Stability of the Semi-Implicit Discrete Duality Finite Volume Scheme for the Curvature Driven Level Set Equation in 2D

Handlovičová, Angela; Kotorová, Dana

doi:10.2478/tmmp-2014-0031

Abstract

Stability of the linear semi-implicit discrete duality finite volume (DDFV) numerical scheme for the solution of the regularized curvature driven level set equation is proved. Our scheme is linear, it is efficient regarding computational times. Numerical experiments confirm accuracy of the proposed scheme

References

[1] ANDREIANOV, B.-BOYER,F.-HUBERT, F.: Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D meshes, Numer. Methods Partial Differential Equations 23 (2007), 145-195.10.1002/num.20170
Search in Google Scholar
[2] ALVAREZ, L.-GUICHARD, F.-LIONS, P. L.-MOREL, J. M.: Axioms and fundamental equations of image processing, Arch. Ration. Mech. Anal. 123 (1993), 200-257.10.1007/BF00375127
Search in Google Scholar
[3] ANGENENT, S. B.-GURTIN, M. E.: Multiphase thermomechanics with an interfacial structure 2. Evolution of an isothermal interface, Arch. Ration. Mech. Anal. 108 (1989), 323-391.10.1007/BF01041068
Search in Google Scholar
[4] CASELLES, V.-KIMMEL, R.-SAPIRO, G.: Geodesic active contours, Int. J. Comput. Vis. 22 (1997), 61-79.10.1023/A:1007979827043
Search in Google Scholar
[5] CRANDALL, M. G.-ISHII, H.-LIONS, P. L.: User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (NS) 27 (1992), 1-67.10.1090/S0273-0979-1992-00266-5
Search in Google Scholar
[6] DECKELNICK, K.-DZIUK, G.: Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs, Interfaces Free Bound. 2 (2000), 341-359.10.4171/IFB/24
Search in Google Scholar
[7] EVANS, L. C.-SPRUCK, J.: Motion of level sets by mean curvature I, J. Differential Geom. 33 (1991), 635-681.10.4310/jdg/1214446559
Search in Google Scholar
[8] EYMARD, R.-GALLOU¨ ET, T.-HERBIN, R.: The finite volume method, in: Handbook of Numerical Analysis, Vol. 7 (Ph. Ciarlet et al., eds.), pp. 715-1022, North-Holland/ Elsevier, Amsterdam, 2000, pp. 715-1022.
Search in Google Scholar
[9] HANDLOVIČOVÁ, A.-MIKULA, K.: Convergenca of the semi-implicit co-volume scheme for the curvature driven level set equation (to appear).
Search in Google Scholar
[10] HANDLOVIČOVÁ,A.-MIKULA, K.-SARTI, A.: Numerical solution of parabolic equations related to level set formulation of mean curvature flow, Comput. Vis. Sci. 1 (1998), 179-182.10.1007/s007910050016
Search in Google Scholar
[11] HANDLOVIČOVÁ, A.-MIKULA, K.-SGALLARI, F.: Variational numerical methods for solving nonlinear diffusion equations arising in image processing, J. Visual Communication and Image Representation 13 (2002), 217-237.10.1006/jvci.2001.0479
Search in Google Scholar
[12] KICHENASSAMY, S.-KUMAR, A.-OLVER, P.-TANNENBAUM, A.-YEZZI, A.: Conformal curvature flows: from phase transitions to active vision, Arch. Ration. Mech. Anal. 134 (1996), 275-301.10.1007/BF00379537
Search in Google Scholar
[13] KOTOROVÁ, D.: Discrete duality finite volume scheme for the curvature-driven level set equation, Acta Polytech. Hungarica 8 (2011), 7-12
Search in Google Scholar
[14] MIKULA, K.-RAMAROSY, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing, Numer. Math. 89 (2001), 561-590.10.1007/PL00005479
Search in Google Scholar
[15] OBERMAN, A. M.: A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math. 99 (2004), 365-379.10.1007/s00211-004-0566-1
Search in Google Scholar
[16] OSHER, S.-FEDKIW, R.: Level Set Methods and Dynamic Implicit Surfaces, in: Appl. Math. Sci., Vol. 153, Springer-Verlag, New York, 2003.10.1007/b98879
Search in Google Scholar
[17] SARTI, A.-MALLADI, R.-SETHIAN, J. A.: Subjective surfaces: A method for completing missing boundaries, Proc. Natl. Acad. Sci. USA 97 (2000), 6258-6263.10.1073/pnas.1101357971858910823902
Search in Google Scholar
[18] SETHIAN, J. A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, in: Cambridge Monogr. Appl. Comput. Math., Vol. 3, Cambridge University Press, New York, 1999
Search in Google Scholar

Stability of the Semi-Implicit Discrete Duality Finite Volume Scheme for the Curvature Driven Level Set Equation in 2D

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