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We propose a numerical validation of a probabilistic approach applied to estimate the relative accuracy between two Lagrange finite elements Pk and Pm,(k < m). In particular, we show practical cases where finite element Pk gives more accurate results than finite element Pm. This illustrates the theoretical probabilistic framework we recently derived in order to evaluate the actual accuracy. This also highlights the importance of the extra caution required when comparing two numerical methods, since the classical results of error estimates concerns only the asymptotic convergence rate. [ABSTRACT FROM AUTHOR]

We derive an explicit k-dependence in W m , p error estimates for P k Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between P k 1 and P k 2 finite elements, ( k 1 < k 2 ), in terms of W m , p -norms. We further prove a weak asymptotic relation in D ′ (R) between these probabilistic laws when difference k 2 − k 1 goes to infinity. Moreover, as expected, one finds that P k 2 finite element is surely more accurate than P k 1 , for sufficiently small values of the mesh size h. Nevertheless, our results also highlight cases where P k 1 is more likely accurate than P k 2 , for a range of values of h. Hence, this approach brings a new perspective on how to compare two finite elements, which is not limited to the rate of convergence. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble–Hilbert lemma, we derive a probability law that evaluates the relative accuracy, considered as a random variable, between two finite elements Pk and Pm, k < m. We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes, the second one is dedicated to a sequence of simplexes constituting a given mesh. Both of these applications could be considered as a first step toward application for adaptive mesh refinement with probabilistic methods. [ABSTRACT FROM AUTHOR]

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 the 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 {P_{k}} and P m {P_{m}} ( k < m {k