A new probabilistic interpretation of Bramble-Hilbert lemma

Joël Chaskalovic, Franck Assous

Published 2018-03-26, updated 2018-10-03Version 2

The aim of this paper is to provide new perspectives on relative finite element accuracy which is usually based on the asymptotic speed of convergence comparison when the mesh size $h$ goes to zero. Starting from a geometrical reading of the error estimate due to Bramble-Hilbert lemma, we derive two probability distributions that estimate the relative accuracy, considered as a random variable, between two Lagrange finite elements $P_k$ and $P_m$, ($k < m$). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that $P_k$ or $P_m$ is more likely accurate than the other, depending on the value of the mesh size $h$.