A New Mixed Functional-probabilistic Approach for Finite Element Accuracy

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 {k<m}). Then we analyze the asymptotic relation between these two probabilistic laws when the difference m - k {m-k} goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.