Hamilton–Jacobi equations on an evolving surface

Deckelnick, Klaus; Elliott, Charles; Miura, Tatsu-Hiko; Styles, Vanessa

doi:10.1090/mcom/3420

Hamilton–Jacobi equations on an evolving surface
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by Klaus Deckelnick, Charles M. Elliott, Tatsu-Hiko Miura and Vanessa Styles;

Math. Comp. 88 (2019), 2635-2664

DOI: https://doi.org/10.1090/mcom/3420

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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