Interpolation Error Estimates for Mean Value Coordinates over Convex Polygons

TL;DR

This paper establishes interpolation error estimates for mean value coordinates on convex polygons, demonstrating their stability and suitability for finite element analysis, especially compared to Wachspress coordinates.

Contribution

It provides the first rigorous proof of uniform gradient bounds for mean value coordinates on convex polygons under simple geometric conditions.

Findings

Mean value coordinates have bounded gradients on convex polygons with certain restrictions.

Unlike Wachspress coordinates, mean value coordinate gradients do not blow up near large interior angles.

The results support the use of mean value coordinates in finite element methods for convex polygons.

Abstract

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in [Gillette et al., AiCM, doi:10.1007/s10444-011-9218-z], we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach pi.