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We prove the optimal convergence estimate for first order interpolants usedin finite element methods based on three major approaches for generalizingbarycentric interpolation functions to convex planar polygonal domains. TheWachspress approach explicitly constructs rational functions, the Sibsonapproach uses Voronoi diagrams on the vertices of the polygon to define thefunctions, and the Harmonic approach defines the functions as the solution of aPDE. We show that given certain conditions on the geometry of the polygon, eachof these constructions can obtain the optimal convergence estimate. Inparticular, we show that the well-known maximum interior angle conditionrequired for interpolants over triangles is still required for Wachspressfunctions but not for Sibson functions.

Error Estimates for Generalized Barycentric Interpolation