as:
Kosinka, J. & Bartoň, M. Adv Comput Math (2015) 41: 489. doi:10.1007/s10444-014-9370-3

Abstract

Given a smooth, strictly convex planar domain, we investigate point-wise convergence of the sequence of Wachspress coordinates defined over finer and finer inscribed polygonal approximations of the domain. Based on a relation between the discrete Wachspress case and the limit smooth case given by the Wachspress kernel defined by Warren et al., we show that the corresponding sequences of Wachspress interpolants and mappings converge as 𝓞(h2) for a sampling step size h of the boundary curve of the domain as h → 0. Several examples are shown to numerically validate the results and to visualise the behaviour of discrete interpolants and mappings as they converge to their smooth counterparts. Empirically, the same convergence order is observed also for mean value coordinates. Moreover, our numerical tests suggest that the convergence of interpolants and mappings is uniform both in the Wachspress and mean value cases.

Keywords

Barycentric coordinates Barycentric mapping Interpolation Convergence 

Mathematics Subject Classification (2010)

65D17 26B10 

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Computer LaboratoryUniversity of CambridgeCambridgeUK
  2. 2.Numerical Porous Media CenterKing Abdullah University of Science and TechnologyThuwalSaudi Arabia

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