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: December 2016

Revised: September 2017

Published: July 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

[1]	G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motion by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500. doi: 10.1137/0732020.
[2]	J. Bence, B. Merriman and S. Osher, Diffusion generated motion by mean curvature, in "Computational Crystal Growers Workshop", J. Taylor ed., Selected Lectures in Math., Amer. Math. Soc., Province, 1992.
[3]	A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), 195-218.
[4]	A. Chambolle and M. Novaga, Approximation of the anisotropic mean curvature flow, Math. Models Methods Appl. Sci., 17 (2007), 833-844. doi: 10.1142/S0218202507002121.
[5]	A. Chambolle and M. Novaga, Implicit time discretization of the mean curvature flow with a discontinuous forcing term, Interfaces Free Bound., 10 (2008), 283-300.
[6]	M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. A. M. S., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[7]	S. Esedoḡlu, S.J. Ruuth and R. Tsai, Diffusion generated motion using the signed distance function, J. Comp. Phys., 229 (2010), 1017-1042. doi: 10.1016/j.jcp.2009.10.002.
[8]	T. Eto, Y. Giga and K. Ishii, An area minimizing scheme for anisotropic mean curvature flow, Adv. Differential Equations, 17 (2012), 1031-1084.
[9]	L.C. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), 533-557. doi: 10.1512/iumj.1993.42.42024.
[10]	L.C. Evans and J. Spruck, Motion of level sets by mean curvature Ⅱ, Trans. Amer. Math. Soc., 330 (1992), 321-332. doi: 10.1090/S0002-9947-1992-1068927-8.
[11]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.
[12]	K. Ishii and M. Kimura, Convergence of a threshold-type algorithm using the signed distance function, Interfaces Free Bound., 18 (2016), 479-522. doi: 10.4171/IFB/371.
[13]	M. Kimura and H. Notsu, A level set method using the signed distance function, Japan J. Indust. Appl. Math., 19 (2002), 415-446. doi: 10.1007/BF03167487.
[14]	S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Memoirs, 13. Mathematical Society of Japan, Tokyo, 2004.
[15]	F. Leoni, Convergence of an approximation scheme for curvature-dependent motion of sets, SIAM J. Numer. Anal., 39 (2001), 1115-1131. doi: 10.1137/S0036142900370459.
[16]	R.Z. Mohammad and K. Švadlenka, Multiphase volume-preserving interface motion via localized signed distance vector scheme, Discrete and Continuous Dynamical Systems, Series S, 8 (2015), 969-988. doi: 10.3934/dcdss.2015.8.969.
[17]	L. Vivier, Convergence of an approximation scheme for computing motions with curvature dependent velocities, Differential Integral Equations, 13 (2000), 1263-1288.