  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.
