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Evolving surface finite element method for the Cahn–Hilliard equation

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Abstract

We use the evolving surface finite element method to solve a Cahn–Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the paper by deriving error estimates and present various numerical examples.

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Acknowledgments

The authors would like to thank Andrew Stuart and Endre Sülli for thoughtful comments and discussion which have improved this work greatly.

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Author notes

  1. Thomas Ranner

    Present address: School of Computing, University of Leeds, EC Stoner Building, Leeds,  LS2 9JT, UK

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Zeeman Building, Coventry,  CV4 7AL, UK

    Charles M. Elliott & Thomas Ranner

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  1. Charles M. Elliott
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  2. Thomas Ranner
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Correspondence to Thomas Ranner.

Additional information

The work of C. M. Elliott was supported by the UK Engineering and Physical Sciences Research Council EPSRC Grant EP/G010404 and the work of T. Ranner was supported by a EPSRC Ph.D. studentship (Grant EP/P504333/1 and EP/P50516X/1) and the Warwick Impact Fund.

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Elliott, C.M., Ranner, T. Evolving surface finite element method for the Cahn–Hilliard equation. Numer. Math. 129, 483–534 (2015). https://doi.org/10.1007/s00211-014-0644-y

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  • DOI: https://doi.org/10.1007/s00211-014-0644-y

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