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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.
Gillette, Andrew 1 ; Rand, Alexander 2
@article{M2AN_2016__50_3_651_0,
author = {Gillette, Andrew and Rand, Alexander},
title = {Interpolation error estimates for harmonic coordinates on polytopes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {651--676},
publisher = {EDP-Sciences},
volume = {50},
number = {3},
year = {2016},
doi = {10.1051/m2an/2015096},
zbl = {1343.31004},
mrnumber = {3507268},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015096/}
}
TY - JOUR AU - Gillette, Andrew AU - Rand, Alexander TI - Interpolation error estimates for harmonic coordinates on polytopes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 651 EP - 676 VL - 50 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015096/ DO - 10.1051/m2an/2015096 LA - en ID - M2AN_2016__50_3_651_0 ER -
%0 Journal Article %A Gillette, Andrew %A Rand, Alexander %T Interpolation error estimates for harmonic coordinates on polytopes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 651-676 %V 50 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015096/ %R 10.1051/m2an/2015096 %G en %F M2AN_2016__50_3_651_0
Gillette, Andrew; Rand, Alexander. Interpolation error estimates for harmonic coordinates on polytopes. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 651-676. doi: 10.1051/m2an/2015096
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