Sharp geometric requirements in the Wachspress interpolation error estimate

TL;DR

This paper investigates the geometric conditions necessary for accurate Wachspress interpolation, establishing that certain commonly assumed conditions are both necessary and sufficient for error estimate validity.

Contribution

It proves that the maximum angle condition and minimum edge length property are sharp and essential geometric restrictions for Wachspress interpolation error estimates.

Findings

MAC and melp are equivalent and necessary for error bounds

Violating these conditions invalidates error guarantees

Sharp geometric requirements are identified for Wachspress interpolation

Abstract

Geometric conditions on general polygons are given in [9] in order to guarantee the error estimate for interpolants built from generalized barycentric coordinates, and the question about identifying sharp geometric restrictions in this setting is proposed. In this work, we address the question when the construction is made by using Wachspress coordinates. We basically show that the imposed conditions: bounded aspect ratio property (barp), maximum angle condition (MAC) and minimum edge length property (melp) are actually equivalent to [MAC,melp], and if any of these conditions is not satisfied, then there is no guarantee that the error estimate is valid. In this sense, MAC and melp can be regarded as sharp geometric requirements in the Wachspress interpolation error estimate.