### Abstract

We prove the optimal convergence estimate for first-order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.

Original language | English (US) |
---|---|

Pages (from-to) | 417-439 |

Number of pages | 23 |

Journal | Advances in Computational Mathematics |

Volume | 37 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Barycentric coordinates
- Finite element method
- Interpolation

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*Advances in Computational Mathematics*,

*37*(3), 417-439. https://doi.org/10.1007/s10444-011-9218-z

**Error estimates for generalized barycentric interpolation.** / Gillette, Andrew; Rand, Alexander; Bajaj, Chandrajit.

Research output: Contribution to journal › Article

*Advances in Computational Mathematics*, vol. 37, no. 3, pp. 417-439. https://doi.org/10.1007/s10444-011-9218-z

}

TY - JOUR

T1 - Error estimates for generalized barycentric interpolation

AU - Gillette, Andrew

AU - Rand, Alexander

AU - Bajaj, Chandrajit

PY - 2012/9

Y1 - 2012/9

N2 - We prove the optimal convergence estimate for first-order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.

AB - We prove the optimal convergence estimate for first-order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.

KW - Barycentric coordinates

KW - Finite element method

KW - Interpolation

UR - http://www.scopus.com/inward/record.url?scp=84866544617&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866544617&partnerID=8YFLogxK

U2 - 10.1007/s10444-011-9218-z

DO - 10.1007/s10444-011-9218-z

M3 - Article

AN - SCOPUS:84866544617

VL - 37

SP - 417

EP - 439

JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

SN - 1019-7168

IS - 3

ER -