# Discrete duality finite volume method for mean curvature flow of surfaces

## Main Article Content

Lukáš Tomek Karol Mikula Mariana Remešíková

## Abstract

In this paper we propose a new numerical method for solving the mean curvature flow of surfaces. Two-dimensional surface is usualy approximated by a triangular mesh. Widely used discretization of Laplace-Beltrami operator over triangulated surfaces is the so-called cotangent scheme [7,8]. In the cotangent scheme the unknowns are the vertices of the triangulation. The basic idea of our new approach is to include a representative point of each triangle (vertex of the dual mesh) in the scheme as a supplementary unknown and generalize the discrete duality finite volume method [3] from~\$R^2\$ to 2D~surfacesembedded in~\$R^3\$. We derive the numerical scheme and present numerical experiments illustrating the basic properties of the meth od.

## Article Details

How to Cite
TOMEK, Lukáš; MIKULA, Karol; REMEŠÍKOVÁ, Mariana. Discrete duality finite volume method for mean curvature flow of surfaces. Proceedings of the Conference Algoritmy, [S.l.], p. 33-43, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/392>. Date accessed: 22 sep. 2017.
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## References

[1] J. W. Barrett, H. Garcke and R. Nurnberg, On the parametric finite element approximation of evolving hypersurfaces in R 3, J. Comput. Phys., 227 (2008), pp. 42814307.

[2] Y. Coudire, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493516.

[3] K. Domelevo and P. Omnes, A finite volume method for the laplace equation on almost arbitrary two-dimensional grids., M2AN, 39(6):1203-1249, 2005.

[4] G. Dziuk and C.M. Elliot, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), pp. 262-292.

[5] R. Eymard, T. Gallout and R. Herbin, Handbook of Numerical Analysis Vol. 7, P.G. Ciarlet and J.-L. Lions, Eds., NorthHolland/Elsevier, Amsterdam (2000) 7131020.

[6] C. Mantegazza, Lecture Notes on Mean Curvature Flow, Addison-Wesley publishing company, Birkhauser Verlag, (2011), 166 s. ISBN 978-3-0348-0144-7

[7] K. Mikula, M. Remešíková, P. Sarkoci and D. Ševčovič, Manifold evolution with tangential redistribution of points, SIAM J. Scientific Computing, Vol. 36, No.4 (2014), pp. A1384A1414

[8] M. Meyer, M. Desbrun , P. Schroeder and A.H. Barr, Discrete differential geometry operators for triangulated 2-manifolds, Visualization and Mathematics III (H.-C. Hege and K. Polthier, eds.) (2003), pp. 35-57

[9] L. Tomek, Discrete duality finite volume method for mean curvature flow of surfaces, Project of PhD thesis. (2015)

[10] H. A. Van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631644.