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$ L^2$-estimates for the evolving surface finite element method


Authors: Gerhard Dziuk and Charles M. Elliott
Journal: Math. Comp. 82 (2013), 1-24
MSC (2010): Primary 65M60, 65M15; Secondary 35K99, 35R01, 35R37, 76R99
DOI: https://doi.org/10.1090/S0025-5718-2012-02601-9
Published electronically: April 13, 2012
MathSciNet review: 2983013
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the evolving surface finite element meth-
od for the advection and diffusion of a conserved scalar quantity on a moving surface. In an earlier paper using a suitable variational formulation in time dependent Sobolev space we proposed and analysed a finite element method using surface finite elements on evolving triangulated surfaces. An optimal order $ H^1$-error bound was proved for linear finite elements. In this work we prove the optimal error bound in $ L^2(\Gamma (t))$ uniformly in time.


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Additional Information

Gerhard Dziuk
Affiliation: Abteilung für Angewandte Mathematik, University of Freiburg, Hermann-Herder-Straße 10, D–79104 Freiburg i. Br., Germany
Email: gerd@mathematik.uni-freiburg.de

Charles M. Elliott
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: c.m.elliott@warwick.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2012-02601-9
Keywords: Surface finite elements, advection diffusion, moving surface, error analysis
Received by editor(s): July 15, 2010
Received by editor(s) in revised form: July 5, 2011, and July 29, 2011
Published electronically: April 13, 2012
Additional Notes: The work was supported by Deutsche Forschungsgemeinschaft via SFB/TR 71
This research was also supported by the UK Engineering and Physical Sciences Research Council (EPSRC), Grant EP/G010404.
Article copyright: © Copyright 2012 American Mathematical Society

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