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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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AFEM for the Laplace-Beltrami operator on graphs: Design and conditional contraction property
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by Khamron Mekchay, Pedro Morin and Ricardo H. Nochetto PDF
Math. Comp. 80 (2011), 625-648 Request permission

Abstract:

We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Beltrami operator on $C^1$ graphs $\Gamma$ in $\mathbb {R}^d ~(d\ge 2)$. We first derive residual-type a posteriori error estimates that account for the interaction of both the energy error in $H^1(\Gamma )$ and the surface error in $W^1_\infty (\Gamma )$ due to approximation of $\Gamma$. We devise a marking strategy to reduce the total error estimator, namely a suitably scaled sum of the energy, geometric, and inconsistency error estimators. We prove a conditional contraction property for the sum of the energy error and the total estimator; the conditional statement encodes resolution of $\Gamma$ in $W^1_\infty$. We conclude with one numerical experiment that illustrates the theory.
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Additional Information
  • Khamron Mekchay
  • Affiliation: Department of Mathematics, Faculty of Science, Chulalongkorn University, Phyathai, Bangkok 10330, Thailand – and – University of Maryland, College Park, Maryland 20742
  • Email: k.mekchay@gmail.com
  • Pedro Morin
  • Affiliation: Instituto de Matemática Aplicada del Litoral, Universidad Nacional del Litoral, CONICET, Güemes 3450, S3000GLN Santa Fe, Argentina
  • Email: pmorin@santafe-conicet.gov.ar
  • Ricardo H. Nochetto
  • Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
  • MR Author ID: 131850
  • Email: rhn@math.umd.edu
  • Received by editor(s): July 20, 2009
  • Received by editor(s) in revised form: February 26, 2010
  • Published electronically: November 16, 2010
  • Additional Notes: The first author was partially supported by NSF Grants DMS-0204670, DMS-0505454, and INT-0126272.
    The second author was partially supported by CONICET through Grants PIP 5478, PIP 112-200801-02182, by Universidad Nacional del Litoral through Grants CAI+D 008-054 and CAI+D PI 062-312, and by NSF Grant DMS-0204670.
    The third author was partially supported by NSF Grants DMS-0204670, DMS-0505454, DMS-0807811, and INT-0126272, and the General Research Board of the University of Maryland
  • © Copyright 2010 American Mathematical Society
  • Journal: Math. Comp. 80 (2011), 625-648
  • MSC (2010): Primary 65N30, 65N50
  • DOI: https://doi.org/10.1090/S0025-5718-2010-02435-4
  • MathSciNet review: 2772090