Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces

Katsuyuki Ishii; Takahiro Izumi

doi:10.3934/dcds.2018046

Article Contents

2018, Volume 38, Issue 3: 1103-1125. Doi: 10.3934/dcds.2018046

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Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces

Katsuyuki Ishii^1, , and
Takahiro Izumi^2,

1.
Graduate School of Maritime Sciences, Kobe University, Higashinada, Kobe 658-0022, Japan
2.
Yasuna Machine Designing, Hojo-Umehara, Himeji 670-0945, Japan

^*Corresponding author
^*Corresponding author

Received: November 30, 2016

Revised: August 31, 2017

Published: June 2018

The first author is supported by JSPS KAKENHI Grant Numbers JP17K05364 and JP252470080.

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Abstract

We consider a threshold-type algorithm for curvature-dependent motions (CDM for short) of hypersurfaces. This algorithm was numerically studied by Kimura - Notsu [13], Esedoḡlu - Ruuth - Tsai [7] and Mohammad - Švadlenka [16], where they used the signed distance function as the level set function for CDM. The convergence of this algorithm and its optimal rate have been considered in Ishii - Kimura [12]. In this paper we give different approaches to the optimal rate of convergence to the smooth and compact CDM from [12]. As for the optimality, we give a more precise estimate than that in [12].

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Mathematics Subject Classification: Primary: 35K15, 35K55; Secondary: 65M12.

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Figure 1. Graphs of $ w_k $, $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w}_k $ and $ \bar{w}_k $ (thick curves)

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Figure 2. Graphs of of $ w^k $, $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w}^k $ and $ \bar{w}^k $ (thick curves)

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References

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