An Analysis of the Saturation Assumption for Polynomial Interpolation with Application to Adaptive Finite Element Methods

Abstract

The saturation assumption plays a central role in much of the analysis of a posteriori error estimates and refinement algorithms for adaptive finite element methods. In this work we provide an analysis of this assumption in the simple setting of interpolation. We have proved elsewhere [Bank and Yserentant, Numer. Math., 131:1 (2015)] that interpolation error is both reliable and efficient as an a posteriori error estimate. Thus behavior of interpolation error is indicative of the behavior of the error in the exact finite element solution of a PDE as well as any practical a posteriori error estimate that is also reliable and efficient.