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On generalized binomial laws to evaluate finite element accuracy: preliminary probabilistic results for adaptive mesh refinement

Joël Chaskalovic and Franck Assous

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 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.

MSC 2010: 65Gxx

Acknowledgment

The authors want to warmly dedicate this research to pay homage to the memory of Professors André Avez and Gérard Tronel who largely promote the passion of research and teaching in mathematics.

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Received: 2019-01-03
Revised: 2019-12-25
Accepted: 2020-02-24
Published Online: 2020-06-13
Published in Print: 2020-06-25

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