# On generalized binomial laws to evaluate finite element accuracy: toward applications for adaptive mesh refinement

## TL;DR

This paper introduces a probabilistic framework based on generalized binomial laws to assess finite element accuracy, providing new insights for adaptive mesh refinement strategies.

## Contribution

It develops a novel probabilistic approach to evaluate finite element accuracy using geometric error estimates, extending to applications in mesh refinement.

## Key findings

- Probabilistic law for relative finite element accuracy derived
- Application of the law to mesh families and sequences of simplexes
- Potential relevance for adaptive mesh refinement

## Abstract

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.