A new probabilistic interpretation of Bramble-Hilbert lemma

TL;DR

This paper introduces a probabilistic framework for analyzing finite element accuracy, providing new insights into the relative performance of different polynomial degree elements based on mesh size.

Contribution

It offers a novel probabilistic interpretation of the Bramble-Hilbert lemma, connecting geometric error estimates with probability distributions to compare finite element accuracy.

Findings

Probabilistic distributions estimate the likelihood of one finite element being more accurate than another.

The likelihood depends on the mesh size, influencing which element is more accurate.

New mathematical properties of these distributions are established.

Abstract

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 $h$.