A new probabilistic interpretation of the bramble-Hilbert lemma

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 the 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.