On generalized binomial laws to evaluate finite element accuracy: toward applications for adaptive mesh refinement
Provides a new probabilistic perspective for finite element accuracy, potentially aiding adaptive mesh refinement, but remains theoretical without empirical validation.
The paper derives a probabilistic law to evaluate relative accuracy between finite elements, extending it to mesh families and simplex sequences for adaptive mesh refinement. No concrete numerical results are provided.
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 $P_k$ and $P_m$, ($k < m$). We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes and the second one is dedicated to a sequence of simplexes which constitute a given mesh. Both of this applications might be relevant for adaptive mesh refinement.