A new probabilistic interpretation of Bramble-Hilbert lemma
Provides a new probabilistic perspective on finite element accuracy for numerical analysts, but the results are theoretical and incremental.
The paper introduces probability distributions to compare the accuracy of Lagrange finite elements P_k and P_m, showing that which element is more accurate depends on mesh size h.
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 $h$.