Error Estimates for Generalized Barycentric Interpolation

TL;DR

This paper establishes optimal convergence estimates for various generalized barycentric interpolation methods on convex polygons, highlighting differences in geometric conditions needed for each approach.

Contribution

It proves that different barycentric interpolation methods can achieve optimal convergence under specific geometric conditions, extending finite element analysis to polygons.

Findings

Wachspress functions require maximum interior angle condition.

Sibson functions do not require maximum interior angle condition.

All methods can achieve optimal convergence under certain geometric constraints.

Abstract

We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.