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March  2018, 38(3): 1103-1125. doi: 10.3934/dcds.2018046

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

 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

Received  December 2016 Revised  September 2017 Published  December 2017

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

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].

Citation: Katsuyuki Ishii, Takahiro Izumi. Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1103-1125. doi: 10.3934/dcds.2018046
##### 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.

show all references

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