Handle URI:
http://hdl.handle.net/10754/597875
Title:
Convergence Rates of AFEM with H −1 Data
Authors:
Cohen, Albert; DeVore, Ronald; Nochetto, Ricardo H.
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<s≤1/2. In contrast to previous work, it is shown that it is not necessary to assume a compatible decay s<1/2 of the data estimator, but rather that this is automatically guaranteed by the approximability assumptions on the solution by adaptive meshes, without further assumptions on f; the borderline case s=1/2 yields an additional factor logN. Computable surrogates for the data indicators are introduced and shown to also yield optimal convergence rates N-s with s≤1/2. © 2012 SFoCM.
Citation:
Cohen A, DeVore R, Nochetto RH (2012) Convergence Rates of AFEM with H −1 Data. Foundations of Computational Mathematics 12: 671–718. Available: http://dx.doi.org/10.1007/s10208-012-9120-1.
Publisher:
Springer Nature
Journal:
Foundations of Computational Mathematics
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
29-Jun-2012
DOI:
10.1007/s10208-012-9120-1
Type:
Article
ISSN:
1615-3375; 1615-3383
Sponsors:
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).
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Full metadata record

DC FieldValue Language
dc.contributor.authorCohen, Alberten
dc.contributor.authorDeVore, Ronalden
dc.contributor.authorNochetto, Ricardo H.en
dc.date.accessioned2016-02-25T12:58:11Zen
dc.date.available2016-02-25T12:58:11Zen
dc.date.issued2012-06-29en
dc.identifier.citationCohen A, DeVore R, Nochetto RH (2012) Convergence Rates of AFEM with H −1 Data. Foundations of Computational Mathematics 12: 671–718. Available: http://dx.doi.org/10.1007/s10208-012-9120-1.en
dc.identifier.issn1615-3375en
dc.identifier.issn1615-3383en
dc.identifier.doi10.1007/s10208-012-9120-1en
dc.identifier.urihttp://hdl.handle.net/10754/597875en
dc.description.abstractThis 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<s≤1/2. In contrast to previous work, it is shown that it is not necessary to assume a compatible decay s<1/2 of the data estimator, but rather that this is automatically guaranteed by the approximability assumptions on the solution by adaptive meshes, without further assumptions on f; the borderline case s=1/2 yields an additional factor logN. Computable surrogates for the data indicators are introduced and shown to also yield optimal convergence rates N-s with s≤1/2. © 2012 SFoCM.en
dc.description.sponsorshipThis 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).en
dc.publisherSpringer Natureen
dc.subjectAdaptivityen
dc.subjectApproximation classesen
dc.subjectCardinalityen
dc.subjectContractionen
dc.subjectH-1 dataen
dc.subjectRatesen
dc.titleConvergence Rates of AFEM with H −1 Dataen
dc.typeArticleen
dc.identifier.journalFoundations of Computational Mathematicsen
dc.contributor.institutionUniversite Pierre et Marie Curie, Paris, Franceen
dc.contributor.institutionLaboratoire Jacques-Louis Lions, Paris, Franceen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionUniversity of Maryland, College Park, United Statesen
kaust.grant.numberKUS-C1-016-04en
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