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Hamilton-Jacobi equations on an evolving surface

Authors: Klaus Deckelnick, Charles M. Elliott, Tatsu-Hiko Miura and Vanessa Styles
Journal: Math. Comp. 88 (2019), 2635-2664
MSC (2010): Primary 65M08, 35F21, 35D40
Published electronically: March 26, 2019
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Abstract: We consider the well-posedness and numerical approximation of a Hamilton-Jacobi equation on an evolving hypersurface in $ \mathbb{R}^3$. Definitions of viscosity sub- and supersolutions are extended in a natural way to evolving hypersurfaces and provide uniqueness by comparison. An explicit in time monotone numerical approximation is derived on evolving interpolating triangulated surfaces. The scheme relies on a finite volume discretisation which does not require acute triangles. The scheme is shown to be stable and consistent leading to an existence proof via the proof of convergence. Finally an error bound is proved of the same order as in the flat stationary case.

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Additional Information

Klaus Deckelnick
Affiliation: Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, 39106 Magdeburg, Germany

Charles M. Elliott
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom

Tatsu-Hiko Miura
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan

Vanessa Styles
Affiliation: School of Mathematical and Physical Sciences, University of Sussex, Brighton, BN1 9QH, United Kingdom

Received by editor(s): November 17, 2017
Received by editor(s) in revised form: October 4, 2018, October 8, 2018, and December 12, 2018
Published electronically: March 26, 2019
Additional Notes: The work of the second author was partially supported by the Royal Society via a Wolfson Research Merit Award.
The work of the third author was partially supported by Grant–in–Aid for JSPS Fellows No. 16J02664 and the Program for Leading Graduate Schools, MEXT, Japan.
Article copyright: © Copyright 2019 American Mathematical Society