A new mixed functional-probabilistic approach for finite element accuracy

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.