From a Geometrical Interpretation of Bramble-Hilbert Lemma to a Probability Distribution for Finite Element Accuracy

Joel Chaskalovic; Franck Assous

doi:10.1007/978-3-030-11539-5_1

From a Geometrical Interpretation of Bramble-Hilbert Lemma to a Probability Distribution for Finite Element Accuracy

Joel Chaskalovic, Franck Assous

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

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 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 and P:m, (k < m ). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that P:k or P:m is more likely accurate than the other, depending on the value of the mesh size h.

Original language	English
Title of host publication	Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers
Editors	Ivan Dimov, István Faragó, Lubin Vulkov
Pages	3-14
Number of pages	12
DOIs	https://doi.org/10.1007/978-3-030-11539-5_1
State	Published - 2019
Event	7th International Conference on Finite Difference Methods, FDM 2018 - Lozenetz, Bulgaria Duration: 11 Jun 2018 → 16 Jun 2018

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	11386 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	7th International Conference on Finite Difference Methods, FDM 2018
Country/Territory	Bulgaria
City	Lozenetz
Period	11/06/18 → 16/06/18

Keywords

Bramble-Hilbert lemma
Error estimates
Finite elements
Probability

Access to Document

10.1007/978-3-030-11539-5_1

Cite this

Chaskalovic, J., & Assous, F. (2019). From a Geometrical Interpretation of Bramble-Hilbert Lemma to a Probability Distribution for Finite Element Accuracy. In I. Dimov, I. Faragó, & L. Vulkov (Eds.), Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers (pp. 3-14). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11386 LNCS). https://doi.org/10.1007/978-3-030-11539-5_1

Chaskalovic, Joel ; Assous, Franck. / From a Geometrical Interpretation of Bramble-Hilbert Lemma to a Probability Distribution for Finite Element Accuracy. Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers. editor / Ivan Dimov ; István Faragó ; Lubin Vulkov. 2019. pp. 3-14 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "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 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 and P:m, (k < m ). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that P:k or P:m is more likely accurate than the other, depending on the value of the mesh size h.",

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Chaskalovic, J & Assous, F 2019, From a Geometrical Interpretation of Bramble-Hilbert Lemma to a Probability Distribution for Finite Element Accuracy. in I Dimov, I Faragó & L Vulkov (eds), Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11386 LNCS, pp. 3-14, 7th International Conference on Finite Difference Methods, FDM 2018, Lozenetz, Bulgaria, 11/06/18. https://doi.org/10.1007/978-3-030-11539-5_1

From a Geometrical Interpretation of Bramble-Hilbert Lemma to a Probability Distribution for Finite Element Accuracy. / Chaskalovic, Joel; Assous, Franck.
Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers. ed. / Ivan Dimov; István Faragó; Lubin Vulkov. 2019. p. 3-14 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11386 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Chaskalovic J, Assous F. From a Geometrical Interpretation of Bramble-Hilbert Lemma to a Probability Distribution for Finite Element Accuracy. In Dimov I, Faragó I, Vulkov L, editors, Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers. 2019. p. 3-14. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-030-11539-5_1

From a Geometrical Interpretation of Bramble-Hilbert Lemma to a Probability Distribution for Finite Element Accuracy

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