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

Mekchay, Khamron; Morin, Pedro; Nochetto, Ricardo

doi:10.1090/S0025-5718-2010-02435-4

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;

Math. Comp. 80 (2011), 625-648

DOI: https://doi.org/10.1090/S0025-5718-2010-02435-4

Published electronically: November 16, 2010

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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 $\mathbb {R}^d ~(d\ge 2)$. 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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