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AFEM for the Laplace-Beltrami operator on graphs: Design and conditional contraction property


Authors: Khamron Mekchay, Pedro Morin and Ricardo H. Nochetto
Journal: Math. Comp. 80 (2011), 625-648
MSC (2010): Primary 65N30, 65N50
DOI: https://doi.org/10.1090/S0025-5718-2010-02435-4
Published electronically: November 16, 2010
MathSciNet review: 2772090
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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\ge2)$. 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
Email: rhn@math.umd.edu

DOI: https://doi.org/10.1090/S0025-5718-2010-02435-4
Keywords: Laplace-Beltrami operator, graphs, adaptive finite element method, a posteriori error estimate, energy and geometric errors, bisection, contraction.
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
Article copyright: © Copyright 2010 American Mathematical Society