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Foundations of Computational Mathematics
Published in: Foundations of Computational Mathematics 6/2016

01-12-2016

High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates

Authors: Andrea Bonito, J. Manuel Cascón, Khamron Mekchay, Pedro Morin, Ricardo H. Nochetto

Published in: Foundations of Computational Mathematics | Issue 6/2016

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Abstract

We present a new AFEM for the Laplace–Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally \(W^1_\infty \) and piecewise in a suitable Besov class embedded in \(C^{1,\alpha }\) with \(\alpha \in (0,1]\). The idea is to have the surface sufficiently well resolved in \(W^1_\infty \) relative to the current resolution of the PDE in \(H^1\). This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in \(W^1_\infty \) and PDE error in \(H^1\).

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Metadata
Title
High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates
Authors
Andrea Bonito
J. Manuel Cascón
Khamron Mekchay
Pedro Morin
Ricardo H. Nochetto
Publication date
01-12-2016
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 6/2016
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-016-9335-7

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