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

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
Publisher	Springer Verlag
Pages	3-14
Number of pages	12
ISBN (Print)	9783030115388
DOIs	https://doi.org/10.1007/978-3-030-11539-5_1
State	Published - 2019
Externally published	Yes
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

Bibliographical note

Publisher Copyright:
© 2019, Springer Nature Switzerland AG.

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). Springer Verlag. 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. Springer Verlag, 2019. pp. 3-14 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{bc2a9804684f471c9f02c3bc591df442,

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

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.",

keywords = "Bramble-Hilbert lemma, Error estimates, Finite elements, Probability",

author = "Joel Chaskalovic and Franck Assous",

note = "Publisher Copyright: {\textcopyright} 2019, Springer Nature Switzerland AG.; 7th International Conference on Finite Difference Methods, FDM 2018 ; Conference date: 11-06-2018 Through 16-06-2018",

year = "2019",

doi = "10.1007/978-3-030-11539-5_1",

language = "אנגלית",

isbn = "9783030115388",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "3--14",

editor = "Ivan Dimov and Istv{\'a}n Farag{\'o} and Lubin Vulkov",

booktitle = "Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers",

address = "גרמניה",

}

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, Springer Verlag, 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. Springer Verlag, 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

TY - GEN

T1 - From a Geometrical Interpretation of Bramble-Hilbert Lemma to a Probability Distribution for Finite Element Accuracy

AU - Chaskalovic, Joel

AU - Assous, Franck

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Bramble-Hilbert lemma

KW - Error estimates

KW - Finite elements

KW - Probability

UR - http://www.scopus.com/inward/record.url?scp=85066140353&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-11539-5_1

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

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:85066140353

SN - 9783030115388

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 3

EP - 14

BT - Finite Difference Methods. Theory and Applications - 7th International Conference, FDM 2018, Revised Selected Papers

A2 - Dimov, Ivan

A2 - Faragó, István

A2 - Vulkov, Lubin

PB - Springer Verlag

T2 - 7th International Conference on Finite Difference Methods, FDM 2018

Y2 - 11 June 2018 through 16 June 2018

ER -

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. Springer Verlag. 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

Abstract

Publication series

Conference

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this