A new mixed functional-probabilistic approach for finite element accuracy

TL;DR

This paper introduces a novel mixed functional-probabilistic framework for assessing finite element accuracy, providing new insights especially for high-order elements through asymptotic analysis.

Contribution

It presents a new perspective based on probabilistic laws derived from a geometrical interpretation, analyzing their asymptotic behavior for high-order finite elements.

Findings

Probabilistic laws estimate relative accuracy between finite elements.

Asymptotic relations reveal behavior for high-order elements.

New insights into finite element accuracy qualification.

Abstract

The aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble-Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements $P_k$ and $P_m$, ($k < m$). Then, we analyze the asymptotic relation between these two probabilistic laws when the difference $m-k$ goes to infinity. New insights which qualified the relative accuracy in the case of high order finite elements are correspondingly obtained.