### 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 language | English (US) |
---|---|

Pages (from-to) | 651-676 |

Number of pages | 26 |

Journal | ESAIM: Mathematical Modelling and Numerical Analysis |

Volume | 50 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2016 |

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### 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.

Research output: Contribution to journal › Article

*ESAIM: Mathematical Modelling and Numerical Analysis*, vol. 50, no. 3, pp. 651-676. https://doi.org/10.1051/m2an/2015096

}

TY - JOUR

T1 - Interpolation error estimates for harmonic coordinates on polytopes

AU - Gillette, Andrew

AU - Rand, Alexander

PY - 2016/5/1

Y1 - 2016/5/1

N2 - 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.

AB - 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.

KW - Generalized barycentric coordinates

KW - Harmonic coordinates

KW - Interpolation error estimates

KW - Polygonal finite elements

KW - Shape quality

UR - http://www.scopus.com/inward/record.url?scp=84971447520&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84971447520&partnerID=8YFLogxK

U2 - 10.1051/m2an/2015096

DO - 10.1051/m2an/2015096

M3 - Article

AN - SCOPUS:84971447520

VL - 50

SP - 651

EP - 676

JO - Mathematical Modelling and Numerical Analysis

JF - Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 3

ER -