Interpolation error estimates for harmonic coordinates on polytopes

Andrew Gillette, Alexander Rand

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Interpolation error estimates in terms of geometric quality measures are established for harmonic coordinates on polytopes in two and three dimensions. First we derive interpolation error estimates over convex polygons that depend on the geometric quality of the triangles in the constrained Delaunay triangulation of the polygon. This characterization is sharp in the sense that families of polygons with poor quality triangles in their constrained Delaunay triangulations are shown to produce large error when interpolating a basic quadratic function. Non-convex polygons exhibit a similar limitation: large constrained Delaunay triangles caused by vertices approaching a non-adjacent edge also lead to large interpolation error. While this relationship is generalized to convex polyhedra in three dimensions, the possibility of sliver tetrahedra in the constrained Delaunay triangulation prevent the analogous estimate from sharply reflecting the actual interpolation error. Non-convex polyhedra are shown to be fundamentally different through an example of a family of polyhedra containing vertices which are arbitrarily close to non-adjacent faces yet the interpolation error remains bounded.

Original languageEnglish (US)
Pages (from-to)651-676
Number of pages26
JournalESAIM: Mathematical Modelling and Numerical Analysis
Volume50
Issue number3
DOIs
StatePublished - May 1 2016

Fingerprint

Interpolation Error
Polytopes
Constrained Delaunay Triangulation
Error Estimates
Interpolation
Harmonic
Polygon
Triangulation
Triangle
Polyhedron
Three-dimension
Convex polyhedron
Quality Measures
Delaunay
Convex polygon
Triangular pyramid
Quadratic Function
Two Dimensions
Face
Estimate

Keywords

  • Generalized barycentric coordinates
  • Harmonic coordinates
  • Interpolation error estimates
  • Polygonal finite elements
  • Shape quality

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Modeling and Simulation
  • Numerical Analysis

Cite this

Interpolation error estimates for harmonic coordinates on polytopes. / Gillette, Andrew; Rand, Alexander.

In: ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, No. 3, 01.05.2016, p. 651-676.

Research output: Contribution to journalArticle

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