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Keywords:
Willmore flow; method of lines; curvature minimization; gradient flow; Laplace–Beltrami operator; Gauss curvature; central differences; numerical viscosity
Willmore flow; method of lines; curvature minimization; gradient flow; Laplace–Beltrami operator; Gauss curvature; central differences; numerical viscosity
Summary:
In this article we discuss numerical scheme for the approximation of the Willmore flow of graphs. The scheme is based on the finite difference method. We improve the scheme we presented in Oberhuber [Obe-2005-2,Obe-2005-1] which is based on combination of the forward and the backward finite differences. The new scheme approximates the Willmore flow by the central differences and as a result it better preserves symmetry of the solution. Since it requires higher regularity of the solution, additional numerical viscosity is necessary in some cases. We also present theorem showing stability of the scheme together with the EOC and several results of the numerical experiments.
References:
[1] Beneš M.: Numerical solution for surface diffusion on graphs. In: Proc. Czech–Japanese Seminar in Applied Mathematics 2005 (M. Beneš, M. Kimura and T. Nakaki, eds.), COE Lecture Note Vol. 3, Faculty of Mathematics, Kyushu University Fukuoka 2006, pp. 9–25 MR 2279046
[2] Deckelnick K., Dziuk G.: Error estimates for the Willmore flow of graphs. Interfaces Free Bound. 8 (2006), 21–46 DOI 10.4171/IFB/134 | MR 2231251
[3] Droske M., Rumpf M.: A level set formulation for Willmore flow. Interfaces Free Bound. 6 (2004), 3, 361–378 DOI 10.4171/IFB/105 | MR 2095338 | Zbl 1062.35028
[4] Du Q., Liu C., Ryham, R., Wang X.: A phase field formulation of the Willmore problem. Nonlinearity (2005), 18, 1249–1267 MR 2134893 | Zbl 1125.35366
[5] Du Q., Liu, C., Wang X.: A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. (2004), 198, 450–468 MR 2062909
[6] Dziuk G., Kuwert, E., Schätzle R.: Evolution of elastic curves in $\mathbb R\mathnormal ^n$: Existence and Computation. SIAM J. Math. Anal. 41 (2003), 6, 2161–2179 MR 2034610
[7] Kuwert E., Schätzle R.: The Willmore flow with small initial energy. J. Differential Geom. 57 (2001), 409–441 MR 1882663 | Zbl 1035.53092
[8] Minárik V., Kratochvíl, J., Mikula K.: Numerical simulation of dislocation dynamics by means of parametric approach. In: Proc. Czech Japanese Seminar in Applied Mathematics (M. Beneš, J. Mikyška, and T. Oberhuber, eds.), Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague 2005, pp. 128–138
[9] Oberhuber T.: Computational study of the Willmore flow on graphs. Accepted to the Proc. Equadiff 11, 2005
[10] Oberhuber T.: Numerical solution for the Willmore flow of graphs. In: Proc. Czech–Japanese Seminar in Applied Mathematics 2005 (M. Beneš, M. Kimura and T. Nakaki, eds.), COE Lecture Note Vol. 3, Faculty of Mathematics, Kyushu University Fukuoka, October 2006, ISSN 1881-4042, pp. 126–138 MR 2279053 | Zbl 1145.65323
[11] Simonett G.: The Willmore flow near spheres. Differential and Integral Equations 14 (2001), No. 8, 1005–1014 MR 1827100 | Zbl 1161.35429
[12] Willmore T. J.: Riemannian Geometry. Oxford University Press, Oxford 1997 MR 1261641 | Zbl 1117.53017
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