Kybernetika 49 no. 6, 829-854, 2013

Numerical analysis of a semi-implicit DDFV scheme for the regularized curvature driven level set equation in 2D

Angela Handlovičová and Dana Kotorová


Stability and convergence of the linear semi-implicit discrete duality finite volume (DDFV) numerical scheme in 2D for the solution of the regularized curvature driven level set equation is proved. Numerical experiments concerning comparison with exact solution and image filtering problem using proposed scheme are included.


stability, numerical solution, convergence, mean curvature flow, level set equation, semi-implicit scheme, discrete duality finite volume method


35K65, 65M08, 65M12


  1. B. Andreianov, F. Boyer and F. Hubert: Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D meshes. Num. Methods PDE 23 (2007), 1, 145-195. Zbl 1111.65101   CrossRef
  2. G. Barles and P. E. Souganidis: Convergence of approximation schemes for fully nonlineae second order equations. Asymptotic Anal. 4 (1991), 3, 271-283. Zbl 0729.65077   CrossRef
  3. S. Corsaro, K. Mikula, A. Sarti and F. Sgallari: Semi-implicit covolume method in 3D image segmentation. SIAM J. Sci. Comput Vol. 28 (2006), 6, 2248-2265. Zbl 1126.65088   CrossRef
  4. L. C. Evans and J. Spruck: Motion of level sets by mean curvature I. J. Differential Geometry 33 (1991), 635-681. Zbl 0726.53029   CrossRef
  5. and : The finite volume method. In: Handbook of Numerical Analysis, Ph. Ciarlet J.L. Lions eds 2000, pp. 715-1022. Zbl 0981.65095   CrossRef
  6. R. Eymard, A. Handlovičová and K. Mikula: Study of a finite volume scheme for the regularized mean curvature flow level set equation. IMA Journal of Numerical Analysis 31 (2011), 3, 813-846. Zbl 1241.65072   CrossRef
  7. A. Handlovičová and K. Mikula: Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation. Appl. Math., Praha 53 (2008), 2, 105-129. Zbl 1199.35197   CrossRef
  8. A. Handlovičová and D. Kotorová: Stability of the semi-implicit discrete duality finite volume scheme for the curvature driven level set equation in 2D. Accepted in Tatra mountains mathematical publications.   CrossRef
  9. A. Handlovičová, K. Mikula and F. Sgallari: Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution. Numer. Math.93 (2003), No. 4, 675-695. Zbl 1065.65105   CrossRef
  10. A. Handlovičová, K. Mikula and F. Sgallari: Variational numerical methods for solving nonlinear diffusion equations arising in image processing. J. Visual Communication and Image Representation 13 (2002), 217-237.   CrossRef
  11. D. Kotorová: Discrete duality finite volume scheme for the curvature-driven level set equation in 3D. In: Advances in architectural, civil and environmental engineering: 22nd Annual PhD Student Conference. Bratislava 2012   CrossRef
  12. D. Kotorová: Comparison of the 3D numerical scheme for solving curvature-driven level set equation based on discrete duality finite volumes. Accepted to proceedings of ODAM conference Olomouc 2013   CrossRef
  13. K. Mikula, A. Sarti and F. Sgallari: Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation. Comput. Visual. Sci. 9 (2006), 1, 23-31.   CrossRef
  14. A. M. Oberman: A convergent monotone difference scheme for motion of level sets by mean curvature. Numer. Math. 99 (2004), 2, 365-379. Zbl 1070.65082   CrossRef
  15. S. Osher and R. Fedkiw: Level set methods and dynamic implicit surfaces. Springer-Verlag 2003. Zbl 1026.76001   CrossRef
  16. J. A. Sethian: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science. Cambridge University Press, New York 1999. Zbl 0973.76003   CrossRef
  17. N. Walkington: Algorithms for computing motion by mean curvature SIAM J. Numer. Anal. 33 (1996), 6, 2215-2238. Zbl 0863.65061   CrossRef