Convergence Rates of AFEM with H −1 Data

Albert Cohen; Ronald DeVore; Ricardo H. Nochetto

doi:10.1007/s10208-012-9120-1

Convergence Rates of AFEM with H −1 Data

Albert Cohen, Ronald DeVore, Ricardo H. Nochetto

Research output: Contribution to journal › Article › peer-review

29 Scopus citations

Abstract

This paper studies adaptive finite element methods (AFEMs), based on piecewise linear elements and newest vertex bisection, for solving second order elliptic equations with piecewise constant coefficients on a polygonal domain Ω⊂ℝ2. The main contribution is to build algorithms that hold for a general right-hand side f∈H-1(Ω). Prior work assumes almost exclusively that f∈L2(Ω). New data indicators based on local H-1 norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or Dörfler marking) combined with a procedure that adapts the mesh to reduce these new indicators. An analysis of our AFEM is given which establishes a contraction property and optimal convergence rates N-s with 0

Original language	English (US)
Pages (from-to)	671-718
Number of pages	48
Journal	Foundations of Computational Mathematics
Volume	12
Issue number	5
DOIs	https://doi.org/10.1007/s10208-012-9120-1
State	Published - Jun 29 2012
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: This research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107; the AFOSR Contract FA95500910500; the ARO/DoD Contract W911NF-07-1-0185; the NSF Grants DMS-0915231, DMS-0807811, and DMS-1109325; the Agence Nationale de la Recherche (ANR) project ECHANGE (ANR-08-EMER-006); the excellence chair of the Fondation "Sciences Mathematiques de Paris" held by Ronald DeVore. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

Access to Document

10.1007/s10208-012-9120-1

Cite this

@article{2eabbb822ce4432e9e60c1b7d5d77a27,

title = "Convergence Rates of AFEM with H −1 Data",

abstract = "This paper studies adaptive finite element methods (AFEMs), based on piecewise linear elements and newest vertex bisection, for solving second order elliptic equations with piecewise constant coefficients on a polygonal domain Ω⊂ℝ2. The main contribution is to build algorithms that hold for a general right-hand side f∈H-1(Ω). Prior work assumes almost exclusively that f∈L2(Ω). New data indicators based on local H-1 norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or D{\"o}rfler marking) combined with a procedure that adapts the mesh to reduce these new indicators. An analysis of our AFEM is given which establishes a contraction property and optimal convergence rates N-s with 0",

author = "Albert Cohen and Ronald DeVore and Nochetto, {Ricardo H.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: This research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107; the AFOSR Contract FA95500910500; the ARO/DoD Contract W911NF-07-1-0185; the NSF Grants DMS-0915231, DMS-0807811, and DMS-1109325; the Agence Nationale de la Recherche (ANR) project ECHANGE (ANR-08-EMER-006); the excellence chair of the Fondation {"}Sciences Mathematiques de Paris{"} held by Ronald DeVore. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

year = "2012",

month = jun,

day = "29",

doi = "10.1007/s10208-012-9120-1",

language = "English (US)",

volume = "12",

pages = "671--718",

journal = "Foundations of Computational Mathematics",

issn = "1615-3375",

publisher = "Springer New York",

number = "5",

}

TY - JOUR

T1 - Convergence Rates of AFEM with H −1 Data

AU - Cohen, Albert

AU - DeVore, Ronald

AU - Nochetto, Ricardo H.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: This research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107; the AFOSR Contract FA95500910500; the ARO/DoD Contract W911NF-07-1-0185; the NSF Grants DMS-0915231, DMS-0807811, and DMS-1109325; the Agence Nationale de la Recherche (ANR) project ECHANGE (ANR-08-EMER-006); the excellence chair of the Fondation "Sciences Mathematiques de Paris" held by Ronald DeVore. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2012/6/29

Y1 - 2012/6/29

N2 - This paper studies adaptive finite element methods (AFEMs), based on piecewise linear elements and newest vertex bisection, for solving second order elliptic equations with piecewise constant coefficients on a polygonal domain Ω⊂ℝ2. The main contribution is to build algorithms that hold for a general right-hand side f∈H-1(Ω). Prior work assumes almost exclusively that f∈L2(Ω). New data indicators based on local H-1 norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or Dörfler marking) combined with a procedure that adapts the mesh to reduce these new indicators. An analysis of our AFEM is given which establishes a contraction property and optimal convergence rates N-s with 0

AB - This paper studies adaptive finite element methods (AFEMs), based on piecewise linear elements and newest vertex bisection, for solving second order elliptic equations with piecewise constant coefficients on a polygonal domain Ω⊂ℝ2. The main contribution is to build algorithms that hold for a general right-hand side f∈H-1(Ω). Prior work assumes almost exclusively that f∈L2(Ω). New data indicators based on local H-1 norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or Dörfler marking) combined with a procedure that adapts the mesh to reduce these new indicators. An analysis of our AFEM is given which establishes a contraction property and optimal convergence rates N-s with 0

UR - http://hdl.handle.net/10754/597875

UR - http://link.springer.com/10.1007/s10208-012-9120-1

UR - http://www.scopus.com/inward/record.url?scp=84866541819&partnerID=8YFLogxK

U2 - 10.1007/s10208-012-9120-1

DO - 10.1007/s10208-012-9120-1

M3 - Article

SN - 1615-3375

VL - 12

SP - 671

EP - 718

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

IS - 5

ER -

Convergence Rates of AFEM with H −1 Data

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this